J. Nonlinear Sci. Vol. 9: pp. 525–573 (1999)
©
1999 Springer-Verlag New York Inc.
An Existence Theorem of Smooth Nonlocal Center
Manifolds for Systems Close to a System with a
Homoclinic Loop
M. V. Shashkov1 and D. V. Turaev2
1
2
Department of Differential Equations, Institute for Applied Mathematics and Cybernetics,
10 Uljanova Street, Nizhny Novgorod 603005, Russia
The Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, Berlin
10117, Germany
Received June 4, 1997; final revision received April 24, 1998
Communicated by Stephen Wiggins
Summary. In this paper we give a proof of the existence of smooth nonlocal center
manifolds for systems close to a system with a homoclinic orbit to a saddle-type equilibrium point. Our proof is based on a consideration of some class of the boundary value
problems (see Section 3). We obtain estimates for solutions of the boundary value problems that allow us to prove the theorem on the center manifolds at the C 1 -assumptions
for the smoothness of systems.
1. Introduction
It is well known that in neighborhoods of equilibrium points and periodic orbits of
C k -smooth dynamical systems there exist C k -smooth invariant center manifolds. This
result goes back to Pliss [1964], Kelley [1967], and then to Fenichel [1971], Hirsch et
al. [1977], and Shoshitaishvili [1975]. For equilibria, the dimension of such manifolds
is determined by the number of roots of the characteristic equation with zero real parts,
and for periodic orbits it is determined by the number of multipliers that lie on the unit
circle. Due to the existence of such manifolds, the investigation of local bifurcations
(bifurcations of equilibrium points and periodic orbits) can be reduced to the study of
the systems on the center manifolds.
However, a large number of the models of multidimensional dynamical systems is
provided by nonlocal bifurcations and, in particular, by bifurcations of homoclinic and
heteroclinic contours consisting of equilibria and orbits asymptotic to them. The investigations of such bifurcations were pioneered by Shilnikov. He studied the principal
nonlocal bifurcations (bifurcations of homoclinic orbits to a saddle and to a saddle-node)
of multidimensional systems (see Shilnikov [1963], [1968]) and discovered the complex
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M. V. Shashkov and D. V. Turaev
structure near a homoclinic loop to a saddle-focus and near a homoclinic bunch to a
saddle-saddle (see Shilnikov [1965], [1967], [1969], [1970]). By now bifurcations of
homoclinic and heteroclinic contours have been developed intensively. For instance,
codimension-two bifurcations of heteroclinic contours with two equilibrium states were
studied in Bykov [1978], [1980], [1993]; Chow et al. [1990b]; Deng [1989], [1991];
Shashkov [1991b], [1992], [1994]; Shashkov and Turaev [1996]; bifurcations of a pair
of homoclinic orbits to a saddle, in Turaev [1984], [1991]; Turaev and Shilnikov [1986];
Gambaudo et al. [1988]; bifurcations of two separatrices of a saddle, one of which forms
a homoclinic loop and the other one tends to the loop, in Homburg, [1993], [1996]; and
bifurcations of a homoclinic loop to a saddle at the resonant eigenvalues, in Chow et al.
[1990a].
In connection with the study of the global bifurcations of multidimensional systems, a
number of papers have been devoted, in recent years, to the extension of center-manifold
theory to homoclinic and heteroclinic contours. First, the existence of a Lipschitz twodimensional nonlocal invariant manifold was pointed out in [Turaev, 1984], where bifurcations of a system with two homoclinic trajectories to a saddle were studied. A proof
of the existence of a C 1 -smooth center manifold for C 4 -smooth systems was done in
[Turaev, 1991]. Note that in [Turaev, 1996] the center-manifold theory was developed
up to more complex contours that consist of finite number of equilibria, limit cycles,
homoclinic and heteroclinic orbits and, moreover, ω- and α-limit trajectories. The analogous theorems on the existence of smooth center manifolds for C 4 -smooth systems,
close to a system with a heteroclinic contour, have been proved in Shashkov [1991a],
[1994]. Independently, Homburg [1993], [1996] has shown the existence of a smooth
two-dimensional center manifold for sufficiently smooth systems with a homoclinic loop
at the nonresonant eigenvalues. By now, the theorem on existence of a smooth center
manifold near a homoclinic loop has been proved at C k+ε assumptions (k ≥ 1, ε > 0)
for the smoothness of vector fields Sandstede [1994]. Note that Sandstede [1994] has
extended his results to an infinitely dimensional case.
In this paper we develop a tool that permits us to prove the presence of nonlocal center
manifolds at minimal restrictions to the smoothness of the system. This tool involves
solutions of some class of boundary value problems (see Section 3). Using the estimates
for the solutions of one such problem near an equilibria, we prove the existence of the
smooth center manifold for C 1 -systems close to a system with a homoclinic orbit to a
saddle equilibrium state. Finally (in Section 9) we give a C 1 example of a vector field
with a smooth global center manifold.
2. Main Theorems
Let us consider a family of vector fields X µ on an (n + m)-dimensional manifold,
Ẋ = F(X, µ),
X ∈ R n+m ,
n ≥ 1,
m ≥ 1,
µ ∈ Rl ,
l ≥ 0.
We assume the function F(X, µ) to be C k -smooth (k ≥ 1) with respect to the phase
variables X and the parameter µ.
An Existence Theorem of Smooth Nonlocal Center Manifolds
527
Fig. 1. The stable manifold W s intersects the unstable
manifold W u along the orbit 0, forming a homoclinic loop
L = O ∪ 0. The orbit 0 does not belong to the strongunstable manifold W uu .
We make the following assumptions:
(A) at µ = 0, the system X 0 has a saddle equilibrium point O and the roots λn , . . . , λ1 ,
γ1 , . . . , γm of the characteristic equation of the linearized system at the point O
satisfy the following inequalities:
Re λn ≤ · · · ≤ Re λ1 < 0 < γ1 < Re γ2 ≤ · · · ≤ Re γm .
In this case, the dimension of the stable manifold W s of the equilibrium point O
is equal to n and the unstable manifold W u of O is an m-dimensional surface. Since
γ1 < Re γi , (i = 2, . . . , m), there exists an (m−1)-dimensional strong-unstable invariant
submanifold W uu in W u . The main feature characterizing W uu is that all its orbits tend
to O, as t → −∞, being tangent to the subspace corresponding to the eigenvalues
γ2 , . . . , γn , whereas all orbits of W u \W uu are tangent, as t → −∞, to the eigendirection
corresponding to the principle eigenvalue γ1 . We shall assume that
(B) at µ = 0, the system X 0 has an orbit 0 that is doubly asymptotic to the equilibrium
point O, i.e., (W s ∩ W u )/O ⊃ 0,
and
(C) the orbit 0 does not belong to the strong-unstable manifold W uu (see Figure 1).
Denote by E s+ ⊂ R n+1 the invariant subspace of the linearization matrix of the
system X 0 at the point O, which corresponds to the eigenvalues λn , . . . , λ1 , γ1 . It is well
known (Fenichel [1971]; Hirsch et al. [1977]) that, under the assumption (A), there exists
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M. V. Shashkov and D. V. Turaev
Fig. 2. There exists a C 1 -smooth invariant manifold
W s+ , containing W s , that is tangent at O to the subspace corresponding to the eigenvalues λn , . . . , λ1 , γ 1 .
The manifold W s+ is not uniquely defined, but any
two of such manifolds have the common tangent
everywhere on W s . The strong-unstable manifold
W uu is uniquely embedded into a smooth invariant
codimension-one foliation F u on W u .
an invariant C 1 -smooth manifold W s+ tangential to E s+ at the point O (see Figure 2).
The manifold W s+ contains entirely the stable manifold W s . It is not uniquely defined,
but any two of them are tangent at all points of W s . It is known that the manifold W uu
is uniquely included in a smooth invariant foliation F u on the manifold W u . We require
the following condition to be fulfilled.
(D) The manifold W s+ is transverse to the leaves of the foliation F u at each point of
the homoclinic orbit 0 (see Figure 3).
Notice that condition (D) must be verified only at one point on the trajectory 0, because
the manifold W s+ and the foliation F u are invariant with respect to the flow defined by
the system X 0 . It should also be noted that the dimension of the manifold W s+ and the
dimension of the leaves of the foliation F u complement each other; therefore condition
(D), as well as conditions (A) and (C), are conditions of the general position.
Theorem 2.1. If conditions (A), (B), (C), and (D) are fulfilled, then there exists a
small neighborhood U of the loop L = O ∪ 0 such that, for all µ small enough, the
system X µ has an (n + 1)-dimensional invariant C 1 -smooth manifold Mcs that depends
smoothly on µ and such that any orbit not lying in Mcs leaves U as t tends to +∞
An Existence Theorem of Smooth Nonlocal Center Manifolds
529
Fig. 3. The manifold W s+ transversely intersects the
leaves of the foliation F u .
(see Figure 4).1 The manifold Mcs is tangent at the point O to the invariant subspace
corresponding to the eigenvalues λn , . . . , λ1 , γ1 .
Reversing the time, Theorem 2.1 immediately implies the following corollary. Let
(A0 ). the eigenvalues at the point O satisfy the following conditions:
Re λn ≤ · · · ≤ Re λ2 < λ1 < 0 < Re γ1 ≤ · · · ≤ Re γm .
In this case, since λ1 > Re λi , (i = 2, . . . , n), there exists an (n − 1)-dimensional
strong-stable invariant submanifold W ss lying entirely in the stable manifold W s . We
also modify the conditions (C) and (D). Namely,
(C0 ). the homoclinic orbit 0, which exists at µ = 0, does not lie in the strong-stable
manifold W ss (see Figure 5).
Denote by E u+ ⊂ R m+1 the invariant subspace of the linearization matrix of the
system X 0 at the point O, which corresponds to the eigenvalues γm , . . . , γ1 , λ1 . Under
the condition (A0 ), there exists an invariant C 1 -smooth manifold W u+ , tangential to E u+
at the point O (see Figure 6). The manifold W u+ is not uniquely defined, but any two
of them contain W u entirely and are tangent at all points of W u . The strong-unstable
manifold W ss is uniquely included in a smooth invariant foliation F s on W s . We require
the following condition to be fulfilled.
1
In fact, Mcs ∈ C k+ε if F(X, µ) ∈ C k+ε and Re γ2 /γ1 > k + ε.
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M. V. Shashkov and D. V. Turaev
Fig. 4. The system X µ has an (n + 1)-dimensional C 1 smooth invariant manifold Mcs for all µ small enough.
Any orbit not lying in Mcs escapes the neighborhood of
the loop L = O ∪ 0, which exists at µ = 0, as t + ∞.
Fig. 5. The homoclinic orbit 0, which exists at µ = 0,
does not belong to W ss , i.e., 0 tends to the equilibrium
O, if t → +∞, along the principle direction corresponding to λ1 .
An Existence Theorem of Smooth Nonlocal Center Manifolds
531
Fig. 6. There exists an invariant C 1 -manifold W u+ , containing W u , that is tangent at O to the subspace corresponding to the eigenvalues λ1 , γ 1 , . . . , γ m . The manifold W u+ is not unique, but any two of them have the
common tangent everywhere on W u . The strong-stable
manifold W ss is uniquely embedded into a smooth invariant codimension-one foliation F s on W s .
(D0 ). At each point of 0, the manifold W u+ is transverse to the leaves of the foliation
F s (see Figure 7).
Theorem 2.2. If the conditions (A0 ), (B), (C0 ), and (D0 ) are fulfilled, then there exists
a small neighborhood U of the homoclinic loop L = O ∪ 0 such that, for all µ small
enough, the system X µ has an (m + 1)-dimensional invariant C 1 -smooth manifold Mcu
that depends smoothly on µ and such that any orbit, not lying in Mcu , leaves U as t
tends to −∞ (see Figure 8). The manifold Mcu is tangent at the point O to the subspace
corresponding to the eigenvalues γm , . . . , γ1 , λ1 .
If the conditions of both Theorem 2.1 and Theorem 2.2 are fulfilled, then we have the
following result.
Theorem 2.3. The manifolds Mcu and Mcs intersect each other transversally along a
two-dimensional invariant C 1 -manifold Mc that depends smoothly on µ. The manifold
Mc contains all orbits of X µ lying in U entirely for all t ∈ (−∞, +∞) and it is tangent
at the point O to the subspace corresponding to the principle eigenvalues γ1 , λ1 .
By definition, in the situation of Theorem 2.1, the n-dimensional stable manifold of
the point O belongs to Mcs . The invariant manifold Mcs is (n + 1)-dimensional and,
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M. V. Shashkov and D. V. Turaev
Fig. 7. The manifold W u+ is transverse to the leaves of
the foliation F s .
Fig. 8. The system X µ has an (m +1)-dimensional
C 1 -smooth invariant manifold Mcu for all µ small
enough. Any orbit not lying in Mcu escapes the
neighborhood of the loop L = O ∪ 0, which exists
at µ = 0, as t → −∞.
hence, the unstable manifold of O is one-dimensional for the restriction of the system
X µ onto Mcs . Thus, the restriction onto the center-stable manifold Mcs reduces the
dimension of the unstable manifold. Analogously, in the situation of Theorem 2.2, the
restriction onto the center-unstable manifold Mcu reduces the dimension of the stable
manifold.
An Existence Theorem of Smooth Nonlocal Center Manifolds
533
Notice that the manifolds Mcs , Mcu , and Mc are in general only C 1 -smooth.2 Therefore, in contrast to the local bifurcation theory, we cannot apply the reduction to the center
manifold directly for studying subtle bifurcational problems requiring higher smoothness of the system. Theorems 2.1–2.3 are, thus, qualitative results that allow one to
evaluate possible dynamical behavior in a neighborhood of the homoclinic loop. For
instance, they provide restrictions on the possible dimensions of the stable and unstable manifolds of the orbits in U or, which is the same, on the number of positive and
negative Lyapunov exponents. Moreover, it is possible to use the restriction onto the
invariant manifold for the study of a bifurcation problem that actually requires only
C 1 -smoothness. For instance, we believe that the bifurcation problems considered by
Shilnikov [1963], [1965], [1968], [1970]; Turaev [1984], [1991]; Turaev and Shilnikov
[1986]; Chow et al. [1990b]; Deng [1989], [1991]; Shashkov [1991b], [1992], [1994];
Shashkov and Turaev [1996]; Homburg [1993]; and Gambaudo [1988] actually require
only C 1 -smoothness.
Since both Theorems 2.2 and 2.3 follow from main Theorem 2.1, all of our considerations below will be focused on the proof of Theorem 2.1. We prove Theorem 2.1,
reducing the problem on a Poincaré map (see Sections 4 and 5) and applying the standard arguments used in proving the contractibility for the graph transformations (see
Section 6). In order to construct the Poincaré map, we use an appropriate boundary
value problem that gives the proper estimates for the solutions of the system near the
equilibrium point (see Section 3). A proof of the smoothness of the invariant manifold
Mcs is carried out in Sections 7 and 8. We give an example of a C 1 -smooth vector field
with the nonlocal center manifold in Section 9.
3. On a Class of Boundary Value Problems
In order to prove Theorem 2.1, we need proper estimates for the orbits of the system
near the homoclinic loop L. Notice that the study of solutions near the equilibrium
point is the most complicated because the flight time of orbits near O is unbounded
and, therefore, we need estimates that are fulfilled for unbounded times. If the system
can be linearized in the neighborhood of the equilibrium point, the question about the
estimates does not arise. However, the smooth linearization requires that the superfluous
additional resonance restrictions be satisfied. Moreover, the system should be extra
smooth. Therefore, the general way to find the suitable estimates near the equilibrium
point is to use Shilnikov’s method, which is based on a consideration of some boundary
value problem (see Shilnikov [1967]). This method gives the proper estimates (see
Ovsyannikov and Shilnikov [1986], [1991], Turaev [1991]) for the solutions and their
derivatives up to order k if the initial system is C k+3 -smooth. There are many nonlocal
bifurcational problems that were solved by this method (see, for instance, Chow et al.
[1990a], [1990b]; Deng [1989], [1991]; Fiedler and Turaev [1996]; Shashkov [1991a],
[1991b], [1994]; Turaev [1984], [1991]; Turaev and Shilnikov [1986]).
Since we consider here vector fields that are only C 1 -smooth, we cannot apply the
results mentioned above and, therefore, we develop Shilnikov’s method. Namely, we
2
For instance, smoothness of Mcs is not higher than the integer part of Re γ2 /γ1 > 1.
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M. V. Shashkov and D. V. Turaev
consider the boundary value problem at the other assumptions for the boundary conditions (formulas (3.61)–(3.64) below). Moreover, since the boundary value problems
are of interest in its own, in this section we consider a sufficiently wide class of these
problems (see (3.1), (3.2), (3.5), and (3.6)). For C k -smooth (k ≥ 1) vector fields, we
obtain estimates for the derivatives of the solutions up to order k. We also investigate the
convergence of these derivatives when the flight time of the solutions tends to infinity.
3.1. The Existence Theorem
Consider a system of ordinary differential equations,
½
u̇ = Au + f (u, v, µ, t),
v̇ = Bv + g(u, v, µ, t),
(3.1)
where u ∈ R n , v ∈ R m , t is time, and µ is a vector of parameters from some compact
set D ⊂ R l , (l ≥ 0). We assume that the functions f and g are C k -smooth (k ≥ 1) with
respect to all variables (u, v, µ, t). Let the following conditions hold for the matrices A
and B:
Spectr B = {β1 , . . . , βm } ,
Spectr A = {α1 , . . . , αn } ,
(3.2)
max Re αi < α < β < min Re βi .
i=1,...,n
i=1,...,m
In this case, it is possible to choose the norms of the vectors u and v in such a way that
for s ≥ 0,
° A s°
°e ° ≤ eα̃s ≤ eαs ,
(3.3)
° −B s °
° ≤ e−β̃s ≤ e−βs ,
°e
where constants α̃, β̃ satisfy the conditions
max Re αi < α̃ < α
i=1,...,n
and
β < β̃ < min Re βi .3
i=1,...,m
We also require that, for any (u, v) ∈ R n+m and µ ∈ D,
°
°
° ∂( f, g) °
°
°
° ∂(u, v) ° < ξ,
(3.4)
(3.5)
where ξ is a small enough constant.4
We are interested in solutions of the system (3.1) that satisfy the following boundary
conditions:
u(0) = u 0 ,
v(τ ) = v 1 ,
τ > 0.
(3.6)
Notice that the difference between our boundary value problem (3.1), (3.6) and Shilnikov’s
is that we do not require the conditions α < 0 and β > 0.
Here and below, the symbol k · k denotes a norm in R p if it is applied to a vector x = (x1 , . . . , x p ), and
denotes the compatible operator norm if it is applied to an operator.
3
4
The exact value of the constant ξ can be extracted from the proofs of the theorems below.
An Existence Theorem of Smooth Nonlocal Center Manifolds
535
Theorem 3.1. A solution of the boundary value problem (3.1), (3.6),
u(t) = u(t; u 0 , v 1 , τ, µ),
v(t) = v(t; u 0 , v 1 , τ, µ)
(3.7)
exists. It is uniquely defined and depends C k -smoothly on (t; u 0 , v 1 , τ, µ). Moreover, the
following estimates hold:
°
°
°
°
° ∂(u, v) °
° ∂(u, v) °
β(t−τ )
°
° ≤ Ceαt ,
°
°
,
(3.8)
° ∂v 1 ° ≤ Ce
° ∂u 0 °
where C is some constant.
Before the proof of Theorem 3.1, we notice that vector fields in neighborhoods of
equilibrium points give us one of the main examples of systems of the kind (3.1). Indeed,
let us consider a family of C k -smooth dynamical systems that depends on a parameter
µ and is given on an (n + m)-dimensional manifold:
Ẋ = F(X, µ),
µ ∈ R l (l ≥ 0),
X ∈ R n+m (n ≥ 1, m ≥ 1),
F(X, µ) ∈ C k (k ≥ 1).
(3.9)
Assume that at µ = 0 the system (3.9) has an equilibrium point O in the origin of the
coordinates and the spectrum of the linear part of the system (3.9) in the point O can be
divided onto two parts, i.e.,
¶¯
µ
∂ F ¯¯
= {α1 , . . . , αn , β1 , . . . , βm } ,
max Re αi < min Re β j .
Spectr
i=1,...,n
j=1,...,m
∂ X ¯(X,µ)=0
In this case, it is possible to introduce coordinates u ∈ R n and v ∈ R m such that, in a
neighborhood of the point O for µ small enough, the system (3.9) is given by
½
u̇ = Au + f (u, v, µ),
(3.10)
v̇ = Bv + g(u, v, µ),
where the matrices A and B satisfy conditions (3.2), (3.3), and the functions f , g are
C k -smooth and satisfy the following conditions:
¯
∂( f, g) ¯¯
= 0.
f (0, 0, 0) = 0,
g(0, 0, 0) = 0,
(3.11)
∂(u, v) ¯(u,v,µ)=0
We are interested in solutions (u(t), v(t)), at t ∈ [0, τ ], lying entirely (t ∈ [0, τ ]) in a
small neighborhood of the point O. Therefore, the transition to a new system is justified
½
u̇ = Au + f˜(u, v, µ),
(3.12)
v̇ = Bv + g̃(u, v, µ),
where the functions f˜, g̃ ∈ C k are given by the following formulas:
f˜(u, v, µ) = f (ϑ(k(u, v)k/ρ)u, ϑ(k(u, v)k/ρ)v, µ) ,
g̃(u, v, µ) = g (ϑ(k(u, v)k/ρ)u, ϑ(k(u, v)k/ρ)v, µ) .
(3.13)
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M. V. Shashkov and D. V. Turaev
Here ρ is a small positive constant and the function ϑ(t) ∈ C ∞ has the following
properties:
¯
¯
½
¯ ∂ϑ(t) ¯
1, if t ≤ 1,
¯
¯
(3.14)
ϑ(t) =
¯ ∂t ¯ < 2.
0, if t ≥ 2,
Notice that the functions f˜ and g̃ satisfy inequality (3.5) for any (u, v) ∈ R n+m and small
µ. By (3.11), (3.13), and (3.14), and by choosing ρ small, the constant ξ can be made
arbitrarily small. Therefore, the system (3.12) is a system of the kind (3.1). The functions
f˜ and g̃ coincide with f and g correspondingly for k(u, v)k ≤ ρ. So, if the boundary
value problem (3.12), (3.6) has a solution (u(t), v(t)) lying entirely (t ∈ [0, τ ]) in the
neighborhood k(u, v)k ≤ ρ, then (u(t), v(t)) is also the solution of the boundary value
problem (3.6), (3.10).5
Let us pass to a proof of Theorem 3.1. Consider the space H of continuous functions
(u(t), v(t)), which are given on the segment t ∈ [0, τ ]. Define a γ -norm by the following
formula:
¡
¢
(3.15)
k(u, v)kγ = sup k(u(t), v(t))ke−γ t ,
t∈[0,τ ]
where
α < γ < β.
(3.16)
Obviously, H with the γ -norm is a complete metric space.6
Let us introduce an integral operator T , which maps any function (u(t), v(t)) ∈ H
into the function (u(t), v(t)) ∈ H , by the following rule:

Zt



At 0

u(t) = e u + e A(t−s) f (u(s), v(s), µ, t)ds,





0


Zt



B(t−τ ) 1

v(t) = e
v + e B(t−s) g(u(s), v(s), µ, t)ds.



(3.17)
τ
It is easy to check that any solution of the boundary value problem (3.1), (3.6) is a fixed
point of the integral operator (3.17). It is also true that any fixed point of T is a solution
of the boundary value problem (3.1), (3.6). Therefore, the question of the existence and
uniqueness of the solution of the problem (3.1), (3.6) is reduced to the question of the
existence and uniqueness of the fixed point of the operator T .
In order to show that T has a unique fixed point, we shall establish that T is a
contraction operator in the space H . Let us check it. Consider any functions (u 1 , v1 ) ∈ H
5 Notice the solution of the boundary value problem (3.12), (3.6), generally speaking, can fall outside the
limits of the neighborhood of O. Therefore, in this case, the solution is not the solution of the boundary value
problem (3.10), (3.6).
6
Note that the γ -norm is equivalent to the usual uniform norm.
An Existence Theorem of Smooth Nonlocal Center Manifolds
537
and (u 2 , v2 ) ∈ H . Let T (u 1 , v1 ) = (u 1 , v 1 ) and T (u 2 , v2 ) = (u 2 , v 2 ); then, by (3.2)–
(3.5) and (3.15)–(3.17), we have the following relations:
° t
°
°Z
°
°
°
A(t−s)
°
e
(
f
(u
(s),
v
(s),
µ,
t)
−
f
(u
(s),
v
(s),
µ,
t))ds
ku 1 − u 2 k = °
1
1
2
2
°
°
°
°
0
Zt
≤
eα(t−s) ξ k(u 1 − u 2 , v1 − v2 )kγ eγ s ds
0
¢
¡
ξ
eαt e(γ −α)t − 1 k(u 1 − u 2 , v1 − v2 )kγ
γ −α
ξ
≤
eγ t k(u 1 , v1 ) − (u 2 , v2 )kγ ,
γ −α
=
(3.18)
° t
°
°Z
°
°
°
B(t−s)
°
(g(u 1 (s), v1 (s), µ, t) − g(u 2 (s), v2 (s), µ, t))ds °
kv 1 − v 2 k = ° e
°
°
°
τ
Zτ
≤
eβ(t−s) ξ k(u 1 − u 2 , v1 − v2 )kγ eγ s ds
t
¡
¢
ξ
eβt e(γ −β)t − e(γ −β)τ k(u 1 − u 2 , v1 − v2 )kγ
β −γ
ξ
≤
eγ t k(u 1 , v1 ) − (u 2 , v2 )kγ .
β −γ
=
(3.19)
By (3.15)–(3.19), we obtain
¡
¢
k(u 1 , v 1 ) − (u 2 , v 2 )kγ ≤ sup ku 1 − u 2 ke−γ t + kv 1 − v 2 ke−γ t
t∈[0,τ ]
µ
≤ ξ
1
1
+
γ −α
β −γ
¶
k(u 1 , v1 ) − (u 2 , v2 )kγ . (3.20)
We assume that the constant ξ is so small that ξ (1/ (γ − α) + 1/ (β − γ )) is less than
1. In this case T : H → H is a contraction operator and, therefore, T has a unique
fixed point (u(t), v(t)) ∈ H . Moreover, any sequence (u 0 , v0 ), (u 1 , v1 ), (u 2 , v2 ), . . . ,
obtained by the iterations
(u n+1 (t), vn+1 (t)) = T (u n (t), vn (t)) ,
(3.21)
with any initial function (u 0 (t), v0 (t)) ∈ H , converges to the fixed point.
Thus, for any fixed values (u 0 , v 1 , τ, µ), the boundary value problem (3.1), (3.6) has
a unique solution. Depending on the boundary conditions (u 0 , v 1 , τ ) and µ, we obtain
different solutions. Below we show the existence of a constant C such that the solution
(3.7) satisfies the following Lipschitz conditions with respect to u 0 and v 1 :
°¡
¢ ¡
¢°
° u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ) − u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ) °
1
1
2
2
°
°
(3.22)
≤ Ceαt °u 01 − u 02 ° ,
538
M. V. Shashkov and D. V. Turaev
°¡
¢ ¡
¢°
° u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ) − u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ) °
1
1
2
2
°
°
(3.23)
≤ Ceβ(t−τ ) °v11 − v21 ° .
Consider the space H̃ of continuous functions (u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ)) that
satisfy Lipschitz conditions (3.22) and (3.23). Here µ ∈ D, τ ≥ 0, 0 ≤ t ≤ τ , u 0 ∈ R n
and v 1 ∈ R m . Let us define an operator T̃ in the space H̃ by the relations (3.17).7
Below, we show that H̃ is an invariant space with respect to T̃ . Consider any function
(u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ)) ∈ H̃ . Now, we shall check that the function
¢
¡
¢
¡
u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ) = T̃ u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ)
satisfies inequality (3.23). Fix any (u 0 , τ, µ) and take any two values v 1 : v 1 = v11 and
v 1 = v21 . Below, we use the following denotations:
¢
¡
u(t; u 0 , v11 , τ, µ), v(t; u 0 , v11 , τ, µ) ≡ (u 1 (t), v1 (t)),
¡
¢
u(t; u 0 , v21 , τ, µ), v(t; u 0 , v21 , τ, µ) ≡ (u 2 (t), v2 (t)),
¢
¡
(3.24)
u(t; u 0 , v11 , τ, µ), v(t; u 0 , v11 , τ, µ) ≡ (u 1 (t), v 1 (t)),
¢
¡
u(t; u 0 , v21 , τ, µ), v(t; u 0 , v21 , τ, µ) ≡ (u 2 (t), v 2 (t)).
By (3.2)–(3.5), (3.17), (3.23), and (3.24), we have the following relations:
° t
°
°Z
°
°
°
A(t−s)
°
( f (u 1 , v1 , µ, t) − f (u 2 , v2 , µ, t))ds °
ku 1 − u 2 k = ° e
°
°
°
0
Zt
≤
eα(t−s) ξ Ceβ(s−τ ) kv11 − v21 kds
0
¢
ξ ¡
1 − e(α−β)t Ceβ(t−τ ) kv11 − v21 k
β −α
ξ
Ceβ(t−τ ) kv11 − v21 k,
≤
β −α
=
(3.25)
kv 1 − v 2 k ≤ e B(t−τ ) kv11 − v21 k
°
° t
°
°Z
°
°
B(t−s)
°
(g(u 1 , v1 , µ, t) − g(u 2 , v2 , µ, t))ds °
+ ° e
°
°
°
τ
≤ eβ(t−τ ) kv11 − v21 k +
µ
ξ
³
Zτ
eβ̃(t−s) ξ Ceβ(s−τ ) kv11 − v21 kds
t
(β̃−β)(t−τ )
´ ¶
C eβ(t−τ ) kv11 − v21 k
1−e
β̃ − β
¶
µ
ξ
C eβ(t−τ ) kv11 − v21 k.
≤ 1+
β̃ − β
=
7
1+
The difference between T̃ and T is that the operators are defined on different spaces of functions.
(3.26)
An Existence Theorem of Smooth Nonlocal Center Manifolds
539
Therefore, by inequalities (3.25) and (3.26),
k(u 1 , v 1 ) − (u 2 , v 2 )k ≤ ku 1 − u 2 k + kv 1 − v 2 k
¶ ¶
µ
µ
1
1
+
C eβ(t−τ ) kv11 − v21 k.
≤ 1+ξ
β −α
β̃ − β
We assume that ξ is so small that ξ(1/(β − α) + 1/(β̃ − β)) < 1. In this case, if C ≥ 1/
(1 − ξ(1/(β − α) + 1/(β̃ − β))), we have
k(u 1 , v 1 ) − (u 2 , v 2 )k ≤ Ceβ(t−τ ) kv11 − v21 k,
i.e., the function (u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ)) satisfies estimate (3.23). In the
same way, it is possible to check that the function (u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ))
satisfies inequality (3.22). This means that H̃ is an invariant space with respect to the
operator T̃ .
Notice, if u 0 , v 1 , τ , and µ are fixed, the sequence of functions
¢
¡
u n (t; u 0 , v 1 , τ, µ), vn (t; u 0 , v 1 , τ, µ) = T̃ n (0, 0)
coincides with the sequence (3.21) with the initial element (u 0 (t), v0 (t)) = (0, 0) ∈ H
and, therefore, converges to the solution (3.7) of the boundary value problem (3.1), (3.6)
in H . Due to the theorem on the passage to the limit in inequalities, the solution also
satisfies relations (3.22) and (3.23).
¢
¡
Below we shall show that the solution u(t; u 0 , v 1 , τ, µ), v(t; u 0 , v 1 , τ, µ) is a C k smooth function with respect to all variables. To establish this, we consider Cauchy’s
initial value problem for the system (3.1) with the following initial conditions:
u(0) = u 0 ,
v(0) = v 0 .
(3.27)
Smoothness of the functions f and g in the right-hand side of the system (3.1) guarantees
the existence and uniqueness of the solution
¢
¡ ∗
(3.28)
u (t; u 0 , v 0 , µ), v ∗ (t; u 0 , v 0 , µ) ,
which is C k -smooth with respect to (t; u 0 , v 0 , µ). The phase trajectories of the system
(3.1) determine the one-to-one correspondence between the initial (3.27) and the boundary (3.6) conditions. This correspondence can be specified by the following formulas:
¢
¡ 0 0¢ ¡
(3.29)
u , v = u(0; u 0 , v 1 , τ, µ), v(0; u 0 , v 1 , τ, µ) ,
¡
¢
¢ ¡
u 0 , v 1 = u ∗ (0; u 0 , v 0 , µ), v ∗ (τ ; u 0 , v 0 , µ) .
(3.30)
Further, we use the following corollary of the implicit function theorem.
Lemma 3.1. Let a function F(x, y) satisfy the Lipschitz condition with respect to x,
i.e., for any x1 , x2 , and y from the domain of definition
kF(x1 , y) − F(x2 , y)k ≤ Lkx1 − x2 k.
540
M. V. Shashkov and D. V. Turaev
Let the equation z = F(x, y) be resolved with respect to x, i.e., x = 8(y, z), and,
moreover, let 8(y, z) be a C k -smooth function with respect to (y, z). Then F(x, y) is
also a C k -smooth function with respect to (x, y) and
°
°
°∂F °
°
°
° ∂x ° ≤ L.
The functions in (3.30) depend C k -smoothly on the variables because the functions in
(3.28) are smooth. The functions in (3.29) satisfy the Lipschitz condition (3.23) with
respect to v 1 . Applying Lemma 3.1, we obtain that the functions in equality (3.29) are
C k -smooth. Now, C k -smoothness of the solution (3.7) follows from the fact that it can
be represented as a superposition of smooth functions:
¢
¢
¡
¡
u t; u 0 , v 1 , τ, µ = u ∗ t; u 0 , v(0, u 0 , v 1 , τ, µ), µ ,
¢
¢
¡
¡
v t; u 0 , v 1 , τ, µ = v ∗ t; u 0 , v(0, u 0 , v 1 , τ, µ), µ .
The smoothness of the solution (3.7) and the Lipschitz conditions (3.22), (3.23) imply
estimates (3.8). The theorem is proved.
Notice that our boundary value problem (3.1), (3.6) makes sense with the following
conditions:
v 1 = 0,
τ = ∞.
In this case, we define the solution as a fixed point of the following integral operator:

Zt



At
0

u(t) = e u + e A(t−s) f (u(s), v(s), µ, t) ds,



0
Zt




v(t) = e B(t−s) g(u(s), v(s), µ, t)ds.



