FINITE-TIME ATTRACTIVITY AND BIFURCATION
FOR NONAUTONOMOUS DIFFERENTIAL EQUATIONS
MARTIN RASMUSSEN
DEPARTMENT OF MATHEMATICS
IMPERIAL COLLEGE
LONDON SW7 2AZ
UNITED KINGDOM
E-MAIL: [email protected]
Abstract. The aim of this article is to introduce nonautonomous and finitetime versions of central concepts from the theory of dynamical systems such as
attractivity and bifurcation. The discussion includes an appropriate spectral
theory for linear systems as well as finite-time analogues of the well-known
transcritical and pitchfork bifurcation.
This work is dedicated to the memory of my supervisor Professor Bernd Aulbach
I am grateful for his consequent support while writing my diploma thesis and
dissertation in Augsburg. He was able both to explain the beautiful structures of
dynamical systems very clearly and to encourage his students by having a positive
attitude towards them. As a scientist, he strongly influenced with various ideas the
research on the qualitative theory of nonautonomous dynamical systems.
1. Introduction
Both attractivity and bifurcation are classical fields in the theory of dynamical
systems. The first concepts of attractivity have been introduced by the Russian
mathematician and engineer Aleksander M. Lyapunov in his thesis [22], whereas the
fundamental ideas and elements in bifurcation theory go back to both the French
mathematician Henri Poincaré [25] and Lyapunov [23]. The fields of attractivity
and bifurcation are strongly related, since a bifurcation is often associated with a
loss or gain of attractivity.
The notions of attractivity which have been discussed in the last one hundred
years are primarily based on asymptotic convergence of solutions by letting some
time period of the system tend to infinity. From the viewpoint of applications,
however, one is interested in the behavior of the system on finite time intervals.
This is not a conflict in a purely autonomous context, since the underlying rules of
a dynamical system do not change in time, and thus, the behavior of the system in
the future (or the past) is useful to describe properties of the system within a finite
time range. On the other hand, nonautonomous systems have gained renewed
Date: November 1, 2008.
2000 Mathematics Subject Classification. 34A30, 34D09, 34D45, 37B55, 37G34.
Key words and phrases. Attractor, dichotomy, dichotomy spectrum, finite-time dynamics,
nonautonomous differential equation, nonautonomous bifurcation, pitchfork bifurcation, repeller,
transcritical bifurcation.
Research supported by a Marie Curie Intra European Fellowship of the European Community
(Grant Agree Number: 220638).
1
2
MARTIN RASMUSSEN
and growing interest in the last years (see, e.g., the conference proceedings [9]),
which in turn implies a strong need to understand the finite-time dynamics of
nonautonomous dynamical systems.
The purpose of this article is to provide suitable and efficient concepts of finitetime attractivity and bifurcation in the context of nonautonomous differential equations. These concepts are perhaps not the final answers to the questions which arise
from the applications, but we hope that they will stimulate further discussion within
this field. We will also treat basic properties of the introduced notions, e.g., by providing a spectral theory for linear systems on finite-time intervals and by discussing
basic one-dimensional bifurcation patters.
We like to emphasize that the study of finite-time behavior in Lagrangian coherent structures has become a very active field of research in the last ten years
(to mention only a few references, we refer to [6, 14, 21, 33], and see the references
therein). We also like to add that the development of nonautonomous bifurcation
theory (without a finite-time viewpoint) has been fast-paced in the last ten years
(see [11, 12, 15, 16, 17, 18, 19, 20, 24, 26, 27, 29, 28, 30])
This paper is organized as follows. In the ensuing section, some basic definitions
are given. Section 3 is devoted to the introduction of the notions of finite-time
attractivity and bifurcation. Several examples illustrate the concepts. In Section 4,
a notion of spectrum is introduced, and we prove an analogue to the Theorem
of Linearized Stability in Section 5. Finally, the last section contains finite-time
versions of the transcritical and pitchfork bifurcation.
Notation. Given a metric space (X, d), we write Uε (x0 ) = {x ∈ X : d(x, x0 ) < ε}
for the ε-neighborhood of a point x0 ∈ X. For arbitrary nonempty sets A, B ⊂ X
and x ∈ X, let d(x, A) := inf{d(x, y) : y ∈ A} be the distance of x to A and
d(A|B) := sup{d(x, B) : x ∈ A} be the Hausdorff semi-distance of A and B. In
−
addition, we set R+
κ := [κ, ∞), Rκ := (−∞, κ] for κ ∈ R and R = R ∪ {±∞}.
N ×N
We denote by R
the set of all real N × N matrices, and we use the symbol 1
for the unit matrix. The Euclidean space RN is equipped with the Euclidean norm
PN
k · k, which is induced by the scalar product h·, ·i, defined by hx, yi := i=1 xi , yi .
Throughout this article, we will use the metric d associated with this norm.
Let f : X → Y be a© function from a set X toªa set Y . Then the graph of f is
defined by graph f := (x, y) ∈ X × Y : y = f (x) .
2. Preliminaries
Throughout this paper, we consider an interval I ⊂ R and a nonautonomous
differential equation of the form
(2.1)
ẋ = f (t, x) ,
where the right hand side f : D ⊂ I × RN → RN fulfills conditions for the local
existence and uniqueness of solutions. Let ϕ stand for the general solution of
(2.1), i.e., ϕ(·, τ, ξ) is the uniquely determined non-continuable solution of (2.1)
satisfying the initial condition ϕ(τ, τ, ξ) = ξ. This means that the cocycle property
ϕ(t, τ, ξ) = ϕ(t, s, ϕ(s, τ, ξ)) holds, and for simplicity in notation, we write ϕ(t, τ )ξ
instead of ϕ(t, τ, ξ).
A subset M of the extended phase space I × RN is called nonautonomous set; we
use the term t-fiber of M for the set M (t) := {x ∈ RN : (t, x) ∈ M }, t ∈ I. We call
M closed or compact if all t-fibers are closed or compact, respectively, and we call M
FINITE-TIME ATTRACTIVITY AND BIFURCATION
3
linear if the fibers of M are linear subspaces of RN . Finally, a nonautonomous set
M is called invariant (w.r.t. the differential equation (2.1)) if ϕ(t, τ, M (τ )) = M (t)
for all t, τ ∈ I.
Furthermore, when discussing bifurcation problems, we consider nonautonomous
differential equations
(2.2)
ẋ = f (t, x, α)
depending on a parameter α, where f : D ⊂ I × RN × (α− , α+ ) → RN for some
reals α− < α+ fulfills conditions for the local existence and uniqueness of solutions.
The general solution has then an additional argument, given by the parameter α,
i.e., ϕ(·, τ, ξ, α) is the uniquely determined non-continuable solution of (2.2) for the
parameter value α satisfying the initial condition ϕ(τ, τ, ξ, α) = ξ.
3. Notions of Finite-Time Attractivity and Bifurcation
This section is devoted to the introduction of notions of finite-time attractivity
and bifurcation for ordinary differential equations. We will first explain the concepts
of a finite-time attractor and finite-time repeller.
Definition 3.1 (Finite-time attractivity). We consider the differential equation
(2.1), and let τ ∈ I and T > 0 with τ + T ∈ I.
(i) An invariant and compact nonautonomous set A of (2.1) is called (τ, T )attractor if
¯
¢
1 ¡
lim sup d ϕ(τ + T, τ )Uη (A(τ ))¯A(τ + T ) < 1
η
η &0
(see Figure 1).
(ii) A solution µ : [τ, τ + T ] → RN of (2.1) is called (τ, T )-attractive if graph µ
is a (τ, T )-attractor.
(iii) An invariant and compact nonautonomous set R of (2.1) is called (τ, T )repeller if
¯
¢
1 ¡
lim sup d ϕ(τ, τ + T )Uη (R(τ + T ))¯R(τ ) < 1
η
η &0
(see Figure 2).
(iv) A solution µ : [τ, τ + T ] → RN of (2.1) is called (τ, T )-repulsive if graph µ
is a (τ, T )-repeller.
Remark 3.2.
(i) Note that the notions of finite-time attractivity and repulsivity are not
invariant with respect to a change of the metric d (as defined in the Introduction) to an equivalent metric.
(ii) The Hausdorff semi-distance d in Definition 3.1 can equivalently be replaced
by the Hausdorff distance dH , which for nonempty sets A, B ⊂ RN is
defined by dH (A, B) := max{d(A|B), d(B|A)}.
(iii) The notions of (τ, T )-attractor and (τ, T )-repeller are dual in the sense that
they change their roles under time reversal.
The following two examples illustrate the notions of Definition 3.1. The first
example is given by the following one-dimensional linear system.
4
MARTIN RASMUSSEN
RN
©
η
< η for η → 0
}
A
τ
τ +T
R
τ +T
R
Figure 1. (τ, T )-attractor
RN
R
τ
η >{
for η → 0
ª
η
Figure 2. (τ, T )-repeller
Example 3.3. Let I := [τ, τ + T ] for some τ ∈ R and T > 0, and consider the
linear nonautonomous differential equation
(3.1)
ẋ = a(t)x
with a continuous function a : I → R. Then it is easy to see that each invariant
and compact nonautonomous set is a (τ, T )-attractor if and only if
Z τ +T
a(s) ds < 0
τ
and a (τ, T )-repeller if and only if
Z τ +T
a(s) ds > 0 .
τ
The second example is given by the following nonlinear system.
Example 3.4. Let I := [τ, τ + T ] for some τ ∈ R and T > 0, and consider the
nonautonomous differential equation
¢
¡
(3.2)
ẋ = a(t)x + b(t)x3 = x a(t) + b(t)x2
FINITE-TIME ATTRACTIVITY AND BIFURCATION
5
with continuous functions a : I → R and b : I → R+ . For simplicity, we define
s
a(t)
w(t) := −
for all t ∈ I with a(t) < 0 .
b(t)
Then, for fixed t ∈ I with a(t) < 0, the zero set of the right hand side is {0, ±w(t)};
for all t ∈ I with a(t) ≥ 0, this zero set is the singleton {0}. An elementary
discussion of the sign of the right hand side of (3.2) yields that the trivial solution
is a (τ, T )-attractor if a(t) < 0 for all t ∈ I, and it is (τ, T )-repulsive if a(t) ≥ 0 for
all t ∈ I.
The next definition helps us to get information about the range of attractivity
and repulsivity.
Definition 3.5 (Radii of attraction and repulsion). The radius of (τ, T )-attraction
of a (τ, T )-attractor A is defined by
¯
©
¡
¢
ª
(τ,T )
AA
:= sup η > 0 : d ϕ(τ + T, τ )Uη̂ (A(τ ))¯A(τ + T ) < η̂ for all η̂ ∈ (0, η) ,
and the radius of (τ, T )-repulsion of a (τ, T )-repeller R is defined by
¯
©
¡
¢
ª
(τ,T )
RR
:= sup η > 0 : d ϕ(τ, τ + T )Uη̂ (R(τ + T ))¯R(τ ) < η̂ for all η̂ ∈ (0, η) .
We consider again the situation of Example 3.3 and 3.4.
Example 3.6. In Example 3.3 and 3.4, we have obtained conditions for the attractivity and repulsivity of nonautonomous sets and solutions of (3.1) and (3.2),
respectively. The following can be said about the corresponding ranges of attraction
and repulsion:
(i) Every invariant and compact nonautonomous set M ⊂ [τ, τ + T ] × R of
(3.1) is a
R τ +T
(τ,T )
• (τ, T )-attractor with AM = ∞ if and only if τ
a(s) ds < 0 ,
R
τ +T
(τ,T )
• (τ, T )-repeller with RM = ∞ if and only if τ
a(s) ds > 0 .
(ii) The trivial solution of (3.2) is
• (τ, T )-attractive with
inf
t∈[τ,τ +T ]
(τ,T )
w(t) ≤ A0
≤
sup
w(t)
t∈[τ,τ +T ]
if a(t) < 0 for all t ∈ [τ, τ + T ] ,
(τ,T )
• (τ, T )-repulsive with R0
= ∞ if a(t) ≥ 0 for all t ∈ [τ, τ + T ] .
The following notion of a finite-time bifurcation depends on the above notions
of attractivity and repulsivity.
Definition 3.7 (Finite-time bifurcation). We consider the differential equation
(2.2), which depends on a parameter α. For a given α0 ∈ (α− , α+ ), we say that
(2.2) admits a supercritical (τ, T )-bifurcation at α0 if there exist an α̂ > α0 and a
continuous function µ : [τ, τ + T ] × (α0 , α̂) → RN such that one of the following
two statements is fulfilled:
(i) µ(·, α) is a (τ, T )-attractive solution of (2.2) for all α ∈ (α0 , α̂), and
(τ,T )
lim Aµ(·,α) = 0
α &α 0
is fulfilled.
6
MARTIN RASMUSSEN
(ii) µ(·, α) is a (τ, T )-repulsive solution of (2.2) for all α ∈ (α0 , α̂), and
(τ,T )
lim Rµ(·,α) = 0
α &α 0
holds.
Accordingly, subcritical (τ, T )-bifurcations are defined by considering the limit
α%α0 .
We obtain the following first example of a (τ, T )-bifurcation.