∞
This operator, as well as the operator T (see (3.17)), has a unique fixed point in the
space H
u(t) = u(t; u 0 , µ),
v(t) = v(t; u 0 , µ).8
¡
¢
Since the function u(t; u 0 , µ), v(t; u 0 , µ) is bounded in the γ -norm, the orbit (u(t), v(t))
belongs to an invariant manifold that corresponds to the eigenvalues α1 , . . . , αn and,
therefore, the function v 0 = V (u 0 , µ) ≡ v(0; u 0 , µ) specifies that
1. the extended stable manifold if 0 < α < β;
2. the stable manifold if α < 0 < β;
3. the strong stable manifold if α < β < 0.
The functions u(t; u 0 , µ) and v(t; u 0 , µ) depend C k -smoothly on (t; u 0 , µ) if α ≤ 0, and the smoothness is
bounded by the integer part of β/α if α > 0.
8
An Existence Theorem of Smooth Nonlocal Center Manifolds
541
By virtue of the symmetry with respect to the changes
t → τ − t,
u → v,
A → −B,
u 0 → v1,
v → u,
B → −A,
α → −β,
v1 → u 0,
β → −α,
there exists an invariant manifold9 that corresponds to the eigenvalues β1 , . . . , βm , i.e.,
1. the strong unstable manifold if 0 < α < β;
2. the unstable manifold if α < 0 < β;
3. the extended unstable manifold if α < β < 0.
3.2. Estimates for the Derivatives
Theorem 3.1 gives us the estimates (3.8) for the derivatives of the first order with respect
to u 0 and v 1 . In this subsection we estimate any derivatives for the solution (3.7) of the
boundary value problem (3.1), (3.6) up to order k.
We use the following denotations for the derivatives of a vector function φ =
(φ1 , . . . , φq ) ∈ R q with respect to a vector argument x = (x1 , . . . , x p ) ∈ R p :
Ã
!
∂ s1 +···+sp φq
∂ s1 +···+sp φ1
∂ |s| φ
≡
.
s , ... ,
s
∂xs
∂ x1s1 · · · ∂ x pp
∂ x1s1 · · · ∂ x pp
Here, the vector s = (s1 , . . . , s p ) consists of nonnegative integer-valued components
and |s| = s1 + · · · + s p .
Theorem 3.2. Let the solution (3.7) of the boundary value problem (3.1), (3.6) lie in a
bounded domain; then the following estimates hold.
1. Let 0 < α < β; then

C if |k1 | = |k2 | = 0,
° 

°

°

°
∂ |k1 |+|k2 |+|k3 | (u, v)
°
°
C e|k1 |αt if |k2 | = 0 and |k1 |α − β < 0,
≤
°
° ¡
° ∂ u 0 , µ¢k1 ∂ ¡v 1 , τ ¢k2 ∂t k3 ° 



 β(t−τ )+|k1 |ατ
if |k2 | 6= 0 or |k1 |α − β > 0.
Ce
(3.31)
2. Let α < 0 < β; then

C if |k1 | = |k2 | = 0,





° 
 αt
°
if |k2 | = 0 and |k1 | 6= 0,
° 
C e
°
|k1 |+|k2 |+|k3 |
(u,
v)
∂
°
°
(3.32)
≤
°
° ¡ ¢k1 ¡
¢
 β(t−τ )
° ∂ u 0 ∂ v 1 , τ k2 ∂(t, µ)k3 ° 
if
|k
|
=
0
and
|k
|
=
6
0,
C
e

1
2





 αt+β(t−τ )
if |k1 | 6= 0 and |k2 | 6= 0.
Ce
9
The smoothness of this manifold is bounded by the integer part of α/β if β < 0.
542
M. V. Shashkov and D. V. Turaev
3. Let α < β < 0; then

C if |k1 | = |k2 | = 0,
° 

°
° 

°
∂ |k1 |+|k2 |+|k3 | (u, v)
°
°
C e|k2 |β(t−τ ) if |k1 | = 0 and α − |k2 |β < 0,
≤
°
° ¡ ¢k1 ¡
¢
k
° ∂ u 0 ∂ v 1 , τ, µ 2 ∂t k3 ° 



 αt−|k2 |βτ
if |k1 | 6= 0 or α − |k2 |β > 0.
Ce
(3.33)
Here C is some positive constant.
Notice that our boundary value problem is symmetric with respect to the following
change:
t → τ − t,
α → −β,
β → −α,
u → v,
v1 → u 0,
k1 → k2 ,
v → u,
u 0 → v1,
k2 → k1 .
(3.34)
Therefore, by Theorem 3.2, we have
Theorem 3.3. Let the solution (3.7) of the boundary value problem (3.1), (3.6) lie in a
bounded domain, then the following estimates hold.
1. Let 0 < α < β; then

C if |k1 | = |k2 | = 0,
° 

°

°

°
|k1 |+|k2 |+|k3 |
(u, v)
∂
°
°
C e|k1 |αt if |k2 | = 0 and |k1 |α − β < 0,
≤
°
° ¡
¢
¡
¢
° ∂ u 0 , τ, µ k1 ∂ v 1 k2 ∂(τ − t)k3 ° 



 β(t−τ )+|k1 |ατ
if |k2 | 6= 0 or |k1 |α − β > 0.
Ce
(3.35)
2. Let α < 0 < β; then

C if |k1 | = |k2 | = 0,





° 

°

°
C eαt if |k2 | = 0 and |k1 | 6= 0,
°
|k1 |+|k2 |+|k3 |
(u, v)
∂
°
°
≤
°
° ¡
° ∂ u 0 , τ ¢k1 ∂ ¡v 1 ¢k2 ∂(τ − t, µ)k3 ° 

C eβ(t−τ ) if |k1 | = 0 and |k2 | 6= 0,






 αt+β(t−τ )
if |k1 | 6= 0 and |k2 | 6= 0.
Ce
(3.36)
3. Let α < β < 0; then

C if |k1 | = |k2 | = 0,
° 

°

°

°
∂ |k1 |+|k2 |+|k3 | (u, v)
°
°
C e|k2 |β(t−τ ) if |k1 | = 0 and α − |k2 |β < 0,
≤
°
° ¡
° ∂ u 0 , τ ¢k1 ∂ ¡v 1 , µ¢k2 ∂(τ − t)k3 ° 



 αt−|k2 |βτ
if |k1 | 6= 0 or α − |k2 |β > 0.
Ce
(3.37)
Here C is some positive constant.
An Existence Theorem of Smooth Nonlocal Center Manifolds
543
Let us prove Theorem 3.2 for the first case, i.e. under the conditions 0 < α < β. Note
that the estimates for the derivatives with respect to µ can be reduced to the estimates for
the derivatives with respect to u 0 . In order to see that it is sufficient to add the equation
µ̇ = 0 to the initial system (3.1) and the condition µ(0) = µ to the boundary conditions
(3.6).
Note also that the differentiation with respect to t does not change the estimates.
Indeed, by (3.1), we have the following recurrence relations:
∂ |k1 |+|k2 |+|k3 |+|k4 |+|k5 −1| (A u + f (u, v, µ, t))
∂ |k1 |+|k2 |+|k3 |+|k4 |+|k5 | u
=
,
0
k
1
k
k
k
k
∂(u ) 1 ∂(v ) 2 ∂µ 3 ∂τ 4 ∂t 5
∂(u 0 )k1 ∂(v 1 )k2 ∂µk3 ∂τ k4 ∂t k5 −1
∂ |k1 |+|k2 |+|k3 |+|k4 |+|k5 −1| (B v + g(u, v, µ, t))
∂ |k1 |+|k2 |+|k3 |+|k4 |+|k5 | v
=
.
∂(u 0 )k1 ∂(v 1 )k2 ∂µk3 ∂τ k4 ∂t k5
∂(u 0 )k1 ∂(v 1 )k2 ∂µk3 ∂τ k4 ∂t k5 −1
(3.38)
So, by (3.38),
µ |k1 |+|k2 |+|k3 |+|k4 |
¶
∂
(u, v)
∂ |k1 |+|k2 |+|k3 |+|k4 |+|k5 | (u, v)
=
O
.
∂(u 0 )k1 ∂(v 1 )k2 ∂µk3 ∂τ k4 ∂t k5
∂(u 0 )k1 ∂(v 1 )k2 ∂µk3 ∂τ k4
In order to find the estimates for the derivatives with respect to τ we use the following
trick. By the definition of the solution of the boundary value problem we have the
following identities:
u(t; u 0 , v 1 , τ, µ) ≡ u(t; u 0 , v(τ + δ; u 0 , v 1 , τ, µ), τ + δ, µ) ,
v(t; u 0 , v 1 , τ, µ) ≡ v(t; u 0 , v(τ + δ; u 0 , v 1 , τ, µ), τ + δ, µ) ,
(3.39)
The differentiation of the identities (3.39) with respect to δ gives us the following equality:
¯
∂(u, v)
∂(u, v) ∂v ¯¯
+
≡ 0.
(3.40)
∂v 1 ∂t ¯t=τ
∂τ
This formula implies that the differentiation with respect to τ is “analogous” to the
differentiation with respect to v 1 , i.e.
µ |k1 |+|k6 |+|k3 |
¶
∂
(u, v)
∂ |k1 |+|k2 |+|k3 |+|k4 | (u, v)
=O
,
(3.41)
∂(u 0 )k1 ∂(v 1 )k2 ∂µk3 ∂τ k4
∂(u 0 )k1 ∂(v 1 )k6 ∂µk3
where |k6 | = |k2 | + |k4 |.
So, to prove Theorem 3.2, under the condition 0 < α < β, we need to check the
following estimates:
° 
°
if |k2 | = 0 and |k1 |α − β < 0,
° ∂ |k1 |+|k2 | (u, v) ° C e|k1 |αt
°
°
(3.42)
≤
° ¡ ¢k1 ¡ ¢k2 °
° ∂ u 0 ∂ v 1 °  β(t−τ )+|k1 |ατ
if |k2 | 6= 0 or |k1 |α − β > 0.
Ce
We prove the validity of these estimates by induction on |k1 | + |k2 |. By virtue of
Theorem 3.1 (see (3.8)), the estimates are fulfilled for |k1 | + |k2 | = 1. Assume that the
estimates (3.42) are fulfilled for any k1 and k2 such that |k1 | + |k2 | ≤ q. Let us show the
validity of the estimates for any k1 and k2 such that |k1 | + |k2 | = q + 1.
544
M. V. Shashkov and D. V. Turaev
Take any k1 and k2 such that 2 ≤ |k1 | + |k2 | = q + 1 ≤ k. Since the solution of the
boundary value problem is a fixed point of the operator T (see (3.17)), the derivative
∂ |k1 |+|k2 | (u,v)
k
k satisfies the following equation:
∂ (u 0 ) 1 ∂ (v 1 ) 2

Zt


∂ |k1 |+|k2 | f (u, v, µ, s)
∂ |k1 |+|k2 | u


=
e A(t−s)

¡ ¢k1 ¡ ¢k2
¡ ¢k ¡ ¢k ds,


∂ u 0 1 ∂ v1 2
 ∂ u 0 ∂ v1
0
(3.43)

Zt

|k1 |+|k2 |
|k1 |+|k2 |

v
∂
g(u,
v,
µ,
s)
∂


= e B(t−s)

¡ ¢k ¡ ¢k ds,

 ∂ ¡u 0 ¢k1 ∂ ¡v 1 ¢k2
∂ u 0 1 ∂ v1 2
τ
where the derivatives of the composite functions f (u(s; u 0 , v 1 , τ, µ), v(s; u 0 , v 1 , τ, µ),
µ, s) and g(u(s; u 0 , v 1 , τ, µ), v(s; u 0 , v 1 , τ, µ), µ, s) are calculated by the following
formula:
∂( f, g) ∂ |k1 |+|k2 | (u, v)
∂ |k1 |+|k2 | ( f, g)
=
∂(u 0 )k1 ∂(v 1 )k2
∂(u, v) ∂(u 0 )k1 ∂(v 1 )k2


+
|k1X
|+|k2 | 
 |i|
|i|=2
 ∂ ( f, g)
 ∂(u, v)i

Ã
X
C
|l1 |+···+|l|i| |=|k1 |
| p1 |+···+| p|i| |=|k2 |
|lm |+| pm |≥1
l1 ...l|i|
p1 ... p|i|
|i|
Y
∂ |l j |+| pj | (u, v)
∂(u 0 )l j ∂(v 1 ) pj
j=1
!

.


(3.44)
l1 ...l|i| are some constants. Notice, by the proposition of induction, the derivatives
p1 ... p|i|
∂
(u,v)
satisfy the estimates (3.42) because |l j | + | p j | ≤ q.
∂(u 0 )l j ∂(v 1 ) p j
Here C
|l j |+| p j |
Let us consider the space N of continuous functions (η1 (t), η2 (t)) which are defined
on the segment t ∈ [0; τ ]. The space N with the γ -norm (see (3.15), (3.16)) is a complete
metric space. Formulas (3.43) and (3.44) associate a map P : N → N . Namely, we
define (η̄1 , η̄2 ) = P [(η1 , η2 )] by the following relations:

¶
Zt
Zt
|k1X
|+|k2 | µ


∂ |i| f

A(t−s) ∂ f
A(t−s)

· · · ds,
η̄1 = e
(η1 , η2 ) ds + e



∂(u, v)
∂(u, v)i

|i|=2
0
0
(3.45)

¶
Zt
Zt
|k1X
|+|k2 | µ

|i|

∂
∂g
g


· · · ds.
η̄ = e B(t−s)
(η1 , η2 ) ds + e B(t−s)


 2
∂(u, v)
∂(u, v)i
|i|=2
τ
τ
In order to obtain the formulas (3.45), we substituted (η1 , η2 ) for
right-hand side of the equation (3.43) and (η̄1 , η̄2 ) for
∂ |k1 |+|k2 | (u,v)
∂(u 0 )k1 ∂(v 1 )k2
∂ |k1 |+|k2 | (u,v)
∂(u 0 )k1 ∂(v 1 )k2
∂ |k1 |+|k2 | (u,v)
∂(u 0 )k1 ∂(v 1 )k2
in the
in the left-hand side.
is a fixed point of P.
Thus, the derivative
Notice that the map P is a linear operator. It can be represented in the following form:
(η̄1 , η̄2 ) = A (η1 , η2 ) + B,
(3.46)
An Existence Theorem of Smooth Nonlocal Center Manifolds
where
 t
Z

e A(t−s)
A (η1 , η2 ) ≡
0
and

B≡
Zt
e A(t−s)
∂f
(η1 , η2 ) ds,
∂(u, v)
|k1X
|+|k2 | µ
|i|=2
0
Zt
τ
545

∂g
e B(t−s)
(η1 , η2 ) ds , (3.47)
∂(u, v)

¶
¶
Zt
|k1X
|+|k2 | µ
|i|
∂ |i| f
∂
g
· · · ds , e B(t−s)
· · · ds .
i
∂(u, v)i
∂(u,
v)
|i|=2
τ
(3.48)
By (3.2)–(3.5) and (3.47), there exists a constant ρ < 1 such that kAk ≤ ρ, i.e., the
operator P satisfies the contraction property
dist (P(η1 , η2 )) ≤ ρ dist (η1 , η2 ) .
|k1 |+|k2 |
∂
(u,v)
It means, in particular, that the derivative ∂(u
0 )k1 ∂(v 1 )k2 is a unique fixed point of the
operator P and, moreover,
°
° |k1 |+|k2 |
°∂
(u, v) °
−1
°
°
(3.49)
° ∂(u 0 )k1 ∂(v 1 )k2 ° ≤ (1 − ρ) kBk .
Therefore, by (3.49) and (3.48), there exists a constant D such that
°
° |k1 |+|k2 |
°∂
(u, v) °
°
°
° ∂(u 0 )k1 ∂(v 1 )k2 °

 t
°
°
Z Y
Zτ
i °
i °
|l j |+| p j |
Y
° ∂ |l j |+| pj | (u, v) °
°
°
(u,
v)
∂
β(t−s)
°
°
°
° 