Example 3.8 (Nonautonomous pitchfork bifurcation). We consider the nonautonomous differential equation
¢
¡
(3.3)
ẋ = αa(t)x + b(t)x3 = x αa(t) + b(t)x2
depending on a real parameter α with continuous functions a : I := [τ, τ + T ] → R
and b : I → R+ . The equation (3.3) is a nonautonomous version of the well-known
autonomous differential equation
¡
¢
ẋ = αx + x3 = x α + x2 ,
which admits a pitchfork bifurcation (see, e.g., Guckenheimer & Holmes [13, p. 150]).
For fixed α ∈ R, (3.3) has already been discussed in Example 3.4, where we have
derived sufficient conditions concerning the attractivity and repulsivity of the trivial
solution. The following statements are direct consequences of these observations.
The above nonautonomous differential equation admits a
(i) supercritical (τ, T )-bifurcation at α = 0 if
a(t) < 0 for all t ∈ [τ, τ + T ] ,
(ii) subcritical (τ, T )-bifurcation at α = 0 if
a(t) > 0 for all t ∈ [τ, τ + T ] .
A generalization of this equation is discussed in Section 6.
4. Spectral theory
The classical concept of a spectrum, which is based on the notion of an exponential dichotomy, is given by the the so-called Sacker-Sell spectrum (see Sacker &
Sell [32], and we also refer to Aulbach & Siegmund [3, 4] and Siegmund [34]). In
this section, we introduce an appropriate notion of dichotomy which is adapted to
the notions of attractivity and repulsivity from the previous section, and we show
that this concept leads to a finite-time spectrum. The main result of this section is
the Spectral Theorem which says that the spectrum is given by the union of finitely
many compact intervals, whose number is bounded by the dimension of the system.
Note also that recently, a different concept of finite-time spectrum has been
introduced in Berger & Doan & Siegmund [5].
Let I := [τ, τ + T ] be a compact interval for some τ ∈ R and T > 0, and consider
the linear nonautonomous differential equation
(4.1)
ẋ = A(t)x ,
where A : I → RN ×N is a continuous function.
Given linear and invariant nonautonomous sets M1 and M2 of (4.1), the sets
©
ª
M1 ∩ M2 := (t, ξ) ∈ I × RN : ξ ∈ M1 (t) ∩ M2 (t)
and
FINITE-TIME ATTRACTIVITY AND BIFURCATION
7
©
ª
M1 + M2 := (t, ξ) ∈ I × RN : ξ ∈ M1 (t) + M2 (t)
are also linear and invariant nonautonomous sets. A finite sum M1 + · · · + Mn of
linear and invariant nonautonomous sets is called Whitney sum M1 ⊕ · · · ⊕ Mn if
the relation Mi ∩ Mj = I × {0} is satisfied for i 6= j.
Linear and invariant nonautonomous sets can be described via invariant projectors.
Definition 4.1 (Invariant projector). An invariant projector of (4.1) is a function
P : I → RN ×N with
P (t) = P (t)2
for all t ∈ I ,
P (t)Φ(t, s) = Φ(t, s)P (s) for all t, s ∈ I .
The range
©
ª
R(P ) := (t, ξ) ∈ I × RN : ξ ∈ R(P (t))
and the null space
©
ª
N (P ) := (t, ξ) ∈ I × RN : ξ ∈ N (P (t))
of an invariant projector P are linear and invariant nonautonomous sets of (4.1)
such that R(P )⊕N (P ) = I×RN . Since the fibres of R(P ) have the same dimension,
we define the rank of P by
rk P := dim R(P ) := dim R(P (t)) for all t ∈ I ,
and we set
dim N (P ) := dim N (P (t)) for all t ∈ I .
Next, a notion of dichotomy is introduced for the linear system.
Definition 4.2 ((τ, T )-dichotomy). Let P : I → RN ×N be an invariant projector
of (4.1). We say that (4.1) admits a (τ, T )-dichotomy with projector P if we have
kΦ(τ + T, τ )ξk < kξk for all 0 6= ξ ∈ R(P (τ )) ,
kΦ(τ, τ + T )ξk < kξk for all 0 =
6 ξ ∈ N (P (τ + T )) .
Remark 4.3. In the scalar case N = 1, the ODE (4.1) admits a (τ, T )-dichotomy if
and only if |Φ(τ + T, τ )| 6= 1.
Definition 4.4 (Nonhyperbolic dichotomies). We consider for a real value of γ the
linear nonautonomous differential equation
(4.2)
ẋ = (A(t) − γ 1)x .
We say that (4.1) admits a nonhyperbolic (τ, T )-dichotomy with growth rate γ and
projector P if (4.2) admits a (τ, T )-dichotomy with projector P .
Remark 4.5. The linear system (4.1) admits a nonhyperbolic (τ, T )-dichotomy with
growth rate γ = 0 if and only if it admits a (τ, T )-dichotomy.
For future reference, we need the following simple criteria for nonhyperbolic
dichotomies.
Lemma 4.6 (Criteria for nonhyperbolic dichotomies). Suppose that (4.1) admits
a nonhyperbolic (τ, T )-dichotomy with growth rate γ and projector Pγ . Then the
following statements are fulfilled:
(i) If Pγ ≡ 1, then (4.1) admits a nonhyperbolic (τ, T )-dichotomy with growth
rate ζ and projector Pζ ≡ 1 for all ζ > γ.
8
MARTIN RASMUSSEN
(ii) If Pγ ≡ 0, then (4.1) admits a nonhyperbolic (τ, T )-dichotomy with growth
rate ζ and projector Pζ ≡ 0 for all ζ < γ.
Proposition 4.7 (Equivalent characterizations of nonhyperbolic dichotomies). Let
P : I → RN ×N be an invariant projector of (4.1). Then (4.1) admits a nonhyperbolic (τ, T )-dichotomy with growth rate γ ∈ R and projector P if and only if we
have
kΦ(τ + T, τ )ξk < eγT kξk
kΦ(τ, τ + T )ξk < e
−γT
kξk
for all 0 6= ξ ∈ R(P (τ )) ,
for all 0 6= ξ ∈ N (P (τ + T )) .
A projector of a nonhyperbolic (τ, T )-dichotomy is not uniquely determined; this
assertion basically follows from a continuation argument. The following proposition,
however, shows that the ranks cannot be different.
Proposition 4.8. Suppose that both P and P̂ are invariant projectors of a nonhyperbolic (τ, T )-dichotomy with growth rate γ. Then rk P = rk P̂ is fulfilled.
Proof. Arguing negatively, we suppose that (4.1) admits a (τ, T )-dichotomy with
two invariant projectors P and P̂ such that rk P < rk P̂ . Thus,
¡
¡ ¢¢
¡ ¢
¡
¡ ¢¢
dim N (P ) ∩ R P̂ = dim N (P ) + dim R P̂ − dim N (P ) + R P̂
¡
¢
> dim N (P ) + dim R(P ) − dim N (P ) + R(P̂ ) ≥ 0 .