≤D
max
° ∂(u 0 )l j ∂(v 1 ) pj ° ds; e
° ∂(u 0 )l j ∂(v 1 ) pj ° ds .
i=2...|k1 |+|k2 |
j=1
j=1
|l1 |+···+|li |=|k1 |
| p1 |+···+| pi |=|k2 |
|lm |+| pm |≥1
t
0
(3.50)
∂ |l j |+| p j | (u,v)
satisfy
the
estimates
(3.42)
By the proposition of induction, the derivatives ∂(u
l
p
0 ) j ∂(v 1 ) j
and, therefore, the right-hand side of inequality (3.50) can be estimated indeed. Thus,
by the relations (3.50) and (3.42), we must estimate the maximum of the integrals
Zt
e
α(t−s)
e
m 1 αs
e
m 2 β(s−τ )+m 3 ατ
Zτ
ds
and
0
eβ(t−s) em 1 αs em 2 β(s−τ )+m 3 ατ ds,
t
where m 1 + m 3 = |k1 | and, moreover, m 2 = m 3 = 0 if |k2 | = 0 and (|k1 | − 1)α − β < 0.
Obviously, the integrals satisfy the estimates (3.42).
So, Theorem 3.2 is proved for the case 0 < α < β. The statements of the theorem
for the other cases can be proved in the same way.
3.3. Behavior of the Derivatives If τ → ∞
In Section 3.2 we obtained the estimates for the derivatives of solutions of the boundary
value problem. Below we study the convergences of these derivatives if τ → ∞.
546
M. V. Shashkov and D. V. Turaev
By virtue of the inequalities (3.31)–(3.33) and (3.35)–(3.37), some of the derivatives
tend to zero if τ → ∞ because their norms go to zero. It is clear also that some of the
derivatives do not have limits since their norms are unbounded. In this subsection we
consider derivatives that are uniformly bounded by some positive constants. Namely, we
study here the convergences of the following functions:
¯
∂ |k1 |+|k3 | (u, v) ¯¯
¯
¡
¢k
∂ u 0 , µ 1 ∂t k3 ¯t=t0
if 0 < |k1 |α < β,
¯
∂ |k1 |+|k3 | (u, v) ¯¯
¯
¡ ¢k
∂ u 0 1 ∂(t, µ)k3 ¯t=t0
if α < 0 < β,
¯
∂ |k1 |+|k3 | (u, v) ¯¯
¯
¡ ¢k
∂ u 0 1 ∂t k3 ¯
if α < β < 0,
t=t0
¯
∂ |k2 |+|k3 | (u, v) ¯¯
¯
¡ ¢k
∂ v 1 2 ∂(τ − t)k3 ¯
(3.51)
if 0 < α < β,
τ −t=t0
¯
¯
∂ |k2 |+|k3 | (u, v)
¯
¯
¡ ¢k2
1
k
∂ v
∂(τ − t, µ) 3 ¯τ −t=t0
if α < 0 < β,
¯
¯
∂ |k2 |+|k3 | (u, v)
¯
¯
¡
¢k2
1
k
3
∂ v , µ ∂(τ − t) ¯τ −t=t0
if α < |k2 |β < 0.
Unfortunately, in the general case, the derivatives (3.51) do not have limits if τ → ∞,
but, under some additional conditions, convergence takes place. The next two theorems,
3.4 and 3.5, give us these conditions.
Below, for any functions φ(t), ψ(t) and constants %, τ , we use the following denotations:
¡
¢
Dist (φ; ψ)%,τ ≡ sup kφ(t) − ψ(t)k e−%t ,
t∈(0;τ )
¡
¢
−%(τ −t)
.
Dist (φ; ψ)−
%,τ ≡ sup kφ(t) − ψ(t)k e
t∈(0;τ )
(3.52)
Theorem 3.4. Let τ → +∞ and µ → µ∗ . Also let the boundary conditions u 0 , v 1 vary
in such a way that the solution (3.7) of the boundary value problem (3.1), (3.6) lies in a
bounded domain and the point (u 0 , v 0 ) ≡ (u(t = 0; u 0 , v 1 , τ, µ), v(t = 0; u 0 , v 1 , τ, µ))
converges to a point (u 0∗ , v∗0 ). Then there exist functions Ak1 k3 (t), Bk1 k3 (t), and Ck1 k3 (t)
An Existence Theorem of Smooth Nonlocal Center Manifolds
547
such that
Ã
!
∂ |k1 |+|k3 | (u, v)
; A k1 k3
→0
Dist
¡
¢k
∂ u 0 , µ 1 ∂t k3
σ,τ
!
Ã
∂ |k1 |+|k3 | (u, v)
; Bk1 k3
→0
Dist
¡ ¢k
∂ u 0 1 ∂(t, µ)k3
σ,τ
!
Ã
|k1 |+|k3 |
(u, v)
∂
; C k1 k3
→0
Dist
¡ ¢k1
0
∂ u
∂t k3
σ,τ
if 0 < |k1 |α < σ < β,
if α < 0 < β,
α < σ < β,
(3.53)
if α < σ < β < 0,
where |k1 | + |k3 | ≤ k.
Notice, by virtue of the changes (3.34), Theorem 3.4 directly implies the following result.
Theorem 3.5. Let τ → +∞ and µ → µ∗ . Let also the boundary conditions u 0 ,
v 1 vary in such a way that the solution (3.7) of the boundary value problem (3.1),
(3.6) lies in a bounded domain and the point (u 1 , v 1 ) ≡ (u(t = τ ; u 0 , v 1 , τ, µ), v(t =
−
τ ; u 0 , v 1 , τ, µ)) converges to a point (u 1∗ , v∗1 ). Then there exist functions A−
k2 k3 (t), Bk2 k3 (t),
and Ck−2 k3 (t) such that
!−
∂ |k2 |+|k3 | (u, v)
−
; A k2 k3
→0
if 0 < α < σ < β,
Dist
¡ ¢k
∂ v 1 2 ∂(τ − t)k3
σ,τ
!−
Ã
∂ |k2 |+|k3 | (u, v)
−
; Bk2 k3
→ 0 if α < 0 < β, α < σ < β,
Dist
¡ ¢k
∂ v 1 2 ∂(τ − t, µ)k3
σ,τ
!−
Ã
|k2 |+|k3 |
(u, v)
∂
; Ck−2 k3
→ 0 if α < σ < |k2 |β < 0,
Dist
¡
¢k2
∂ v 1 , µ ∂(τ − t)k3
σ,τ
Ã
(3.54)
where |k2 | + |k3 | ≤ k.
Now we shall prove Theorem 3.4 for the first case, i.e., under the condition 0 < |k1 |α <
σ < β. Note that, by the considerations of the previous subsection, the derivatives with
respect to µ and t can be calculated via derivatives with respect to u 0 . So, we must prove
|k1 |
only.
the statement for ∂∂(u(u,v)
0 )k1
. Let the sequences
First, let us prove the theorem for k1 such that |k1 | = 1, i.e., for ∂(u,v)
∂u 0
vi1 , τi , and µi , (i = 1, 2, 3, . . .) satisfy the theorem conditions. In this case, for any
fixed τ0 , the appropriate solutions (u i (t; u i0 , vi1 , τi , µi ), vi (t; u i0 , vi1 , τi , µi )) converge
uniformly on t ∈ [0, τ0 ] to a solution of the initial value problem
(u ∗ (t; u 0∗ , v∗0 , µ∗ ), v∗ (t; u 0∗ , v∗0 , µ∗ )) that starts from the point (u 0∗ , v∗0 ), i.e.,
u i0 ,
sup k(u i (t), vi (t)) − (u ∗ (t), v∗ (t))k → 0.
t∈(0,τ0 )
(3.55)
548
M. V. Shashkov and D. V. Turaev
It means, in particular, that for any δ > 0,
!
Ã
∂( f, g)
∂( f, g)
¡
¢ (u i , vi , µi , s) ; ¡
¢ (u ∗ , v∗ , µ∗ , s)
Dist
0
1
∂ u ,v
∂ u 0, v1
→0
if τi → ∞.
δ,τi
(3.56)
We shall show that the sequence of the derivatives ∂(u∂ui ,v0 i ) is a Cauchy sequence in the
following sense: For any ε > 0, there exists a constant N (ε) such that
µ
∂(u l , vl ) ∂(u p , v p )
;
Dist
∂u 0
∂u 0
¶
σ,τl
≤ ε,
if l > N (ε), p > N (ε), and α < σ < β.10 Since the solution (u i , vi ), (i = l, p) is a
fixed point of the operator T (see (3.17)), the derivative ∂(u∂ui ,v0 i ) satisfies the following
equality:

Zt


 ∂u i = e At + e A(t−s) ¡ ∂ f ¢ (u , v , µ , s) ∂(u i , vi ) ds,

i
i
i


∂u 0
 ∂u 0
∂ u 0, v1
0
Zt


∂(u i , vi )
∂g
∂vi


= e B(t−s) ¡ 0 1 ¢ (u i , vi , µi , s)
ds.


0
 ∂u
∂u 0
∂ u ,v
(3.57)
τi
Therefore, for any fixed t ∈ [0, τl ], by (3.2)–(3.5), (3.31), (3.52), (3.56), and (3.57), we
have
°
°
° ∂u l
∂u p °
°
°
−
° ∂u 0
∂u 0 °
°
°
¯
¯
°
°
Zt
¯
¯
°
°
∂(u
,
v
)
∂
f
∂(u
,
v
)
∂
f
p
p
l
l
¯
¯
α(t−s) °
° ds
− ¡ 0 1¢ ¯
≤ e
° ∂ ¡u 0 , v 1 ¢ ¯¯
°
0
0
¯
∂u
∂u
∂ u ,v
°
°
(u l ,vl ,µl )
(u p ,vp ,µp )
0
°
°
¯
°
°
¶
µ
Zt
¯
°
°
∂f
∂(u l , vl ) ∂(u p , v p )
¯
α(t−s) °
°
;
eσ s ds
≤ e
° ∂ ¡u 0 , v 1 ¢ ¯¯
° Dist
∂u 0
∂u 0
°
°
σ,τ
l
(u l ,vl ,µl )
0


¯
¯
t
°
°
Z
¯
¯
° ∂(u p , v p ) °
∂
f
∂
f
¯
¯
α(t−s)
δs
°
°
 e
Dist  ¡ 0 1 ¢ ¯
; ¡ 0 1¢ ¯
+ e
° ∂u 0 ° ds
∂ u ,v ¯
∂ u ,v ¯
(u l ,vl ,µl )
(u p ,vp ,µp ) δ,τl
0
µ
¶
∂(u l , vl ) ∂(u p , v p )
;
eσ t + ε2 (N )e(α+δ)t ,
(3.58)
≤ ε1 Dist
∂u 0
∂u 0
σ,τl
10
Without loss of generality, we assume that τl ≤ τ p .
An Existence Theorem of Smooth Nonlocal Center Manifolds
549
°
°
° ∂vl
∂v p °
°
°
° ∂u 0 − ∂u 0 °
°
°
¯
¯
°
°
Zτl
¯
¯
°
∂(u
,
v
)
∂g
∂(u
,
v
)
∂g
p
p °
l
l
¯
¯
β(t−s) °
° ds
− ¡ 0 1¢ ¯
≤ e
° ∂ ¡u 0 , v 1 ¢ ¯¯
∂u 0
∂u 0 °
∂ u ,v ¯
°
°
(u l ,vl ,µl )
(u p ,vp ,µp )
t
°
°
¯
°
°
Zτp
¯
°
∂(u p , v p ) °
∂g
¯
β̃(t−s) °
° ds
¡
¢
+ e
¯
° ∂ u 0, v1 ¯
∂u 0 °
°
°
u
,v
,µ
( p p p)
τl
°
°
¯
°
°
¶
µ
Zτl
¯
°
°
∂g
∂(u l , vl ) ∂(u p , v p )
¯
β̃(t−s) °
°
¡
¢
;
eσ s ds
≤ e
° ∂ u 0 , v 1 ¯¯
° Dist
∂u 0
∂u 0
°
°
σ,τ
l
(u l ,vl ,µl )
t


¯
¯
°
°
Zτl
¯
¯
°
°
∂g
∂g
¯
¯
δs ° ∂(u p , v p ) °
β̃(t−s)


¡
¡
¢
¢
Dist
;
e
ds
+ e
¯
¯
°
∂u 0 °
∂ u 0, v1 ¯
∂ u 0, v1 ¯
(u l ,vl ,µl )
(u p ,vp ,µp ) δ,τl
t
°
°
¯
°°
°
°
Zτp
¯
° ° ∂(u p , v p ) °
°
∂g
¯
β̃(t−s) °
°
°
°
+ e
° ° ∂u 0 ° ds
° ∂ ¡u 0 , v 1 ¢ ¯¯
°
°
(u p ,vp ,µp )
τl
µ
¶
∂(u l , vl ) ∂(u p , v p )
;
eσ t + ε4 (N )e(α+δ)t + ε5 eβ(t−τl )+ατl ,
(3.59)
≤ ε3 Dist
∂u 0
∂u 0
σ,τl
where the constants ε1 , ε3 , δ can be made arbitrarily small and ε2 (N ) → 0, ε4 (N ) → 0
if N → ∞. By the inequalities (3.58) and (3.59), the following relation holds:
µ
¶
¶
µ
∂(u l , vl ) ∂(u p , v p )
∂(u l , vl ) ∂(u p , v p )
;
≤
ε
Dist
;
+ ε7 (N ),
Dist
6
∂u 0
∂u 0
∂u 0
∂u 0
σ,τl
σ,τl
(3.60)
where ε6 < 1 and ε7 (N ) → 0 if N → ∞. Formula (3.60) implies that
∂(u ,v )
Dist( ∂(u∂ul ,v0 l ) ; ∂up 0 p )σ,τl → 0 at N → ∞, i.e., the sequence of the derivatives ∂(u∂ui ,v0 i ) is
a Cauchy sequence and, therefore, it has a limit.
|k1 |
So, we have proved the statement of Theorem 3.4 for the derivatives ∂∂(u(u,v)
0 )k1 such that
|k1 | = 1. The statement of the theorem for the higher order derivatives (2 ≤ |k1 | < k)
may be proved by induction on |k1 |.
3.4. A Boundary Value Problem for the System Xµ
In this subsection we evaluate behavior of orbits of the system X µ near equilibria O
using the results obtained above.
It is well known that in a neighborhood of the saddle O one can introduce local
coordinates (x, y, z), x ∈ R n , y ∈ R 1 , z ∈ R m−1 , such that the system X µ takes the form

ẋ = Ax + f x (x, y, z, µ),
ẏ = γ y + f y (x, y, z, µ),
(3.61)