¡
¢
Hence, there exists a nonzero element ξ ∈ N (P (τ )) ∩ R P̂ (τ ) . We obtain
°
°
kΦ(τ + T, τ )ξk < kξk = °Φ(τ, τ + T )Φ(τ + T, τ )ξ ° < kΦ(τ + T, τ )ξk ,
¡
¢
since 0 6= ξ ∈ R P̂ (τ ) and 0 6= Φ(τ + T, τ )ξ ∈ N (P (τ + T )). This contradiction
finishes the proof of this proposition.
¤
The following theorem says that the notions of dichotomy are consistent to the
concepts of attractivity and repulsivity.
Theorem 4.9 (Nonhyperbolic dichotomies and the notions of attractivity and repulsivity). Suppose that (4.1) admits a nonhyperbolic (τ, T )-dichotomy with growth
rate γ and invariant projector P . Then the following statements are fulfilled:
(i) If γ ≤ 0 and rk P ≥ 1, then every solution of (4.1) is not (τ, T )-repulsive.
(ii) If γ ≥ 0 and rk P ≤ N − 1, then every solution of (4.1) is not (τ, T )attractive.
(iii) If γ ≤ 0 and P ≡ 1, then every solution of (4.1) is (τ, T )-attractive with
A0 = ∞.
(iv) If γ ≥ 0 and P ≡ 0, then every solution of (4.1) is (τ, T )-repulsive with
R0 = ∞.
The central definition of this section is as follows.
Definition 4.10. The (τ, T )-dichotomy spectrum of (4.1) is defined by
©
Σ := γ ∈ R : (4.1) does not admit a nonhyperbolic
ª
(τ, T )-dichotomy with growth rate γ .
Moreover, the resolvent set of the spectrum is defined by ρ := R \ Σ.
FINITE-TIME ATTRACTIVITY AND BIFURCATION
9
Lemma 4.11. The resolvent set ρ is open, more precisely, for all γ ∈ ρ, there exists
an ε > 0 such that Uε (γ) ⊂ ρ. Furthermore, the relation rk Pζ = rk Pγ is fulfilled
for all ζ ∈ Uε (γ) and every invariant projector Pγ and Pζ of the nonhyperbolic
dichotomies of (4.1) with growth rates γ and ζ, respectively.
Proof. For γ ∈ ρ, there exists an invariant projector Pγ such that
kΦ(τ + T, τ )ξk < eγT kξk
kΦ(τ, τ + T )ξk < e
We define
½
β := max
max
06=ξ∈R(Pγ (τ ))
−γT
kξk
for all 0 6= ξ ∈ R(Pγ (τ )) ,
for all 0 6= ξ ∈ N (Pγ (τ + T )) .
kΦ(τ + T, τ )ξk
kΦ(τ, τ + T )ξk
,
max
γT
e kξk
e−γT kξk
06=ξ∈N (Pγ (τ +T ))
¾
<1
and set ε := ln β/(2T ). Thus, for all ζ ∈ Uε (γ), we have
kΦ(τ + T, τ )ξk < eζT kξk
kΦ(τ, τ + T )ξk < e
−ζT
kξk
for all 0 6= ξ ∈ R(Pγ (τ )) ,
for all 0 6= ξ ∈ N (Pγ (τ + T )) .
This implies ζ ∈ ρ. The equality of the ranks of the invariant projectors follows
from Proposition 4.8.
¤
Lemma 4.12. Let γ1 , γ2 ∈ ρ with γ1 < γ2 , and choose invariant projectors Pγ1
and Pγ2 for the corresponding nonhyperbolic dichotomies with growth rates γ1 and
γ2 , respectively. Then we have rk Pγ1 ≤ rk Pγ2 . Moreover, [γ1 , γ2 ] ⊂ ρ is fulfilled if
and only if rk Pγ1 = rk Pγ2 .
Proof. We first prove that rk Pγ1 ≤ rk Pγ2 and observe that
R(Pγ1 ) ∩ N (Pγ2 ) = I × {0}
holds, because 0 6= ξ ∈ R(Pγ1 (τ )) ∩ N (Pγ2 (τ )) would satisfy
kξk = kΦ(τ, τ + T )Φ(τ + T, τ )ξk < e−γ2 T kΦ(τ + T, τ )ξk < e−γ2 T e+γ1 T kξk < kξk .
This yields
¡
¢
0 = dim R(Pγ1 ) ∩ N (Pγ2 )
¡
¢
= rk Pγ1 + dim N (Pγ2 ) − dim R(Pγ1 ) + N (Pγ2 ) ,
and therefore,
¡
¢
rk Pγ2 = rk Pγ1 + N − dim R(Pγ1 ) + N (Pγ2 ) ≥ rk Pγ1 .
Assume that [γ1 , γ2 ] ⊂ ρ. Arguing negatively, suppose that rk Pγ1 =
6 rk Pγ2 . We
choose invariant projectors Pγ for the nonhyperbolic dichotomies of (4.1) with
growth rate γ for all γ ∈ (γ1 , γ2 ) and define
©
ª
ζ0 := sup ζ ∈ [γ1 , γ2 ] : rk Pζ 6= rk Pγ2 .
Due to Lemma 4.11, there exists an ε > 0 such that rk Pζ0 = rk Pζ for all ζ ∈ Uε (ζ0 ).
This is a contradiction to the definition of ζ0 . Conversely, let rk Pγ1 = rk Pγ2 . We
have already seen at the beginning of this proof that R(Pγ1 ) ∩ N (Pγ2 ) = I × {0}.
Since rk Pγ1 = rk Pγ2 , this implies the existence of an invariant projector P with
N (P ) = N (Pγ2 ) and R(P ) = R(Pγ1 ). Thus, for all γ ∈ [γ1 , γ2 ], we have
kΦ(τ + T, τ )ξk < eγT kξk
kΦ(τ, τ + T )ξk < e
−γT
kξk
for all 0 6= ξ ∈ R(Pγ1 (τ )) ,
for all 0 6= ξ ∈ N (Pγ2 (τ + T )) .
10
MARTIN RASMUSSEN
This implies [γ1 , γ2 ] ⊂ ρ and finishes the proof of this lemma.
¤
The following Spectral Theorem says that the dichotomy spectrum is the union
of at least one and at most N compact intervals.
Theorem 4.13 (Spectral Theorem). There exists an n ∈ {1, . . . , N } such that
Σ = [a1 , b1 ] ∪ · · · ∪ [an , bn ]
with −∞ < a1 ≤ b1 < a2 ≤ b2 < · · · < an ≤ bn < ∞.
Proof. The set ρ is open due to Lemma 4.11. Therefore, Σ is the disjoint union of
closed intervals, and the boundedness of Σ follows directly. To show the relation
n ≤ N , we assume that n ≥ N + 1. Thus, there exist
ζ1 < ζ 2 < · · · < ζ N ∈ ρ
such that the N + 1 intervals
(−∞, ζ1 ) , (ζ1 , ζ2 ) , . . . , (ζN , ∞)
have nonempty intersection with the spectrum Σ. It follows from Lemma 4.12 that
0 ≤ rk Pζ1 < rk Pζ2 < · · · < rk PζN ≤ N
is fulfilled for invariant projectors Pζi of the nonhyperbolic dichotomy with growth
rate ζi , i ∈ {1, . . . , n}. This implies either rk Pζ1 = 0 or rk PζN = N . Thus,
(−∞, ζ1 ] ∩ Σ = ∅
or
[ζN , ∞) ∩ Σ = ∅ ,
and this is a contradiction. To show n ≥ 1, we assume that Σ = ∅. Obviously,
there exist ζ1 , ζ2 ∈ R such that (4.1) admits a nonhyperbolic dichotomy with growth
rate ζ1 and projector Pζ1 ≡ 0 and a nonhyperbolic dichotomy with growth rate ζ2
and projector Pζ2 ≡ 1. Applying Lemma 4.12, we get (ζ1 , ζ2 ) ∩ Σ 6= ∅. This
contradiction yields n ≥ 1 and finishes the proof of this theorem.
¤
Spectra of scalar linear differential equations can be computed explicitly.
Example 4.14. We consider scalar differential equations of the form
ẋ = a(t)x ,
¡Rt
¢
where a : I → R is a continuous function. We have Φ(t, τ ) = exp τ a(s) ds for all
t, τ ∈ I. The Spectral Theorem says that the (τ, T )-dichotomy spectrum consists of
exactly one closed interval. Furthermore, due to Remark 4.3, the (τ, T )-dichotomy
spectrum fulfills
©
ª
Σ = |Φ(τ + T, τ )| .
It is well-known that an autonomous linear differential equation
(4.3)
ẋ = Ax
with a matrix A ∈ RN ×N admits an exponential dichotomy if and only if the real
part of every eigenvalue λ of A, denoted by <λ, is unequal to zero (see, e.g., Sacker
& Sell [31, p. 430(1)]). The Sacker-Sell spectrum therefore satisfies
©
ª
ΣSS = <λ : λ is an eigenvalue of A .
A relation of this kind does not hold for the (τ, T )-dichotomy spectrum. Nevertheless, by letting T tend to ∞, we obtain the following statement.
FINITE-TIME ATTRACTIVITY AND BIFURCATION
11
Theorem 4.15 (Spectra of autonomous linear systems). Consider the linear system (4.3), and let ΣT be the (0, T )-dichotomy spectrum of (4.3). Then the limit
relation
©
ª
lim ΣT = <λ : λ is an eigenvalue of A
T →∞
holds with respect to the Hausdorff distance.
Proof. There exist n ∈ {1, . . . , N } and reals λ1 < λ2 < · · · < λn with
©
ª
<λ : λ is an eigenvalue of A = {λ1 , . . . , λn } .
It is sufficient to show that for all ε > 0, there exists a τ > 0 with
n
[
¡ T¢
T
(4.4)
{λ1 , . . . , λn } ⊂ Uε Σ
and Σ ⊂
Uε (λi ) for all T ≥ τ .
i=1
Let ε > 0. It is an elementary result in the theory of linear differential equations (see, e.g., Coppel [10, p. 56]) that there exist nontrivial linear subspaces
U1 , . . . , Un ⊂ RN with U1 ⊕ · · · ⊕ Un = RN and a real constant K ≥ 1 such that
for all i ∈ {1, . . . , n},
³³
°
°
ε´ ´
(4.5) °eAt ξ ° ≤ K exp λi +
t kξk for all ξ ∈ U1 ⊕ · · · ⊕ Ui and t ≥ 0 ,
4
³³
´
°
°
ε ´
1
(4.6) °eAt ξ ° ≥
exp λi −
t kξk for all ξ ∈ Ui ⊕ · · · ⊕ Un and t ≥ 0
K
4
is fulfilled. We choose τ > 0 and T ≥ τ with K exp(−ετ /4) < 1.
To prove the first condition of (4.4), we choose an i ∈ {1, . . . , n} and assume to the
contrary that Uε (λi ) ∩ ΣT = ∅. Thus, there exists an invariant projector P(λi −ε/2)
with
° AT °
¡
¢
°e ξ ° < e(λi −ε/2)T kξk for all 0 6= ξ ∈ R P(λ −ε/2) (0)
(4.7)
i
and an invariant projector P(λi +ε/2) with
° −AT °
¡
¢
°e
(4.8)
ξ ° < e−(λi +ε/2)T kξk for all 0 6= ξ ∈ N P(λi +ε/2) (T ) .
Because of Lemma 4.12, we have rk P(λ¡i −ε/2) = rk P(λi +ε/2) =: r. In the
¢ case
r ≥ dim U1 + · · · + dim Ui , we have dim R(P
(0))
∩
(U
⊕
·
·
·
⊕
U
)
≥ 1.
i
n
¡ (λi −ε/2) ¢
Then there exists a nonzero element ξ ∈ R P(λi −ε/2) (0) ∩ (Ui ⊕ · · · ⊕ Un ), and this
leads to the contradiction
°
°
(4.7)
kξk = e−(λi −ε/2)T e(λi −ε/2)T kξk > e−(λi −ε/2)T °eAT ξ °
¡
¢−1
1 (λi −ε/4)T
e
kξk = Ke−εT /4
kξk > kξk .
K
¡
¢
If r < dim U1 + · · · + dim Ui , then dim N (P¡(λi +ε/2) (0)) ∩¢ (U1 ⊕ · · · ⊕ Ui ) ≥ 1.
Thus, there exists a nontrivial element ξ ∈ N P(λi +ε/2) (0) ∩ (U1 ⊕ · · · ⊕ Ui ), and
this also yields the contradiction
°
° (4.8)
°
° (4.5)
kξk = °e−AT eAT ξ ° < e−(λi +ε/2)T °eAT ξ ° ≤ e−(λi +ε/2)T Ke(λi +ε/4)T kξk
(4.6)
≥ e−(λi −ε/2)T
=
Ke−εT /4 kξk < kξk .
To prove the second condition of (4.4), let λ ∈
/ ∪ni=1 Uε (λi ). We set λ0 := −∞ and
λn+1 := ∞. There exists a i ∈ {0, . . . , n} such that
λ ≥ λi + ε
and λ ≤ λi+1 − ε .