ż = Bz + f z (x, y, z, µ),
where A is a matrix (n × n) and Spectr A = {λ1 . . . λn }, B is a matrix (m − 1 × m − 1)
550
M. V. Shashkov and D. V. Turaev
and Spectr B = {γ2 . . . γm }, γ ≡ γ1 . Remember that, by the condition (A), we have the
relations
max Re λi < 0 < γ < min Re γi .
i=1,...,n
(3.62)
i=2,...,m
The functions f x , f y , f z are C r -smooth (k ≥ 1) with respect to the phase variables
(x, y, z) and the parameter µ. Moreover, these functions satisfy the following equalities:
( f x , f y , f z )|(x,y,z)=0 = 0,
¯
∂ ( f x , f y , f z ) ¯¯
= 0.
∂(x, y, z) ¯(x,y,z,µ)=0
(3.63)
Below we are interested in the solutions of the system (3.61) satisfying the following
boundary conditions:
x(0) = x 0 ,
y(0) = y 0 ,
z(τ ) = z 1 ,
τ > 0,
(3.64)
with small x 0 , y 0 , and z 1 . So, the problem (3.61), (3.64) is a boundary value problem of
the kind (3.1), (3.6).11
By virtue of Theorem 3.2 (see (3.31)), the solution
¢
¡
(x(t), y(t), z(t)) ≡ x(t; x 0 , y 0 , z 1 , τ, µ), y(t; x 0 , y 0 , z 1 , τ, µ), z(t; x 0 , y 0 , z 1 , τ, µ)
(3.65)
of the boundary value problem (3.61), (3.64) satisfies the following inequalities:
°
°
° ∂(x, y, z) °
° ≤ C,
°
°
°
∂t
°
°
° ∂(x, y, z) °
αt
°
°
° ∂(x 0 , y 0 , µ) ° ≤ C e ,
°
°
° ∂(x, y, z) °
β(t−τ )
°
°
,
° ∂(z 1 , τ ) ° ≤ C e
(3.66)
where
max Re λi < 0 < γ < α < β < min Re γi .
i=1,...,n
i=2,...,m
(3.67)
Moreover, by Theorems 3.4 and 3.5, if µ → 0, τ → ∞, and
(x 0 , y 0 , z 0 ) ≡ (x(0), y(0), z(0)) → (x + , y + , z + ) ∈ W s ,
(x 1 , y 1 , z 1 ) ≡ (x(τ ), y(τ ), z(τ )) → (x − , y − , z − ) ∈ W u ,
then
¯
∂(x, y, z) ¯¯
¡
¢¯
∂ x 0 , y 0 , µ, t ¯
→ A,
t=0
where A and B are some matrices.
11
We use (x, y) for the variable u and the variable z for v.
¯
∂(x, y, z) ¯¯
¡
¢¯
∂ z 1 , τ, t ¯
t=τ
→ B,
(3.68)
An Existence Theorem of Smooth Nonlocal Center Manifolds
551
4. Local and Global Maps
The proof of the main Theorem 2.1 is based on the investigation of the Poincaré map
along the orbits of the system X µ in a neighborhood of the homoclinic loop. This map
may be represented as a superposition of two maps: Tloc and Tgl , where Tloc is defined
by the flow of the system near the equilibrium point and Tgl is defined by the flow near
the global piece of the homoclinic trajectory 0. Using the form (3.61) for the system X µ ,
we shall construct cross sections S 0 , S 1 near the equilibria O and investigate properties
of the first return maps Tloc : S 0 → S 1 , Tgl : S 1 → S 0 .
By virtue of the system (3.61), the stable manifold W s of the point O is an ndimensional surface that, when µ = 0, is tangent to the plane (y, z) = 0 at the point
O = (0, 0, 0). This means that W s is locally the graph of a smooth function,
¡
¢
(y, z) = y s (x, µ), z s (x, µ) ,
where
¡
¢
y (0, µ), z (0, µ) = 0,
s
s
¯
∂ (y s , z s ) ¯¯
= 0.
¯
∂x
(x,µ)=0
The unstable manifold W u of O is locally the graph of a smooth function
x = x u (y, z, µ),
where
x (0, 0, µ) = 0,
u
¯
∂ x u ¯¯
= 0.
∂ (y, z) ¯(y,z,µ)=0
If µ = 0 and t → +∞, the orbit 0 ⊂ W s ∩ W u tends to O. Therefore, there exist
small enough ε > 0, δ > 0, and µ such that the surface
°¡
©
¢°
ª
(4.1)
S 0 = (x, y, z) | kxk = ε, ° x − x + , y − y + , z − z + ° ≤ δ
is a cross section for the orbits close to 0; here (x + , y + , z + ) are the coordinates of a
point of the first intersection of 0 with the surface kxk = ε (see Figure 9).
Since the orbit 0 ⊂ W s ∩ W u does not belong to the strong-unstable submanifold
uu
W (see condition (C)), 0 leaves the equilibrium point O tending toward the y-axis,
which corresponds to the leading direction. Without loss of generality, we assume that
0 leaves O in the positive direction of the y-axis. In this case, if δ > 0, y − > 0, and µ
are small enough, the surface,
°¡
©
¢°
ª
(4.2)
S 1 = (x, y, z) | y = y − , ° x − x − , z − z − ° ≤ δ ,
is a cross section for the orbits close to 0; here (x − , y − , z − ) are the coordinates of a
point of the first intersection of 0 with the surface y = y − .
Thus, we have constructed two cross sections in a small neighborhood of the equilibrium point O: S 1 and S 0 . It is clear that the dimension of the cross sections equals
(n + m − 1) and, without loss of generality, we may consider the coordinates
(x1 , . . . , xn , z 1 , . . . , z m−1 ) as coordinates (x 1 , z 1 ) on the cross section S 1 and the coordinates (x1 , . . . , xn−1 , y, z 1 , . . . , z m−1 ) as coordinates (x 0 , y 0 , z 0 ) on the cross section S 0 .
552
M. V. Shashkov and D. V. Turaev
Fig. 9. Two cross sections S 0 and S 1 can be constructed
for orbits of X µ near the loop L = O ∪ 0.
The Poincaré map T : S 0 → S 0 is the superposition of Tloc : S 0 → S 1 and Tgl :
S → S 0 . The map Tgl : S 1 → S 0 is a diffeomorphism because the flight time from S 1
to S 0 is bounded. Therefore Tgl−1 : S 0 → S 1 may be represented in the following form:


µ
¶ µ
¶ x 0 − x + (µ)
¶
µ 1
φ(x 0 , y 0 , z 0 , µ)
a11 (µ) a12 (µ)  0
x − x − (µ)
+