12
MARTIN RASMUSSEN
Now, we define the invariant projector P by
R(P (0)) = U1 ⊕ · · · ⊕ Ui
and
N (P (0)) = Ui+1 ⊕ · · · ⊕ Un .
Thus, for all nonzero ξ ∈ R(P (0)), we have
° AT ° (4.5)
°e ξ ° ≤ Ke(λi +ε/4)T kξk ≤ Ke(λ−3ε/4)T kξk < eλT kξk ,
and for all nonzero ξ ∈ N (P (T )),
° −AT ° (4.6)
°e
ξ ° ≤ Ke−(λi+1 −ε/4)T kξk ≤ Ke−(λ+3ε/4)T kξk < e−λT kξk
is fulfilled. Hence, λ ∈
/ ΣT , and this finishes the proof of this theorem.
¤
Remark 4.16. Using Floquet Theory (see, e.g., Coddington & Levinson [8, pp. 78–
80] or Chicone [7, Section 2.4, pp. 162–197]), one can extend the above theorem to
periodic linear differential systems of the form
(4.9)
ẋ = A(t)x ,
N ×N
where A : R → R
fulfills A(t) = A(t + ω) for all t ∈ R with some ω > 0. We
denote the transition operator of (4.9) by Φ. For the Sacker-Sell spectrum, one
obtains
©
ª
ΣSS = ln |λ| : λ is an eigenvalue of Φ(ω, 0) .
The matrix Φ(ω, 0) is called monodromy matrix of (4.9). The (0, T )-dichotomy
spectrum ΣT fulfills the limit relation
©
ª
lim ΣT = ln |λ| : λ is an eigenvalue of Φ(ω, 0)
T →∞
in the sense of Hausdorff distance.
We now look at the (0, T )-dichotomy spectrum of a special autonomous planar
linear system.
Example 4.17. For fixed T > 0, we want to compute the (0, T )-dichotomy spectrum ΣT of the linear autonomous system
(4.10)
ẋ = Ax
where
A :=
µ
1
0
¶
1
.
1
Specifically in this example, we use the norm k · k1 : R2 → R+
0 , defined by
k(x1 , x2 )k1 := |x1 | + |x2 |. Note that for γ ∈ R, the relation
µ (1−γ)T
¶
e
T e(1−γ)T
e(A−γ 1)T =
0
e(1−γ)T
is fulfilled (see, e.g., Aulbach [2]). Hence, for all ξ = (ξ1 , ξ2 ) ∈ R2 with kξk1 = 1,
we have
°
°
°
µ ¶°
µ ¶°
µ ¶°
°
° (A−γ 1)T 1 °
°
°
°
°e
° ≤ °e(A−γ 1)T ξ1 ° ≤ °e(A−γ 1)T 0 °
.
°
°
ξ2 °1
0 °1 °
1 °1
{z
}
{z
}
|
|
= e(1−γ)T
= T e(1−γ)T + e(1−γ)T
The term T e(1−γ)T +e(1−γ)T is strictly monotone decreasing in γ ∈ R, and therefore,
there exists a uniquely determined γ∗ = γ∗ (T ) > 1 with T e(1−γ∗ )T + e(1−γ∗ )T = 1.
Using these observations, it is easy to see that ΣT = {1, γ∗ }, since
FINITE-TIME ATTRACTIVITY AND BIFURCATION
13
(i) for γ < 1, the linear system (4.10) admits a nonhyperbolic (0, T )-dichotomy
with growth rate γ and invariant projector Pγ ≡ 0,
(ii) for γ ∈ (1, γ∗ ), the linear system (4.10) admits a nonhyperbolic (0, T )dichotomy
¡
¢ with growth rate γ and invariant
¡
¢ projector Pγ , determined by
R Pγ (0) = {β(1, 0) : β ∈ R} and N Pγ (0) = {β(0, 1) : β ∈ R}.
(iii) for γ > γ∗ , the linear system (4.10) admits a nonhyperbolic (0, T )-dichotomy
with growth rate γ and invariant projector Pγ ≡ 1,
(iv) for γ ∈ {1, γ∗ }, the linear system (4.10) admits no nonhyperbolic (0, T )dichotomy with growth rate γ.
Note that Theorem 4.15 implies that limT →∞ γ∗ (T ) = 1.
5. Linearized attractivity and repulsivity
In this section, we provide a finite-time version of the well-known Theorem of
Linearized Stability.
Theorem 5.1 (Linearized attractivity and repulsivity). Consider a compact interval I := [τ, τ + T ] for some τ ∈ R and T > 0, and let
(5.1)
ẋ = A(t)x + F (t, x)
be a nonautonomous differential equation with continuous functions A : I → RN ×N
and F : I × U → RN , U ⊂ RN a neighborhood of 0, such that F (t, 0) = 0 for all
t ∈ I. Let ϕ denote the general solution of (5.1) and Φ : I × I → RN ×N denote the
transition operator of the linearized equation ẋ = A(t)x, and define
©
ª
K+ := sup kΦ(t, s)k : τ ≤ s ≤ t ≤ τ + T
and
©
ª
K− := sup kΦ(t, s)k : τ ≤ t ≤ s ≤ τ + T .
Then the following statements are fulfilled:
(i) If
kΦ(τ + T, τ )k < 1
(5.2)
and there exist δ > 0 and β > 1 with
¡
¢
ln β kΦ(τ + T, τ )k
kF (t, x)k ≤ −
kxk
T K+
for all t ∈ I and x ∈ Uδ (0) ,
there exists an η > 0 such that
kϕ(τ + T, τ, ξ)k ≤ β −1 kξk
for all ξ ∈ Uη (0) ,
i.e., the trivial solution of (5.1) is (τ, T )-attractive.
(ii) If
kΦ(τ, τ + T )k < 1
(5.3)
and there exist δ > 0 and β > 1 with
¡
¢
ln β kΦ(τ, τ + T )k
kF (t, x)k ≤ −
kxk
T K−
for all t ∈ I and x ∈ Uδ (0) ,
there exists an η > 0 such that
kϕ(τ, τ + T, ξ)k ≤ β −1 kξk
for all ξ ∈ Uη (0) ,
i.e., the trivial solution of (5.1) is (τ, T )-repulsive.
14
MARTIN RASMUSSEN
Proof. We only prove (i), since (ii) can be shown analogously. Due to the continuity
of the general solution, there exists an η < δ with
kϕ(t, τ, ξ)k < δ
for all t ∈ I and ξ ∈ Uη (0) .
We choose ξ ∈ Uη (0) arbitrarily. Then the solution ϕ(·, τ, ξ) of (5.1) is also a
solution of the linear differential equation
ẋ = A(t)x + F (t, ϕ(t, τ, ξ)) .
Thus, the variation of the constants formula implies
Z t
Φ(t, s)F (s, ϕ(s, τ, ξ)) ds for all t ∈ I .
ϕ(t, τ, ξ) = Φ(t, τ )ξ +
τ
Hence, for all t ∈ I, the relation
Z
t
°
°
kΦ(t, s)k°F (s, ϕ(s, τ, ξ))° ds
τ
¡
¢Z t
ln β kΦ(τ + T, τ )k
≤ kΦ(t, τ )k kξk − K+
kϕ(s, τ, ξ)k ds
T K+
τ
¡
¢Z t
ln β kΦ(τ + T, τ )k
= kΦ(t, τ )k kξk −
kϕ(s, τ, ξ)k ds .
T
τ
kϕ(t, τ, ξ)k ≤ kΦ(t, τ )ξk +
We apply Gronwall’s inequality (cf. Abraham & Marsden & Ratiu [1, Theorem 4.1.7,
p. 242]) and obtain that for all ξ ∈ Uη (0), we have
¡
¡
¢¢
kϕ(τ + T, τ, ξ)k ≤ kΦ(τ + T, τ )k kξk exp −ln β kΦ(τ + T, τ )k
= β −1 kξk .
This finishes the proof of this theorem.
¤
Remark 5.2.
(i) The (τ, T )-dichotomy spectrum of the linearization ẋ = A(t)x is in case (i)
of Theorem 5.1 a subset of R− and in case (ii) a subset of R+ .
(ii) The conditions (5.2) and (5.3) of Theorem 5.1 are fulfilled if we have
lim sup
x→0 t∈I
kF (t, x)k
= 0.
kxk
This limit relation is only sufficient but not necessary for the above mentioned conditions.
6. Bifurcations in dimension one
In this section, we derive nonautonomous analogues of two classical autonomous
bifurcation patterns. We begin with the study of the transcritical bifurcation scenario.
Theorem 6.1 (Nonautonomous transcritical bifurcation). Let x− < 0 < x+ and
α− < α+ be in R and I := [τ, τ + T ], and consider the nonautonomous differential
equation
(6.1)α
ẋ = a(t, α)x + b(t, α)x2 + r(t, x, α)
FINITE-TIME ATTRACTIVITY AND BIFURCATION
15
with continuous functions a : I × (α− , α+ ) → R, b : I × (α− , α+ ) → R and r :
I × (x− , x+ ) × (α− , α+ ) → R fulfilling r(·, 0, ·) ≡ 0. Let Φα : I × I → R denote the
transition operator of the linearized equation ẋ = a(t, α)x. We define
©
ª
K(α) := sup Φα (t, s) : t, s ∈ I
for all α ∈ (α− , α+ )
and assume that there exists an α0 ∈ (α− , α+ ) such that the following hypotheses
hold:
(i) Hypothesis on linear part. We either have
(6.2)
Φα (τ + T, τ ) < 1
Φα (τ + T, τ ) > 1
for all α ∈ (α− , α0 ) and
for all α ∈ (α0 , α+ )
Φα (τ + T, τ ) > 1
Φα (τ + T, τ ) < 1
for all α ∈ (α− , α0 ) and
for all α ∈ (α0 , α+ ) .
or
(6.3)
(ii) Hypothesis on nonlinearity. The quadratic term either fulfills
(6.4)
lim inf inf b(t, α) > 0
α→α0
t∈I
or
(6.5)
lim sup sup b(t, α) < 0 ,
α→α0
t∈I
and the remainder satisfies
(6.6)
lim
sup
sup
x→0 α∈(α −|x|,α +|x|) t∈I
0
0
|r(t, x, α)|
=0
|x|2
and
(6.7)
lim sup lim sup sup −
α→α0
x→0
t∈I
T K(α)|r(t, x, α)|
¡
©
ª¢ < 1 .
|x| ln min Φα (τ + T, τ ), Φα (τ, τ + T )
Then there exist α̂− < 0 < α̂+ such that the following statements are fulfilled:
(i) In case (6.2), the trivial solution is (τ, T )-attractive for α ∈ (α̂− , α0 ) and
(τ, T )-repulsive for α ∈ (α0 , α̂+ ). The differential equation (6.1)α admits
a (τ, T )-bifurcation, since the corresponding radii of (τ, T )-attraction and
repulsion satisfy
lim Aα
0 =0
α%α0
and
lim Rα
0 = 0.
α &α 0
(ii) In case (6.2), the trivial solution is (τ, T )-repulsive for α ∈ (α̂− , α0 ) and
(τ, T )-attractive for α ∈ (α0 , α̂+ ). The differential equation (6.1)α admits
a (τ, T )-bifurcation, since the corresponding radii of (τ, T )-repulsion and
attraction satisfy
lim Rα
0 =0
α%α0
and
lim Aα
0 = 0.
α &α 0
Proof. Let ϕα denote the general solution of (6.1)α . We will only prove assertion
(i), since the proof of (ii) is similar. This means that (6.2) is fulfilled. W.l.o.g., we
only treat the case (6.4). We choose α̂− < 0 < α̂+ such that
(6.8)
inf
α∈(α̂− ,α̂+ ), t∈I
b(t, α) > 0
16
MARTIN RASMUSSEN
and for all α ∈ (α̂− , α̂+ ), we have
¡
©
ª¢
ln min Φα (τ + T, τ ), Φα (τ, τ + T )
|r(t, x, α)|
lim sup sup
≤ −γ
|x|
T K(α)
x→0
t∈I
for some γ ∈ (0, 1). Because of these two relations, Theorem 5.1 can be applied,
and the attractivity and repulsivity of the trivial solutions as stated in the theorem
follows. We define
©
ª
K− := inf Φα (t, s) : t, s ∈ I, α ∈ [α̂− , α0 ] ∈ (0, 1) .
Assume to the contrary that
η := lim sup Aα
0 >0
α%α0
holds. Due to (6.8) and (6.6), there exist α̃− ∈ (α̂− , α0 ), ξ ∈ (0, K− η) and L > 0
with
·
¸
ξ
2
(6.9) b(t, α)x + r(t, x, α) > L for all t ∈ I, α ∈ (α̃− , α0 ) and x ∈ K− ξ,
.
K−
We fix α̂ ∈ (α̃− , α0 ) such that Aα̂
0 > ξ and
(6.10)
Φα̂ (τ + T, τ ) ≥ 1 −
K− LT
.
ξ
For arbitrary τ ∈ I, the solution µτ (·) := ϕα̂ (·, τ, ξ) of (6.1)α̂ is also a solution of
the inhomogeneous linear differential equation
ẋ = a(t, α̂)x + b(t, α̂)(µτ (t))2 + r(t, µτ (t), α̂) .