y − y (µ) +
=
, (4.3)
z 1 − z − (µ)
a21 (µ) a22 (µ)
ψ(x 0 , y 0 , z 0 , µ)
z 0 − z + (µ)
1
where
det
µ
a11
a21
a12
a22
¶
6= 0.
Here a11 , a22 , a12 , a21 are matrices of the dimensions (n ×n), (m −1×m −1), (n ×m −1),
(m−1×n), respectively. The smooth functions φ(x 0 , y 0 , z 0 , µ), ψ(x 0 , y 0 , z 0 , µ) contain
only the nonlinear terms on (x 0 − x + , y 0 − y + , z 0 − z + ) and are defined in the domain
°¡
© 0 0 0
¢°
ª
(x , y , z , µ) | ° x 0 − x + , y 0 − y + , z 0 − z + , µ ° ≤ δ .
(4.4)
Note that we assume that the manifold W s+ transversely intersects the leaves of the
foliation F u (condition (D)). This assumption means exactly the same as the condition
det (a11 ) 6= 0.
(4.5)
Consider the local map Tloc : S 0 → S 1 . The study of this map is not so trivial because
the flight time of orbits that go from S 0 to S 1 is unbounded. Moreover the infimum τ ∗
of the flight times can be made arbitrarily big by choosing small δ. In order to obtain the
local map, we use the boundary value problem (3.61), (3.64). According to this problem,
for given τ and small x 0 , y 0 , z 1 , there exists a unique orbit
¢
¡
(x(t), y(t), z(t)) ≡ x(t; x 0 , y 0 , z 1 , τ, µ), y(t; x 0 , y 0 , z 1 , τ, µ), z(t; x 0 , y 0 , z 1 , τ, µ) ,
An Existence Theorem of Smooth Nonlocal Center Manifolds
553
which, at t = 0, starts with a point (x 0 , y 0 , z 0 ) and reaches a point (x 1 , y 1 , z 1 ) at t = τ .
This means that the following equalities are fulfilled:
x 1 = x(τ ; x 0 , y 0 , z 1 , τ, µ),
y 1 = y(τ ; x 0 , y 0 , z 1 , τ, µ),
z 0 = z(0; x 0 , y 0 , z 1 , τ, µ).
(4.6)
If we fix y 1 = y − > 0 and kx 0 k = ε, the first and third equations of the system (4.6)
give implicitly the map Tloc : (x 0 , y 0 , z 0 ) 7→ (x 1 , z 1 ) from S 0 to S 1 where τ should be
expressed from the second equation of the system (4.6) as a function of (x 0 , y 0 , z 1 , µ):
x 1 = x(τ (x 0 , y 0 , z 1 , µ); x 0 , y 0 , z 1 , τ (x 0 , y 0 , z 1 , µ), µ),
z 0 = z(0; x 0 , y 0 , z 1 , τ (x 0 , y 0 , z 1 , µ), µ),
(4.7)
y − ≡ y(τ (x 0 , y 0 , z 1 , µ); x 0 , y 0 , z 1 , τ (x 0 , y 0 , z 1 , µ), µ).
(4.8)
where
We shall show (see (4.17)) that
¯
¯
∂ y ¯¯
∂ y ¯¯
+
6= 0.
∂t ¯t=τ
∂τ ¯t=τ
Therefore, the flight time τ > τ ∗ (δ) can indeed be found from the equation (4.8) and
the local map Tloc may be represented in the following form:
x 1 = f (x 0 , y 0 , z 1 , µ),
z 0 = g(x 0 , y 0 , z 1 , µ).
(4.9)
Since y − > 0, the functions f , g are defined in the domain
°¡
¢°
ª
© 0 0 1
(x , y , z , µ) | ° x 0 − x + , y 0 − y + , z 1 − z − , µ ° ≤ δ, y 0 > y s (x 0 , µ) , (4.10)
where the function y = y s (x 0 , µ) gives the y-coordinate of the intersection of the stable
manifold W s with the cross section S 0 .
Notice the relation that follows from the identity (4.8):
¯ ¶−1
¯
µ ¯
¯
∂ y ¯¯
∂y
∂ y ¯¯
∂τ
¯ .
+
(4.11)
=
−
¯
¯
0
0
1
0
0
1
∂(x , y , z , µ)
∂t t=τ
∂τ t=τ
∂(x , y , z , µ) ¯t=τ
By formulas (4.7), (4.9), and (4.11) we have
∂x1
∂f
≡
∂(x 0 , y 0 , z 1 , µ)
∂(x 0 , y 0 , z 1 , µ)
¯
¯
µ ¯
¯
¯
¯
∂x
¯ − ∂x ¯ + ∂x ¯
=
¯
¯
0
0
1
∂(x ,y ,z ,µ)
∂t
∂τ ¯
t=τ
t=τ
¶µ
t=τ
∂z 0
∂g
≡
∂(x 0 , y 0 , z 1 , µ)
∂(x 0 , y 0 , z 1 , µ)
¯
¯ µ ¯
¯
∂z ¯¯
∂ y ¯¯
∂z
¯
−
=
¯
¯
0
0
1
∂(x , y , z , µ)
∂τ
∂t ¯
t=0
t=0
t=τ
¯
¯ ¶−1
¯
¯
∂ y ¯¯
∂ y ¯¯
∂y
¯ ,
+
¯
¯
0
0
1
∂t t=τ ∂τ t=τ
∂(x ,y ,z ,µ) ¯t=τ
¯ ¶−1
¯
¯
∂ y ¯¯
∂y
¯ . (4.12)
+
¯
0
0
1
∂τ t=τ
∂(x , y , z , µ) ¯t=τ
554
M. V. Shashkov and D. V. Turaev
By the identity (3.40), we also have the relation
¯
¯
¯
∂ y ¯¯
∂z ¯¯
∂ y ¯¯
+
∂z 1 ¯
∂t ¯
∂τ ¯
t=τ
t=τ
≡ 0.
(4.13)
t=τ
The orbit 0 leaves the equilibria O tending in the leading direction (positive y-axis);
therefore,
¯
µ ¯ ¶
∂ y ¯¯
∂z ¯¯
=o
(4.14)
at y − → 0.
¯
∂t
∂t ¯
t=τ
The estimates (3.66) imply that
t=τ
¯
°
° ∂y ¯
¯
°
° ∂z 1 ¯
t=τ
°
°
° ≤ C.
°
So, by (4.13), (4.14), and (4.15),
¯
µ ¯ ¶
∂ y ¯¯
∂ y ¯¯
=o
at y − → 0.
¯
∂τ t=τ
∂t ¯t=τ
¯
¯
∼ γ y − , formula (4.16) implies that
Since ∂∂ty ¯
t=τ
¯
¯
∂ y ¯¯
∂ y ¯¯
+
∼ γ y − > 0.
∂t ¯t=τ
∂τ ¯t=τ
Since y − is fixed, formula (4.17) implies that
¯ ¶−1
µ ¯
∂ y ¯¯
2
∂ y ¯¯
+
<
,
¯
¯
∂t t=τ
∂τ t=τ
γ y−
(4.15)
(4.16)
(4.17)
(4.18)
∂y
|t=τ )−1 is bounded. Now, according to the estimates (3.66) for the
i.e., ( ∂∂ty |t=τ + ∂τ
boundary value problem (3.61), (3.64), and by (4.12), (4.18), we have the following
relations:
° 1° °
° °
°
°
°
°∂x ° ° ∂ f °
° °
°
°
∂f
∂x1
°
°≡°
° ≤ D eατ ,
°≡°
°
°
° ∂z 1 ° ° ∂z 1 ° ≤ d,
° ∂(x 0 , y 0 , µ) ° ° ∂(x 0 , y 0 , µ) °
(4.19)
° °
°
° 0° °
°
°
0
° °
°
° ∂z ° ° ∂g °
°
∂g
∂z
−βτ
° °
°
°
° °
°
°
° ∂z 1 ° ≡ ° ∂z 1 ° ≤ D e ,
° ∂(x 0 , y 0 , µ) ° ≡ ° ∂(x 0 , y 0 , µ) ° ≤ d,
where D and d are some constants and the constants α and β satisfy inequalities (3.67).
Moreover, if µ → 0, τ → ∞, (x 0 , y 0 , z 0 ) → (x + , y + , z + ) ∈ W s , and (x 1 , y 1 , z 1 ) →
(x − , y − , z − ) ∈ W u , by (3.68), we have
∂f
∂x1
≡ 1 → Ā,
∂z 1
∂z
∂g
∂z 0
≡
→ B̄,
∂(x 0 , y 0 , µ)
∂(x 0 , y 0 , µ)
(4.20)
where Ā and B̄ are fixed matrices. Without loss of generality, we may assume that Ā ≡ 0
1
0
on S 1 and z 0 7→ z new
and B̄ ≡ 0. Otherwise, we shall change the variables x 1 7→ xnew
0
on S in the following way:
1
= x 1 − Āz 1 ,
xnew
0
z new
= z 0 − B̄(x 0 , y 0 , µ).
(4.21)
So, we have
d = o(1)
if τ → +∞.
(4.22)
An Existence Theorem of Smooth Nonlocal Center Manifolds
555
5. Redefinition of the Local and Global Maps
In order to establish the existence of the center manifold Mcs , we must show that the cross
section S 0 contains an invariant set with respect to the operator T = Tgl ◦Tloc and that this
set is the graph of some vector-function z 0 = h 0∗ (x 0 , y 0 , µ). Below we use the regular
method, which is based on proving the contractibility for the graph transformations (see
Section 6). Namely, we show that the operator T induces a contraction operator P in a
complete metric space H 0 of functions z 0 = h 0 (x 0 , y 0 , µ). However, to construct the
operator P, we must redefine the maps Tgl and Tloc in an extended domain and, by that,
redefine the Poincaré map T = Tgl ◦ Tloc .
Consider first the global map Tgl : S 1 → S 0 . As we have seen in Section 3.4, the
inverse map Tgl−1 : S 0 → S 1 is given by the formula (4.3). We redefine Tgl−1 by the
following formulas:
x 1 = 8(x 0 , y 0 , z 0 , µ)
≡ x − (µ) + a11 (µ)
µ
x 0 − x + (µ)
y 0 − y + (µ)
¶
¡
¢
+ a12 (µ) z 0 − z + (µ)
+ ϑ(k(x 0 − x + , y 0 − y + )k/δ)φ(x 0 , y 0 , z 0 , µ),
z 1 = 9(x 0 , y 0 , z 0 , µ) ≡ z − (µ)
¶
µ 0
¶
µ
¡ 0
¢
x − x + (µ)
+
0
0 0
+ a22 (µ) z − z (µ) + ψ(x , y , z , µ)
+ a21 (µ)
y 0 − y + (µ)
× ϑ(k(x 0 − x + , y 0 − y + )k/δ),
where ϑ(t) is a C ∞ -function such that
½
1,
if t ≤ 1,
ϑ(t) =
0,
if t ≥ 2,
(5.1)
¯
¯
¯ ∂ϑ(t) ¯
¯
¯
¯ ∂t ¯ < 2.
(5.2)
Observe that the redefined map coincides with the initial map if k(x 0 −x + , y 0 −y + )k < δ,
but functions 8 and 9 are defined in the larger domain
°¡
©
¢°
¡
¢
ª
(5.3)
Ägl = (x 0 , y 0 , z 0 , µ) | ° z 0 − z + , µ ° ≤ δ x 0 , y 0 ∈ R n .
Let us consider now the local map Tloc : S 0 → S 1 (see (4.9)). We overwrite Tloc as
x 1 = F(x 0 , y 0 , z 1 , µ)
≡ ϑ(k(x 0 − x + , y 0 − y + )k/δ) f (x 0 , |y 0 − y s (x 0 , µ)| + y s (x 0 , µ), z 1 , µ),
z 0 = G(x 0 , y 0 , z 1 , µ)
≡ ϑ(k(x 0 − x + , y 0 − y + )k/δ)g(x 0 , |y 0 − y s (x 0 , µ)| + y s (x 0 , µ), z 1 , µ). (5.4)
556
M. V. Shashkov and D. V. Turaev
The functions f and g, defining the local map (4.9), coincide with the functions F and
G, respectively, if k(x 0 − x + , y 0 − y + )k < δ and y 0 > y s (x 0 , µ), but the functions F
and G are defined for any (x 0 , y 0 ) ∈ R n , i.e., in the domain
°¡
©
¢°
ª
(5.5)
Äloc = (x 0 , y 0 , z 1 , µ) | ° z 1 − z − , µ ° ≤ δ, (x 0 , y 0 ) ∈ R n .
Moreover, by (4.19), (4.22), (5.2), and (5.4), F and G satisfy the following estimates:
°
°
°
°
°
°∂F °
°
∂F
° ≤ M eατ ,
°
°
°
° ∂z 1 ° = o(1) if τ → ∞,
° ∂(x 0 , y 0 , µ) °
(5.6)
°
°
°
°
°
° ∂G °
°
∂G
−βτ
° = o(1)
°
°
°
if τ → ∞,
° ∂z 1 ° ≤ M e .
° ∂(x 0 , y 0 , µ) °
Here M is a constant and the constants α and β satisfy inequalities (3.67).
6. Existence of the Lipschitz Center Manifold
Introduce the space H 0 of bounded (m − 1)-dimensional vector-functions z 0 =
h 0 (x 0 , y 0 , µ) defined in the domain
©
ª
(6.1)
D 0 = (x 0 , y 0 , µ) | (x 0 , y 0 ) ∈ R n , kµk < δ .
Let the functions h 0 ∈ H 0 satisfy the following Lipschitz condition:
°
°
°
° 0 0 0
°h (x , y , µ1 ) − h 0 (x 0 , y 0 , µ2 )° ≤ ` °(x 0 , y 0 , µ1 ) − (x 0 , y 0 , µ2 )°.
1
1
2
2
1
1
2
2
(6.2)
Here ` > 0 is a small constant, which we define below. The space H 0 with the uniform
norm
°
° 0 0 0
° 0°
°h (x , y , µ)°
°h ° 0 ≡
sup
(6.3)
D
(x 0 ,y 0 ,µ)∈D 0
is a complete metric space.
Introduce also the space H 1 of bounded (m − 1)-dimensional vector-functions z 1 =
1 1
h (x , µ) defined in the domain
©
ª
(6.4)
D 1 = (x 1 , µ) | x 1 ∈ R n , kµk < δ .
Let the functions h 1 ∈ H 1 satisfy the Lipschitz condition
°
°
°
° 1 1
°h (x , µ1 ) − h 1 (x 1 , µ2 )° ≤ L °(x 1 , µ1 ) − (x 1 , µ2 )°,
1
2
1
2
(6.5)
where L is some, maybe big, constant, which we define later. The space H 1 with the
norm
°
°
° 1°
°h ° 1 ≡ sup °h 1 (x 1 , µ)°
(6.6)
D
(x 1 ,µ)∈D 1
is a complete metric space.
An Existence Theorem of Smooth Nonlocal Center Manifolds
557
Lemma 6.1. The map Tgl−1 induces an operator Pgl : H 0 → H 1 , i.e., Tgl−1 transforms
points of the graph of any function h 0 ∈ H 0 to points of the graph of some function
h 1 ∈ H 1 . Moreover the map Pgl satisfies the property of the limited expansion, i.e., for
any h 01 ∈ H 0 , h 02 ∈ H 0 , the following estimate holds:
°
°
° ¡ 0¢
¡ ¢°
° Pgl h − Pgl h 0 ° 1 ≤ Q °h 0 − h 0 ° 0 ,
(6.7)
1
2 D
1
2 D
where Q is some constant.
To prove the lemma we must show that Tgl−1 induces an operator Pgl : H 0 → H 1 .
Let a point with the coordinates (x 0 , y 0 , h 0 (x 0 , y 0 , µ)) be mapped by Tgl−1 to a point
(x 1 , h 1 (x 1 , µ)). By virtue of formulas (5.1), we have the equalities
x 1 = 8(x 0 , y 0 , h 0 (x 0 , y 0 , µ), µ),
h 1 (x 1 , µ) = 9(x 0 , y 0 , h 0 (x 0 , y 0 , µ), µ).
(6.8)
Let us fix h 0 ∈ H 0 . We assume that the constant ` in (6.2) is small enough. In this case
the function h 0 ∈ H 0 satisfies the Lipschitz condition with the small constant ` and,
since det(a11 ) 6= 0 (see (4.5)), the first equation in (6.8) may be resolved with respect to
x 0 and y 0 , i.e.,
x 0 = x 0 (x 1 , µ),
y 0 = y 0 (x 1 , µ).
(6.9)
Substituting x (x , µ) and y (x , µ) into the right-hand side of the second equation in
(6.8), we obtain the function h 1 (x 1 , µ). Therefore, the operator Pgl is correctly defined.
It is easy to check that there exists a constant L such that the function h 1 (x 1 , µ) satisfies
the Lipschitz condition (6.5) and that the function h 1 (x 1 , µ) is bounded. This implies
that the operator Pgl maps any function from H 0 into H 1 .
Let us examine the property of the limited expansion. Let the functions h 01 ∈ H 0 and
0
h 2 ∈ H 0 be mapped by Pgl to h 11 ∈ H 1 and h 12 ∈ H 1 , respectively. Let us estimate
the norm kh 11 − h 12 k D1 . Assume that a point (x 0 , y 0 , h 01 (x 0 , y 0 , µ)) is mapped to a point
(x11 , h 11 (x11 , µ)) and a point (x 0 , y 0 , h 02 (x 0 , y 0 , µ)) is mapped to a point (x21 , h 12 (x21 , µ))
(see Figure 10). By (6.8), (6.5), and (5.1) we obtain
0
1
0
1
kh 11 (x11 , µ) − h 12 (x11 , µ)k
≤ kh 11 (x11 , µ) − h 12 (x21 , µ)k + kh 12 (x21 , µ) − h 12 (x11 , µ)k
≤ kh 11 (x11 , µ) − h 12 (x21 , µ)k + L kx21 − x11 k
k9(x 0 , y 0 , h 01 (x 0 , y 0 , µ), µ) − 9(x 0 , y 0 , h 02 (x 0 , y 0 , µ), µ)k
+ L k8(x 0 , y 0 , h 01 (x 0 , y 0 , µ), µ) − 8(x 0 , y 0 , h 02 (x 0 , y 0 , µ), µ)k
ð
°
°
° !
° ∂8 °
°
° ∂9 °
°
°
° 0
°
°
°
°h − h 0 ° 0 ≤ Q °h 0 − h 0 ° 0 , (6.10)
≤ °
1
2 D
1
2 D
° ∂z 0 ° + L ° ∂z 0 °
Ägl
Ägl
where Q is some constant and k·kÄgl denotes sup(x 0 ,y 0 ,z 0 ,µ)∈Ägl k·k. The property of the
limited expansion immediately follows from the estimate (6.10). Lemma 6.1 is proved.
558
M. V. Shashkov and D. V. Turaev
Fig. 10. The map Tgl−1 induces a map Pgl : H 0 → H 1 ,
satisfying the property of the limited expansion.
−1
−1
Lemma 6.2. The map Tloc
induces an operator Ploc : H 1 → H 0 , i.e., Tloc
transforms
1
1
points of the graph of any function h ∈ H to points of the graph of some function
h 0 ∈ H 0 . Moreover the operator Ploc satisfies the property of the strong contraction,
i.e., for any h 11 ∈ H 1 , h 12 ∈ H 1 ,
°
°
°
¡ ¢
¡ ¢°
° Ploc h 1 − Ploc h 1 ° 0 ≤ q °h 1 − h 1 ° 1 ,
1
2 D
1
2 D
(6.11)
where the constant q can be made arbitrarily small.
−1
induces an operator Ploc : H 1 → H 0 .
To prove the lemma, first we check that Tloc
0
0 0
Assume that the points of the graph z = h (x , y 0 , µ) are mapped by Tloc to the points
of the graph z 1 = h 1 (x 1 , µ) ∈ H 1 . By (5.4) we have the equalities
x 1 = F(x 0 , y 0 , h 1 (x 1 , µ), µ),
h 0 (x 0 , y 0 , µ) = G(x 0 , y 0 , h 1 (x 1 , µ), µ),
(6.12)
which give us the operator Ploc : H 1 → H 0 . First, let us check that the formulas (6.12)
determine the operator Ploc correctly, i.e., it is possible to calculate z 0 = h 0 (x 0 , y 0 , µ)
if we know h 1 ∈ H 1 . Then we check that h 0 belongs to H 1 . Let us fix h 1 ∈ H 1 and
(x 0 , y 0 , µ) ∈ D 0 . Since h 1 is a Lipschitz function with respect to x 1 (see (6.5)), the map
x 1 7→ x 1 defined by the formula
x 1 = F(x 0 , y 0 , h 1 (x 1 , µ), µ)
(6.13)
An Existence Theorem of Smooth Nonlocal Center Manifolds
559
is a contraction map from R n to R n . Indeed, it follows from formulas (6.13), (6.5), and
(5.6), that
°
°
°
° 1
°x − x 1 ° ≡ ° F(x 0 , y 0 , h 1 (x 1 , µ), µ) − F(x 0 , y 0 , h 1 (x 1 , µ), µ)°
1
1
1
2
°
°
°
°∂F ° ° 1
° °x − x 1 °,
≤ L°
1
1
° ∂z 1 °
Äloc
where
(6.14)
°
°
°∂F °
°
≤ p < 1.
L°
° ∂z 1 °
Äloc
(6.15)
This implies that the first equation of the system (6.12) may be resolved with respect to
x 1 , i.e.,
x 1 = x 1 (x 0 , y 0 , µ).
(6.16)
Substituting (6.16) into the second equation of the system (6.12), we obtain h (x , y 0 , µ).
Hence, the operator Ploc is correctly defined. By (5.4), (5.2) the function G is bounded
and, therefore, h 0 is also bounded. In order to show that the function h 0 belongs to H 0 ,
we must prove that h 0 satisfies the Lipschitz condition (6.2) in the domain D 0 (see (6.1)).
First, we check this property for τ ∈ [τ 0 ; θ τ 0 ], i.e., in the domain
©
ª
(6.17)
Dτ00 = (x 0 , y 0 , µ) | (x 0 , y 0 ) ∈ R n , kµk < δ, τ ∈ [τ 0 ; θ τ 0 ] ,
0
0
where the constant θ satisfies the following equalities:
1 < θ < β/α.
(6.18)
We denote the corresponding range of the variables (x 0 , y 0 , z 1 , µ) by
°¡
©
¢°
ª
τ0
= (x 0 , y 0 , z 1 , µ) | ° z 1 − z − , µ ° ≤ δ, (x 0 , y 0 ) ∈ R n , τ ∈ [τ 0 ; θ τ 0 ] .
Äloc
Let
(x10 , y10 , µ1 )
(x20 , y20 , µ2 )
(6.19)
∈
and
∈
By (6.12), (6.5), and (6.16), we obtain
°¡ 1 0 0
¢ ¡ 1 0 0
¢°
° x (x , y , µ1 ), µ1 − x (x , y , µ2 ), µ2 °
1
1
2
2
°¡ ¡
¢
¢
≡ ° F x10 , y10 , h 1 (x 1 (x10 , y10 , µ1 ), µ1 ), µ1 , µ1
¢
¢°
¡ ¡
− F x20 , y20 , h 1 (x 1 (x20 , y20 , µ2 ), µ2 ), µ2 , µ1 °
°
°
° ∂(F, µ) ° °¡ 0 0
¢ ¡
¢°
° ° x , y , µ1 − x 0 , y 0 , µ2 °
°
≤°
1
1
2
2
°
0
0
0
∂(x , y , µ) Äτ
loc
°
°
° ∂ F ° °¡ 1 0 0
¢ ¡ 1 0 0
¢°
°
° °
+ L °
° ∂z 1 ° τ 0 x (x1 , y1 , µ1 ), µ1 − x (x2 , y2 , µ2 ), µ2 . (6.20)
Ä
Dτ00
Dτ00 .
loc
Here we use the following notation: k·kÄτ 0 ≡ sup(x 0 ,y 0 ,z 1 ,µ)∈Äτ 0 k·k . It follows from
loc
loc
(6.15) and (6.20) that the function (x 1 (x 0 , y 0 , µ), µ) satisfies the Lipschitz property
°¡ 1 0 0
¢ ¡
¢°
° x (x , y , µ1 ), µ1 − x 1 (x 0 , y 0 , µ2 ), µ2 °
1
1
2
2
°
°
° ∂(F, µ) °
°
≤°
° ∂(x 0 , y 0 , µ) °
Ã
0
τ
Äloc
°
° !−1
°¡ 0 0
°∂F °
¢ ¡
¢°
° x , y , µ1 − x 0 , y 0 , µ2 °. (6.21)
°
1− L °
1
1
2
2
° ∂z 1 ° τ 0
Ä
loc
560
M. V. Shashkov and D. V. Turaev
By virtue of the relations (5.6) (6.5), (6.12), (6.15), (6.16), and (6.21), the function
h 0 (x 0 , y 0 , µ) satisfies the estimate
°
° 0 0 0
°h (x , y , µ1 ) − h 0 (x 0 , y 0 , µ2 )°
1
1
2
2
° ¡
¢
≡ °G x10 , y10 , h 1 (x 1 (x10 , y10 , µ1 ), µ1 ), µ1
¢°
¡
− G x20 , y20 , h 1 (x 1 (x20 , y20 , µ2 ), µ2 ), µ2 °
°
°
°
°
∂G
°
°
≤°
∂(x 0 , y 0 , µ) °
τ0
Äloc
°
°
° ∂G °
°
+ L ° 1°
∂z °Äτ 0
loc
°¡ 0 0
¢ ¡
¢°
° x , y , µ1 − x 0 , y 0 , µ2 °
1
1
2
°
°
° ∂(F, µ) °
°
°
° ∂(x 0 , y 0 , µ) °
2
Ã
0
τ
Äloc
° !−1
°
°∂F °
°
1−L °
° ∂z 1 ° τ 0
Ä
loc
°¡
¢ ¡
¢°
× ° x10 , y10 , µ1 − x20 , y20 , µ2 °
¢ ¡
¢°
0 °¡
≤ M0 e%τ ° x10 , y10 , µ1 − x20 , y20 , µ2 °,
(6.22)
where M0 > 0 and % < 0 are some constants. Note that τ 0 ≥ τ ∗ (δ) and τ ∗ (δ) → ∞
0
if δ → 0. Therefore for any small `, there exists δ such that the constant M0 e%τ will
be less than `, i.e., the function h 0 (x 0 , y 0 , µ) satisfies the Lipschitz property (6.2) in the
domain Dτ00 . Note that h 0 (x 0 , y 0 , µ) is a continuous function, D 0 is a convex domain,
and
[
Dτ00 = D 0 .
τ 0 ≥τ ∗ (δ)
Therefore the function h 0 (x 0 , y 0 , µ) satisfies the Lipschitz property (6.2) in the domain
D 0 , i.e., h 0 belongs to H 0 .
Now let us prove the strong contractibility of the map Ploc : H 1 → H 0 . Assume
the functions h 11 ∈ H 1 and h 12 ∈ H 1 are mapped by Ploc to h 01 ∈ H 0 and h 02 ∈ H 0 ,
respectively. Let us estimate the norm kh 01 − h 02 k D0 . Let a point (x11 , h 11 (x11 , µ)) be
mapped to a point (x 0 , y 0 , h 01 (x 0 , y 0 , µ)) and a point (x21 , h 12 (x21 , µ)) be mapped to a
point (x 0 , y 0 , h 02 (x 0 , y 0 , µ)) (see Figure 11). Then, by (6.12), we have
° 1
°
° ¡
¢
¡
¢°
°x − x 1 ° ≡ ° F x 0 , y 0 , h 1 (x 1 , µ), µ − F x 0 , y 0 , h 1 (x 1 , µ), µ °
1
2
1 1
2 2
° ¡
¢
¡
¢°
≤ ° F x 0 , y 0 , h 11 (x11 , µ), µ − F x 0 , y 0 , h 12 (x11 , µ), µ °
° ¡
¢
¡
¢°
+ ° F x 0 , y 0 , h 12 (x11 , µ), µ − F x 0 , y 0 , h 12 (x21 , µ), µ °
°
°
°
°
°∂F °
°
°
°
°∂F ° ° 1
1°
°
°
°
°
L °x11 − x21 °.
h 1 − h 2 D1 + ° 1 °
(6.23)
≤ ° 1°
°
∂z Äloc
∂z Äloc
Therefore, by (6.23) and (6.15), we obtain
!−1 °
Ã
°
°
°
°
°∂F °
°
°∂F ° ° 1
° 1
1
°
°
° °h − h 1 ° 1 .
°x − x ° ≤ 1 − °
L
1
2
1
2 D
° ∂z 1 °
° ∂z 1 °
Äloc
Äloc
(6.24)
An Existence Theorem of Smooth Nonlocal Center Manifolds
561
−1
Fig. 11. The map Tloc
induces a map Ploc : H 1 → H 0 ,
satisfying the property of the strong contraction.
By virtue of the relations (5.6), (6.5), (6.12), (6.24), and (6.15), the following estimate
takes place:
°
° 0 0 0
°h (x , y , µ) − h 0 (x 0 , y 0 , µ)°
1
2
° ¡
¢
¡
¢°
≡ °G x 0 , y 0 , h 11 (x11 , µ), µ − G x 0 , y 0 , h 12 (x21 , µ), µ °
° ¡
¢
¡
¢°
≤ °G x 0 , y 0 , h 11 (x11 , µ), µ − G x 0 , y 0 , h 12 (x11 , µ), µ °
° ¡
¢
¡
¢°
+ °G x 0 , y 0 , h 12 (x11 , µ), µ − G x 0 , y 0 , h 12 (x21 , µ), µ °
Ã
!−1
°
°
°
°
°
°
° ∂G °
°∂F °
°
° ∂G ° ° 1
1
° °h − h ° 1 + °
°
°
L 1−°
L
≤°
1
2 D
° ∂z 1 °
° ∂z 1 °
° ∂z 1 °
Äloc
Äloc
Äloc
°
°
°∂F ° ° 1
°
°
°
° °h − h 1 ° 1 ≤ M1 e%τ ∗ °h 1 − h 1 ° 1 ,
× °
(6.25)
1
2 D
1
2 D
° ∂z 1 °
Äloc
where M1 > 0 and % < 0 are some constants. Note that τ ∗ (δ) → ∞ if δ → 0. Therefore
∗
for any small q > 0, there exists δ > 0 such that the constant M1 e%τ is less than q, i.e.,
Ploc is the strong contraction operator. Lemma 6.2 is proved.
By virtue of Lemmas 6.1 and 6.2, we have the following statement.
Lemma 6.3. The superposition P = Ploc ◦ Pgl : H 0 → H 0 is the contraction map,
i.e., for any h 01 ∈ H 0 , h 02 ∈ H 0 , the following estimate holds:
°
°
° ¡ 0¢
¡ ¢°
° P h − P h 0 ° 0 ≤ p °h 0 − h 0 ° 0 ,
(6.26)
1
2 D
1
2 D
where the constant p = q Q is strictly less 1.
562
M. V. Shashkov and D. V. Turaev
By virtue of Lemma 6.3 and Banach’s principle, there exists a unique fixed point h 0∗ =
P(h 0∗ ) ∈ H 0 that is a limit of the functions h 00 , h 01 , h 02 , . . ., obtained by the iterations
0
= P(h i0 ), (i = 0, 1, 2, . . .) with any initial function h 00 ∈ H 0 . The graph of the
h i+1
function h 0∗ is an invariant set with respect to the Poincaré map T and, therefore, orbits of
the system X µ passing through the points of this graph form the invariant manifold Mcs .
−1
◦ Tgl−1 ,
Since the contraction map P is induced by the inverse Poincaré map T −1 = Tloc
cs
cs
the manifold M is a repelling manifold, i.e., any orbit not lying in M leaves the
neighborhood U of the homoclinic orbit as t tends to +∞.
7. Smoothness of the Invariant Functions h0∗ and h1∗
In order to establish that Mcs is a smooth manifold, we must prove that h 0∗ (x 0 , y 0 , µ) ∈
H 0 is a smooth function with respect to (x 0 , y 0 , µ).
In our case, whereas the map Tgl−1 is a diffeomorphism, the following statement takes
place.
Lemma 7.1. The operator Pgl maps a smooth function h 0 ∈ H 0 to a smooth function
h1 ∈ H 1.
To calculate the derivatives of the function h 1 (x 1 , µ), we use the formulas (6.8), (6.9).
By these formulas we have the following identity:
¡
¢ ¡ ¡
¢ ¢
x 1 , µ ≡ 8 x 0 (x 1 , µ), y 0 (x 1 , µ), h 0 (x 0 (x 1 , µ), y 0 (x 1 , µ), µ), µ , µ ,
(7.1)
which directly implies the relation
∂(x 0 (x 1 , µ), y 0 (x 1 , µ), µ)
≡
∂(x 1 , µ)
µ
∂(8, µ)
∂(8, µ)
∂h 0
+
∂(x 0 , y 0 , µ)
∂z 0 ∂(x 0 , y 0 , µ)
¶−1
.
(7.2)
Therefore, by the relations (6.8), (6.9), and (7.2),
∂h 1 (x 1 , µ)
≡
∂(x 1 , µ)
×
·µ
µ
∂9
∂9
∂h 0
+ 0
0
0
0
∂(x , y , µ) ∂z ∂(x , y 0 , µ)
¶
∂(8, µ)
∂h 0
∂(8, µ)
+
∂(x 0 , y 0 , µ)
∂z 0 ∂(x 0 , y 0 , µ)
¶−1 #
.
x 0 =x 0 (x 1 ,µ)
y =y (x ,µ)
0
0
1
z 0 =h 0 (x 0 (x 1 ,µ),y 0 (x 1 ,µ),µ)
(7.3)
Notice, by virtue of the relations (5.1), (5.2), (6.2), and (4.5), the operator
µ
∂h 0
∂(8, µ)
∂(8, µ)
+
0
0
0
0
∂(x , y , µ)
∂z
∂(x , y 0 , µ)
¶
is an invertible operator. Thus, the derivative may indeed be calculated via formula (7.3).
An Existence Theorem of Smooth Nonlocal Center Manifolds
563
Consider the space N 0 of continuous functions η0 (x 0 , y 0 , µ) defined in the domain
D 0 (see (6.1)). We assume that the functions η0 ∈ N 0 are uniformly bounded by the
constant ` (see (6.2)), i.e.,
°
° 0 0 0
°η (x , y , µ)° ≤ `.
(7.4)
The space N 0 with the uniform norm
° 0°
°η ° 0 ≡
sup
D
(x 0 ,y 0 ,µ)∈D 0
°
° 0 0 0
°η (x , y , µ)°
(7.5)
is a complete metric space.
Let us introduce the space N 1 of continuous functions η1 (x 1 , µ) defined in the domain
1
D (see (6.4)). Let the functions η1 ∈ N 1 be uniformly bounded by the constant L (see
(6.5)), i.e.,
°
° 1 1
°η (x , µ)° ≤ L .
(7.6)
The space N 1 with the norm
° 1°
°η °
D1
≡
sup
(x 1 ,µ)∈D 1
°
° 1 1
°η (x , µ)°
(7.7)
is also a complete metric space.
h0
. For any h 0 ∈ H 0 , we define the
The relation (7.3) induces a family of operators Sgl
¢
0 ¡
h
η0 7→ η1 by the following rule:
operator Sgl
·µ
η (x , µ) ≡
1
1
µ
×
¶
∂9 0 0 0
∂9
+
η (x , y , µ)
∂(x 0 , y 0 , µ) ∂z 0
¶−1 #
∂(8, µ) 0 0 0
∂(8, µ)
η (x , y , µ)
+
∂(x 0 , y 0 , µ)
∂z 0
x 0 =x 0 (x 1 ,µ)
.
y =y (x ,µ)
0
0
1
z 0 =h 0 (x 0 (x 1 ,µ),y 0 (x 1 ,µ),µ)
(7.8)
Here the functions x 0 (x 1 , µ) and y 0 (x 1 , µ) (see (6.9)) are obtained from the first equation
(6.8). These functions depend continuously on the function h 0 ∈ H 0 . Indeed, let the sequence of functions (xi0 (x 1 , µ), yi0 (x 1 , µ)), (i = 1, 2, 3, . . .) correspond to the sequence
of functions h i0 ∈ H 0 . Let also h i0 → h 0∗ ∈ H 0 , i.e., kh i0 − h 0∗ k D0 → 0. In this case
the sequence (xi0 (x 1 , µ), yi0 (x 1 , µ)) converges to the function (x∗0 (x 1 , µ), y∗0 (x 1 , µ)),
which corresponds to the function h 0∗ ∈ H 0 , i.e., k(xi0 (x 1 , µ), yi0 (x 1 , µ)) − (x∗0 (x 1 , µ),
y∗0 (x 1 , µ))k D1 → 0.
This fact, along with the formulas (7.8), (5.1), and (4.5), implies the following statement.
0
h
maps any function η0 from N 0 to
Lemma 7.2. For any h 0 ∈ H 0 , the operator Sgl
η1 ∈ N 1 . This operator depends continuously on h 0 ∈ H 0 . Namely, if the sequence of
functions h i0 ∈ H 0 converges to h 0∗ ∈ H 0 as i → ∞, i.e., kh i0 − h 0∗ k D0 → 0 then, for any
564
M. V. Shashkov and D. V. Turaev
h0
h0
h0
h0
η0 ∈ N 0 , the sequence Sgli (η0 ) converges to Sgl∗ (η0 ), i.e., kSgli (η0 ) − Sgl∗ (η0 )k D1 → 0.
0
h
satisfies the property of the limited expansion, i.e., for any
Moreover, the operator Sgl
0
0
0
0
0
η1 ∈ N , η2 ∈ N , and h ∈ H 0 , the following estimate holds:
° 0¡ ¢
°
°
¡ 0 ¢°
°
° h 0
h0
(7.9)
η2 ° 1 ≤ Q °η10 − η20 ° D0 ,
°Sgl η1 − Sgl
D
where Q is some constant.
Let us now explore the properties of the operator Ploc .
Lemma 7.3. The operator Ploc maps a smooth function h 1 ∈ H 1 to a smooth function
h0 ∈ H 0.
To prove Lemma 7.3, let us assume first that (x 0 , y 0 ) does not belong to the stable
manifold, i.e., τ 6= ∞. In this case, if h 1 ∈ H 1 is a smooth function, by (6.12), (6.16),
the following relation takes place:
¢ µ
¡
¶−1
∂ x 1 (x 0 , y 0 , µ), µ
∂(F, µ) ∂h 1
∂(F, µ)
¡
¢
=
E
−
,
(7.10)
∂z 1 ∂(x 1 , µ)
∂(x 0 , y 0 , µ)
∂ x 0, y0, µ
where E is the identity matrix. By (5.6) and (6.5), we have
°
° °
°
° ∂(F, µ) ° ° ∂h 1 °
°
° °
°
° ∂z 1 ° ° ∂(x 1 , µ) ° 1 < 1,
Äloc
D
and, therefore,
(7.11)
¶
∂(F, µ) ∂h 1
(7.12)
∂z 1 ∂(x 1 , µ)
is an invertible operator. Hence, formula (7.10) makes sense. By the relations (6.12),
(6.16), and (7.10),
µ
E−
∂G
∂h 0
¢ =
0
0
0
∂(x , y 0 , µ)
∂ x , y ,µ
¡
+
∂G ∂h 1
∂z 1 ∂(x 1 , µ)
µ
E−
∂(F, µ) ∂h 1
∂z 1 ∂(x 1 , µ)
¶−1
∂(F, µ)
. (7.13)
∂(x 0 , y 0 , µ)
Formula (7.13) allows us to extend the domain of definition up to the points that belong
to the stable manifold, i.e., for τ = ∞. In order to show this, we consider the following
function:

∂G
∂G


+ 1 η1

0 , y 0 , µ)

∂(x
∂z


 µ
¶
0
0 1
1
∂(F, µ) 1 −1 ∂(F, µ)
F(x , y , z , η , µ) ≡

× E−
η
,
if τ 6= ∞;


∂z 1
∂(x 0 , y 0 , µ)




0,
if τ = ∞.
(7.14)
An Existence Theorem of Smooth Nonlocal Center Manifolds
565
The continuity of the function F, at the points provided that τ 6= ∞, follows from the
fact that F is an algebraic combination of the continuous functions. The continuity, at
the points provided that τ = ∞, follows from the fact that kFk → 0 if τ → ∞ (see
(5.6), (7.6), and (7.14)). So, the superposition
¶
µ
∂h 1
0
0
1 1
,µ ,
(7.15)
F x , y , h (x , µ),
∂(x 1 , µ)
where x 1 = x 1 (x 0 , y 0 , µ) (see (6.12), (6.16)), depends continuously on (x 0 , y 0 , µ)
because this function is a superposition of continuous functions. Therefore, the function
0
(7.15) is an extension of the function ∂(x 0∂h,y 0 ,µ) up to the points that belong to the stable
manifold, i.e., provided that τ = ∞. This fact, along with the continuity of the function
h 0 (x 0 , y 0 , µ), implies the smoothness of h 0 (x 0 , y 0 , µ). Thus, we have the equality
µ
¶
∂h 1
∂h 0
0
0
1 1
,
y
,
h
(x
,
µ),
=
F
x
,
µ
,
where x 1 = x 1 (x 0 , y 0 , µ),
∂(x 0 , y 0 , µ)
∂(x 1 , µ)
(7.16)
for all points (x 0 , y 0 , µ) ∈ D 0 . Lemma 7.3 is proved.
0
Observe that ∂(x 0∂h,y 0 ,µ) is a uniformly continuous function because it is continuous
and equals zero if k(x − x + , y 0 − y + )k ≥ ρ (see (5.2), (5.4), and (6.12)). By Lemmas 7.1
and 7.3, the following statement holds.
Lemma 7.4. The operator P = Ploc ◦ Pgl maps any smooth function h 0 ∈ H 0 to a
smooth function h̄ 0 ∈ H 0 .
1
h
. For any h 1 ∈ H 1 , we define the
The relation (7.16) induces a family of operators Sloc
h1
1
0
operator Sloc (η ) 7→ η by the formulas
¡
¢
where x 1 = x 1 (x 0 , y 0 , µ).
η0 (x 0 , y 0 , µ) = F x 0 , y 0 , h 1 (x 1 , µ), η1 (x 1 , µ), µ ,
(7.17)
Here we come to the following lemma.
1
h
maps any function η1 ∈ N 1 to η0 ∈ N 0 .
Lemma 7.5. For any h 1 ∈ H 1 , the operator Sloc
1
1
This operator depends continuously on h ∈ H . Namely, if the sequence of functions
h i1 ∈ H 1 (i → ∞) converges to h 1∗ ∈ H 1 , i.e., kh i1 − h 1∗ k D1 → 0, then for any η1 ∈ N 1 ,
h1
h1
h1
h1
the sequence Sloci (η1 ) converges to Sloc∗ (η1 ), i.e., kSloci (η1 )− Sloc∗ (η1 )k D0 → 0. Moreover,
h1
satisfies the property of the strong contraction, i.e., for any η11 ∈ N 1 ,
the operator Sloc
1
1
1
η2 ∈ N , and h ∈ H 1 , the following estimate is valid:
° 1 ¡ ¢
°
°
¡ 1 ¢°
°
° h
h1
(7.18)
η2 ° 0 ≤ q °η11 − η21 ° D1 ,
°Sloc η11 − Sloc
D
where the constant q can be made arbitrarily small.
The function η0 (x 0 , y 0 , µ) (see (7.17)) is a continuous function since it is a superposition
of continuous functions. It follows from the relations (7.14), (5.2), (5.4), (5.6) that
566
M. V. Shashkov and D. V. Turaev
the norm kFk D0 is bounded (it tends to zero as τ ∗ (δ) → ∞). Hence, the function
η0 (x 0 , y 0 , µ) is bounded by the constant ` (see (7.4)) and, therefore, η0 (x 0 , y 0 , µ) ∈
h1
with respect to h 1 follows from the following
N 0 . The continuity of the operator Sloc
considerations. Assume that the sequence of functions h i1 ∈ H 1 converges to h 1∗ ∈ H 1 if
i → ∞, i.e., kh i1 −h 1∗ k D1 → 0. In this case there exists the sequence xi1 = xi1 (x 0 , y 0 , µ),
which corresponds to the sequence h i1 ∈ H 1 (see (6.16)). By virtue of the relation (6.24),
the sequence xi1 = xi1 (x 0 , y 0 , µ) uniformly converges to the function x∗1 = x∗1 (x 0 , y 0 , µ),
which corresponds to h 1∗ . Indeed,
°
° 1 0 0
°x (x , y , µ) − x 1 (x 0 , y 0 , µ)° 0
∗
i
Ã
°
°
°∂F °
°
≤ 1−°
° ∂z 1 °
Äloc
D
!−1 °
°
°∂F ° ° 1
°
°
° °h − h 1 ° 1 → 0 .
L
i
∗ D
° ∂z 1 °
Äloc
(7.19)
The functions h 1∗ and η1 are uniformly continuous. Therefore, the sequence
h i1 (xi1 (x 0 , y 0 , µ)) converges to h 1∗ (x∗1 (x 0 , y 0 , µ)) and η1 (xi1 (x 0 , y 0 , µ)) converges to
η1 (x∗1 (x 0 , y 0 , µ)), i.e.,
°
° 1 1 0 0
°h (x (x , y , µ)) − h 1 (x 1 (x 0 , y 0 , µ))° 0 → 0,
i
i
∗ ∗
D
°
° 1 1 0 0
°η (x (x , y , µ)) − η1 (x 1 (x 0 , y 0 , µ))°
i
∗
D0
→ 0.
Since F is a uniformly continuous function, we obtain that
°
° 0 0 0
°η (x , y , µ) − η0 (x 0 , y 0 , µ)° 0
i
∗
° ¡ 0 0 1 1 0 0D
¢
= °F x , y , h i (xi (x , y , µ), µ), η1 (xi1 (x 0 , y 0 , µ), µ), µ
¢°
¡
− F x 0 , y 0 , h 1∗ (x∗1 (x 0 , y 0 , µ), µ), η1 (x∗1 (x 0 , y 0 , µ), µ), µ ° D0 → 0. (7.20)
1
h
follows from the relations
The property of the strong contraction for the operator Sloc
(7.17), (7.14), (7.6), and (5.6). Indeed,
°
° 0 0 0
°η (x , y , µ) − η0 (x 0 , y 0 , µ)° 0
1
2
D
°
Ã
µ
¶−1
° ∂G
∂(F, µ) 1 1
°
= ° 1 η11 (x 1 , µ) E −
η
(x
,
µ)
1
° ∂z
∂z 1
°
µ
¶−1 !
∂(F, µ) 1 1
∂(F, µ) °
°
1 1
− η2 (x , µ) E −
η2 (x , µ)
°
∂z 1
∂(x 0 , y 0 , µ) ° 0
D
°
°
≤ q °η11 (x 1 , µ) − η21 (x 1 , µ)° D1 ,
(7.21)
where the constant q may be made arbitrarily small. Lemma 7.5 is proved.
By virtue of Lemmas 7.2 and 7.5, the following statement takes place.
0
1
0
h
h
◦ Sgl
, where h 1 =
Lemma 7.6. For any h 0 ∈ H 0 , the superposition operator S h ≡ Sloc
¡ 0¢
Pgl h , maps any function η0 from N 0 to η0 ∈ N 0 . This operator depends continuously
An Existence Theorem of Smooth Nonlocal Center Manifolds
567
0
on the function h 0 ∈ H 0 . Moreover, the operator S h satisfies the contraction property,
i.e., for any η10 ∈ N 0 , η20 ∈ N 0 , and h 0 ∈ H 0 , the following estimate holds:
° 0¡ ¢
°
°
¢°
0 ¡
°
° h 0
(7.22)
°S η1 − S h η20 ° 0 ≤ p °η10 − η20 ° D0 ,
D
where the constant p = q Q is strictly less than 1.
Below we use the following statement, which is a parametric variant of Banach’s
contraction principle.
Lemma 7.7. (“Fiber contraction theorem” [Hirsch & Pugh, 1970]) Let H 0 and N 0
be metric spaces. We assume also that N 0 is a complete space. Let P be an operator in
H 0 , i.e., P : H 0 → H 0 , and let any sequence h 00 , h 01 , h 02 , . . . obtained by the iterations
0
= P(h i0 ), (i = 0, 1, 2, . . .), converge to a unique fixed point h 0∗ = P(h 0∗ ) ∈ H
h i+1
for any initial element h 00 ∈ H 0 . For any element h 0 ∈ H 0 , let there be an operator
0
0
S h : N 0 → N 0 . We assume that the family of the operators S h satisfies the following
conditions.
0
1. For any h 0 ∈ H 0 , the operator S h satisfies the contraction property, i.e.,
´
³ 0
¡
¢
0
dist S h (η10 ), S h (η20 ) ≤ p dist η10 , η20 ,
where η10 ∈ N 0 , η20 ∈ N 0 , and the constant p is less than 1.
0
2. The family of the operators S h depends continuously on h 0 ∈ H 0 , i.e., if a sequence
0
0
h i0 ∈ H 0 (i → ∞) tends to h 0∗ ∈ H 0 , then the sequence S h i (η0 ) tends to S h ∗ (η0 ) for
any η0 ∈ N 0 .
Then, the operator Ä : H 0 × N 0 → H 0 × N 0 defined by the formula Ä(h 0 , η0 ) ≡
0
(P(h 0 ), S h (η0 )) has a unique fixed point, (h 0∗ , η∗0 ) = Ä(h 0∗ , η∗0 ). Moreover, any sequence
0
0
0
0
, ηi+1
) = Ä(h i0 , ηi0 ), (i =
(h 0 , η0 ), (h 01 , η10 ), (h 02 , η20 ), . . ., obtained by the iterations (h i+1
0
0
0, 1, 2, . . .) with any initial element (h 0 , η0 ), converges to the point (h 0∗ , η∗0 ).
By the conditions of the lemma, the operator P has a unique fixed point, i.e., h 0∗ =
0
P(h 0∗ ) ∈ H 0 . Since N 0 is a complete space and S h ∗ is a contractive operator, there exists
0
a unique fixed point η∗0 = S h ∗ (η∗0 ) ∈ N 0 . The point (h 0∗ , η∗0 ) ∈ H 0 × N 0 is a fixed point
0
of the operator Ä because (h 0∗ , η∗0 ) = Ä(h 0∗ , η∗0 ) ≡ (P(h 0∗ ), S h ∗ (η∗0 )). The uniqueness
of the fixed point for the operator Ä follows from the uniqueness of the fixed point for
the operator P. Let us show that any sequence (h 00 , η00 ), (h 01 , η10 ), (h 02 , η20 ), . . ., obtained
0
0
, ηi+1
) = Ä(h i0 , ηi0 ), (i = 0, 1, 2, . . .) with any initial element
by the iterations (h i+1
0
0
(h 0 , η0 ), converges to the point (h 0∗ , η∗0 ). It is clear that
´
³
¢
¡
0
0
= dist η∗0 , S h i (ηi0 )
dist η∗0 , ηi+1
´
³ 0
´
³
0
0
≤ dist η∗0 , S h i (η∗0 ) + dist S h i (η∗0 ), S h i (ηi0 )
´
³
¡
¢
0
≤ dist η∗0 , S h i (η∗0 ) + p dist η∗0 , ηi0 .
(7.23)
568
M. V. Shashkov and D. V. Turaev
0
By the conditions of the lemma, the family of the operators S h i depends continuously
on h i0 and h i0 → h 0∗ if i → ∞. Therefore,
´
³
´
³
0
0
at i → ∞,
dist η∗0 , S h i (η∗0 ) → dist η∗0 , S h ∗ (η∗0 ) ≡ 0
0
i.e., for any ε > 0, there exists i 0 such that, for any i > i 0 , dist(η∗0 , S h i (η∗0 )) < ε. Hence,
for i = i 0 + j, we have the inequality
¢
¡
¢
¡
¢
¡
dist η∗0 , ηi0 ≡ dist η∗0 , ηi00 + j+1 ≤ ε + p dist η∗0 , ηi00 + j
¡
¢
≤ ε + p ε + · · · + p j−1 ε + p j dist η∗0 , ηi00
¡
¢
ε
(7.24)
+ p j dist η∗0 , ηi00 .
≤
1− p
Observe that there exists j0 such that p j dist(η∗0 , ηi00 ) ≤ ε for any j > j0 . Therefore, the
inequality dist(η∗0 , ηi0 ) ≤ ε(1/(1 − p) + 1) holds true for any i > i 0 + j0 , i.e., ηi0 → η∗0 .
Lemma 7.7 is proved.
By Lemmas 6.3, 7.6, and 7.7, any sequence (h 00 , η00 ), (h 01 , η10 ), (h 02 , η20 ), . . ., obtained
0
0
0
, ηi+1
) = (P(h i0 ), S h i (ηi0 )), (i = 0, 1, 2, . . .), converges to the
by the iterations (h i+1
fixed point (h 0∗ , η∗0 ). Let us choose the point (0, 0) as the initial point (h 00 , η00 ). By
Lemma 7.4 and the formulas (7.16), (7.17), the relation
ηi0 =
∂h i0
∂(x 0 , y 0 , µ)
(7.25)
holds true for any i = 0, 1, 2 . . .. Since h i0 uniformly converges to h 0∗ and ηi0 uniformly
converges to η∗0 , the equality
η∗0 =
∂h 0∗
y 0 , µ)
∂(x 0 ,
(7.26)
also holds true, i.e., h 0∗ is a C 1 -smooth function with respect to all of its variables. Notice
that, using the results of Section 3, by induction it is possible to prove that h 0∗ ∈ C k+ε if
X µ ∈ C k+ε and Re γ2 /γ1 > k + ε.
To finish the proof of the main Theorem 2.1 we must show that orbits passing through
the invariant curve h 0∗ (h 1∗ ) form a smooth manifold Mcs . A proof of this fact is in the
following section.
8. Smoothness of the Invariant Manifold Mcs
Here we show that orbits passing through the points of the graphs of functions h 0∗ and
h 1∗ form a smooth invariant manifold Mcs .
Note that the flight time of the orbits is bounded outside any neighborhood of the
equilibrium point O. Therefore, outside the neighbourhood, the smoothness of the manifold follows from the smoothness of the function h 1∗ . The following theorem guarantees
the smoothness of the manifold near the equilibrium point O = (x = 0, y = 0, z = 0).
An Existence Theorem of Smooth Nonlocal Center Manifolds
569
Fig. 12. The functions h 11 and h 12 uniquely determine the
invariant manifolds W1s+ and W2s+ , correspondingly. The
manifolds have a common tangent everywhere on W s .
Theorem 8.1. Let z 1 = h 1 (x 1 , µ) (h 1 ∈ H 1 ) be a smooth function. Then orbits passing
through the points of the graph h 1 form an invariant manifold. This manifold is locally
a graph of some smooth function z 0 = W s+ (x 0 , y 0 , µ). Two different functions h 11 and
h 12 uniquely determine two different manifolds W1s+ and W2s+ , such that the manifolds
have a common tangent space everywhere on the stable manifold (see Figure 12).
In order to find the function z 0 = W s+ (x 0 , y 0 , µ), we shall use the boundary value
problem (3.61), (3.64). Namely, let us consider an orbit (x(t), y(t), z(t)) that starts, at
t = 0, with a point (x 0 , y 0 , z 0 ) on the manifold W s+ and reaches, at t = τ , a point
(x 1 , y 1 , z 1 ) on the cross section S 1 . Since the coordinate x 1 on the cross section S 1
corresponds to the coordinate x 1 − Āz 1 of the initial system (see (4.21)) and (x 1 , y 1 , z 1 )
belongs to the graph h 1 , we have the following equalities:
¢
¡
z 1 = h 1 x 1 − Āz 1 , µ .
(8.1)
y1 = y− ,
According to the boundary value problem (3.61), (3.64), we have the following relations:
x 1 = x(τ ; x 0 , y 0 , z 1 , τ, µ),
y 1 = y(τ ; x 0 , y 0 , z 1 , τ, µ),
(8.2)
z 0 = z(0; x 0 , y 0 , z 1 , τ, µ).
Since y 1 = y − (see (8.1)), by (4.17), the flight time τ may be expressed from the second
equation of the system (8.2) as a function τ = τ (x 0 , y 0 , z 1 , µ). Thus we have
x 1 = x 1 (x 0 , y 0 , z 1 , µ) ≡ x(τ (x 0 , y 0 , z 1 , µ); x 0 , y 0 , z 1 , τ (x 0 , y 0 , z 1 , µ), µ),
z 0 = z 0 (x 0 , y 0 , z 1 , µ) ≡ z(0; x 0 , y 0 , z 1 , τ (x 0 , y 0 , z 1 , µ), µ).
(8.3)
570
M. V. Shashkov and D. V. Turaev
The derivatives of these functions may be expressed by (4.12) and they satisfy the
estimates (4.19), (4.20). Now, by (8.1) and (8.3), we have
¢
¡
(8.4)
z 1 = h 1 x 1 (x 0 , y 0 , z 1 , µ) − Āz 1 , µ .
Using formulas (4.19), (4.20), and (6.5), we obtain the following estimate:
°
µ 1
¶°
°
°
∂x
∂h 1
°
°
¢
− Ā ° ≤ q < 1.
° ¡ 1
1
°
° ∂ x − Āz 1
∂z
(8.5)
By the implicit function theorem, the last inequality means that z 1 can be expressed from
the equation (8.4) as a function
z 1 = z 1 (x 0 , y 0 , µ).
(8.6)
Note that, by (8.4), (4.19), (4.20), and (6.5), we have the following relation:
Ã
µ 1
¶!−1
∂h 1
∂x
∂x1
∂h 1
∂z 1
¡
¡
¢
¢
=
E
−
−
Ā
∂(x 0 , y 0 )
∂z 1
∂ x 1 − Āz 1
∂ x 1 − Āz 1 ∂(x 0 , y 0 )
= O(eατ ).
(8.7)
Substituting the function (8.6) in the second equation of (8.3), we obtain that the invariant
manifold W s+ is a graph of some function
z 0 = z 0 (x 0 , y 0 , z 1 (x 0 , y 0 , µ), µ) ≡ W s+ (x 0 , y 0 , µ).
(8.8)
The derivative of this function
∂z 0
∂z 0
∂z 1
∂ W s+ (x 0 , y 0 , µ)
=
+
(8.9)
∂(x 0 , y 0 )
∂(x 0 , y 0 ) ∂z 1 ∂(x 0 , y 0 )
may be calculated via formulas (4.12) and (8.7). By (4.19), (4.20), and (8.7), we have
∂ W s+ (x 0 , y 0 , µ)
→ B̄,
∂(x 0 , y 0 )
if
τ → ∞,
(8.10)
¢
¡
i.e., the derivative has the fined limit if the initial point x 0 , y 0 , z 0 = W s+ (x 0 , y 0 , µ)
goes to the stable manifold W s . Moreover, the limit does not depend on the function h 1 .
Thus, all of the manifolds W s+ have the same tangent space everywhere on W s . Notice
that the manifold has zero tangent space {z = 0} at the point O, since the stable manifold
W s tangents to the x-axis and orbits, passing through the points S 1 ∩ W u tangent to the
y-axis. Theorem 8.1 is proved.
That also concludes the proof of the main Theorem 2.1.
9. An Example
The following three-dimensional system of differential equations gives an example of a
C 1 -smooth vector field having a global center invariant manifold:
ẋ = y,
ẏ = −x + µ1 y + µ2 x y + µ3 y 2 + µ4 x 2 + µ5 z/ln(z 2 ),
ż = µ6 z + µ7 y/ln(y 2 ).
(9.1)
An Existence Theorem of Smooth Nonlocal Center Manifolds
571
Fig. 13. The phase portrait of system (9.3) if µ1 = µ2 = 0.
Here we assume that


s
µ
¶
1
µ2
µ2 2
µ1 +
−
+ 8  , µ3 < 0, µ4 > 0, µ3 + µ4 > 0. (9.2)
µ1 +
µ6 <
2
µ4
µ4
Note that the right-hand side of (9.1) contains C 1 -smooth functions y/ln(y 2 ), and
z/ln(z 2 ), which are not C 1+ε -smooth.
The system has two equilibria O1 and O2 with coordinates (x1 , y1 , z 1 ) = (0, 0, 0)
and (x 2 , y2 , z 2 ) = (1/µ4 , 0, 0), correspondingly. If µ7 = 0, the system (9.1) has a stable
global center manifold {z = 0}. Therefore, the vector field takes the following form on
this two-dimensional manifold:
ẋ = y,
ẏ = −x + µ1 y + µ2 x y + µ3 y 2 + µ4 x 2 .
(9.3)
Equations (9.3) were studied by Bautin (see Bautin and Leontovich [1976]). If µ1 =
µ2 = 0, the system is conservative (see Figure 13). Moreover, by (9.2), separatrices of
the saddle O2 = (1/µ4 , 0) form a homoclinic orbit. In this case the integral of the system
(9.3) takes the form
¶
µ
µ4 − µ3
µ4 − µ3
µ4 2
x + y2 +
x
+
e−2µ3 x = h.
H (x, y) =
µ3
µ23
2µ33
Therefore, homoclinic orbits to the saddle O2 satisfy the following relation:
H (x, y) = H (1/µ4 , 0).
Thus, if (µ1 , µ2 , µ7 ) = (0, 0, 0), the initial system (9.1) has a homoclinic orbit to O2 .
Moreover, by (9.2), this system satisfies Theorem 2.2. Therefore, by the theorem, we
572
M. V. Shashkov and D. V. Turaev
have that for any µ3 , µ4 , µ5 , µ6 , and small µ1 , µ2 , µ7 , the system has a smooth center
manifold Mcu that depends smoothly on the parameters.
Acknowledgments
The authors are grateful to L. P. Shilnikov, V. N. Belykh, L. M. Lerman, S. V. Gonchenko,
O. V. Stenkin, B. Fiedler, and A. J. Homburg for useful discussions. This research was
supported by the EC-Russia Collaborative Project ESPRIT P 9282–ACTCS, by the ISF
grant R98300, and the grant of INTAS-93-0570-ext. M. Shashkov was also supported
by DAAD (1995) to work at Institut für Mathematik I, Freie Universität, Berlin.
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