(6.11)
Since Aα̂
0 > ξ, we have
(6.12)
µτ (τ + T ) < ξ .
Moreover, from the definition of K− and (6.9), we directly get
(6.13)
µτ (τ + t) ≥ K− ξ
for all t ∈ [0, T ] .
We distinguish two cases.
Case 1. There exists a t̄ ∈ (0, T ] such that
µτ (τ + t̄) =
ξ
.
K−
We choose t̄ maximal with this property. Due to (6.12), this means that µ(τ + t) ≤
ξ/K− for all t ∈ [t̄, T ]. Then the variation of constants formula, applied to (6.11),
implies the relation
µτ (τ + T )
=
ξ
+
K−
¡
¢
Φα̂ (τ + T, t) b(t, α̂)(µτ (t))2 + r(t, µτ (t), α̂) dt
Φα̂ (τ + T, τ + t̄)
Z
τ +T
+
τ +t̄
(6.9)
≥
ξ + K− L(T − t̄) > ξ .
FINITE-TIME ATTRACTIVITY AND BIFURCATION
17
This contradicts (6.12).
Case 2. For all t ∈ (0, T ], we have
µτ (τ + t̄) <
ξ
.
K−
In this case, the variation of constants formula, applied to (6.11), yields
=
(6.9), (6.10)
≥
µτ (τ + T )
Φα̂ (τ + T, τ )ξ +
Z τ +T
¡
¢
+
Φα̂ (τ + T, t) b(t, α̂)(µτ (t))2 + r(t, µτ (t), α̂) dt
τ
¶
µ
K− LT
ξ + K− LT = ξ .
1−
ξ
This contradicts (6.12) also, and thus, limα%α0 Aα
0 = 0 is proved. Analogously, one
can show limα&α0 Rα
=
0
and
treat
the
case
(6.5).
¤
0
Remark 6.2.
(i) The hypothesis on the linear part implies that the (τ, T )-dichotomy spectrum of the linearization ẋ = a(t, α)x converges to {0} in Hausdorff distance
in the limit α → α0 .
(ii) Condition (6.7) is only used to obtain the attractivity or repulsivity of the
trivial solution by applying Theorem 5.1. Alternatively, one can directly
postulate that the trivial solution changes its stability at the parameter
value α0 from, say, attractivity to repulsivity.
Finally, we treat a nonautonomous version of the well-known pitchfork bifurcation. The proof will be omitted, since it is similar to that of Theorem 6.1.
Theorem 6.3 (Nonautonomous pitchfork bifurcation). Let x− < 0 < x+ and
α− < α+ be in R and I := [τ, τ + T ], and consider the nonautonomous differential
equation
ẋ = a(t, α)x + b(t, α)x3 + r(t, x, α)
(6.14)α
with continuous functions a : I × (α− , α+ ) → R, b : I × (α− , α+ ) → R and r :
I × (x− , x+ ) × (α− , α+ ) → R fulfilling r(·, 0, ·) ≡ 0. Let Φα : I × I → R denote the
transition operator of the linearized equation ẋ = a(t, α)x. We define
©
ª
K(α) := sup Φα (t, s) : t, s ∈ I
for all α ∈ (α− , α+ )
and assume, there exists an α0 ∈ (α− , α+ ) such that the following hypotheses hold:
(i) Hypothesis on linear part. We either have
(6.15)
Φα (τ + T, τ ) < 1
Φα (τ + T, τ ) > 1
for all α ∈ (α− , α0 ) and
for all α ∈ (α0 , α+ )
Φα (τ + T, τ ) > 1
Φα (τ + T, τ ) < 1
for all α ∈ (α− , α0 ) and
for all α ∈ (α0 , α+ ) .
or
(6.16)
(ii) Hypothesis on nonlinearity. The cubic term either fulfills
(6.17)
lim inf inf b(t, α) > 0
α→α0
t∈I
18
MARTIN RASMUSSEN
or
(6.18)
lim sup sup b(t, α) < 0 ,
α→α0
t∈I
and the remainder satisfies
(6.19)
lim
sup
sup
x→0 α∈(α −x2 ,α +x2 ) t∈I
0
0
|r(t, x, α)|
=0
|x|3
and
(6.20)
lim sup lim sup sup −
α→α0
x→0
t∈I
T K(α)|r(t, x, α)|
¡
©
ª¢ < 1 .
|x| ln min Φα (τ + T, τ ), Φα (τ, τ + T )
Then there exist α̂− < 0 < α̂+ such that the following statements are fulfilled:
(i) In case (6.15) and (6.17) is fulfilled, the trivial solution is (τ, T )-attractive
for α ∈ (α̂− , α0 ) and (τ, T )-repulsive for α ∈ (α0 , α̂+ ). The differential
equation (6.14)α admits a (τ, T )-bifurcation, since the corresponding radii
of (τ, T )-attraction satisfy
lim Aα
0 = 0.
α%α0
(ii) In case (6.15) and (6.18) is fulfilled, the trivial solution is (τ, T )-attractive
for α ∈ (α̂− , α0 ) and (τ, T )-repulsive for α ∈ (α0 , α̂+ ). The differential
equation (6.14)α admits a (τ, T )-bifurcation, since the corresponding radii
of (τ, T )-repulsion satisfy
lim Rα
0 = 0.
α&α0
(iii) In case (6.16) and (6.17) is fulfilled, the trivial solution is (τ, T )-repulsive
for α ∈ (α̂− , α0 ) and (τ, T )-attractive for α ∈ (α0 , α̂+ ). The differential
equation (6.14)α admits a (τ, T )-bifurcation, since the corresponding radii
of (τ, T )-attraction satisfy
lim Aα
0 = 0.
α&α0
(iv) In case (6.16) and (6.18) is fulfilled, the trivial solution is (τ, T )-repulsive
for α ∈ (α̂− , α0 ) and (τ, T )-attractive for α ∈ (α0 , α̂+ ). The differential
equation (6.14)α admits a (τ, T )-bifurcation, since the corresponding radii
of (τ, T )-repulsion satisfy
lim Rα
0 = 0.
α%α0
Remark 6.4.
(i) The hypothesis on the linear part implies that the (τ, T )-dichotomy spectrum of the linearization ẋ = a(t, α)x converges to {0} in Hausdorff distance
in the limit α → α0 .
(ii) Condition (6.20) is only used to obtain the attractivity or repulsivity of the
trivial solution by applying Theorem 5.1. Alternatively, one can directly
postulate that the trivial solution changes its stability at the parameter
value α0 from, say, attractivity to repulsivity.
FINITE-TIME ATTRACTIVITY AND BIFURCATION
19
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