International Journal of Systems Science
volume 35, number 5, 20 April 2004, pages 287–292
Lyapunov function proof of Poincaré’s theorem
W. M. HADDAD{*, S. G. NERSESOV{ and V.-S. CHELLABOINA{
One of the most fundamental results in analysing the stability properties of periodic
orbits and limit cycles of dynamical systems is Poincare´’s theorem. The proof of this
result involves system analytic arguments along with the Hartman–Grobman theorem.
Using the notions of stability of sets, lower semicontinuous Lyapunov functions are
constructed to provide a Lyapunov function proof of Poincare´’s theorem.
1. Introduction
Poincaré’s theorem (Poincaré 1881) provides a
powerful tool when analysing the stability properties
of periodic orbits and limit cycles of n-dimensional
dynamical systems in the case where the solution of the
system can be relatively easily integrated. Specifically,
Poincaré’s theorem provides necessary and sufficient
conditions for the stability of periodic orbits based on
the stability properties of a fixed point of a discretetime dynamical system constructed from a Poincaré
return map. In particular, for a given candidate periodic
orbit, an (n 1)-dimensional hyperplane is constructed
that is transversal to the periodic orbit and which defines
the Poincaré return map. Solutions starting on the
hyperplane that are sufficiently close to a point on the
periodic orbit will intersect the hyperplane after a
time approximately equal to the period of the periodic
orbit. This mapping traces the system solution from
a point on the hyperplane to its next corresponding intersection with the hyperplane. Hence, using
system analytic arguments along with the somewhat
involved Hartman–Grobman theorem (Hartman 1973),
the Poincaré return map can be used to establish a relationship between the stability properties of a dynamical
system with periodic solutions and the stability properties of an equilibrium point of an (n 1)-dimensional
discrete-time system. Using the notions of Lyapunov
and asymptotic stability of sets (Bhatia and Szegö
Received 16 July 2002. Revised 24 March 2004. Accepted
22 April 2004.
{ School of Aerospace Engineering, Georgia Institute of
Technology, Atlanta, GA 30332-0150, USA.
{ Department of Aerospace Engineering, University of MissouriColumbia, Columbia, MO 65211, USA.
* To whom correspondence should be addressed.
e-mail: [email protected]
1970, Leonessa et al. 2000), we construct lower semicontinuous Lyapunov functions to provide a Lyapunov
function proof of Poincaré’s theorem.
2.
Notation and mathematical preliminaries
This section establishes the notation and definitions
needed. Let R be the set of real numbers, let Rþ be
the set of positive numbers, let Rn be the set of n 1
real column vectors, and let N be the set of non-negative
integers. Furthermore, let @D, D, and D be the boundary, interior and closure of the set D Rn , respectively.
We write k k for the Euclidean vector norm, V 0 ðxÞ
for the gradient of V at x, B ðÞ, 2 Rn , > 0, for the
open ball centered at with radius , and distð p, MÞ
for the smallest distance from a point p to any point
in the set M; that is, distð p, MÞ ¼ inf x2M kp xk.
This paper considers the nonlinear dynamical system:
x_ ðtÞ ¼ f ðxðtÞÞ,
xð0Þ ¼ x0 ,
t 2 I x0 ,
ð1Þ
where xðtÞ 2 D Rn , t 2 I x0 , is the system state vector,
D is an open set, f : D ! Rn , and I x0 ¼ ½0, x0 Þ,
0 < x0 1, is the maximal interval of existence for
the solution xðÞ of (1). A function x : I x0 ! D is said
to be a solution to (1) on the interval I x0 ½0, 1Þ
with initial condition xð0Þ ¼ x0 , if x(t) satisfies (1) for
all t 2 I x0 . It is assumed that the dynamics f ðÞ are
such that the solution to (1) is unique for every initial
condition in D and jointly continuous in t and x0 . A
sufficient condition ensuring this is Lipschitz continuity
of f ðÞ.
Next, we introduce several definitions that are
necessary for the main results of this paper. For these
definitions, we denote the solution xðÞ to (1) with initial
condition x0 2 D by sðt, x0 Þ and let the map st : D ! D
be defined by st ðx0 Þ ¼ sðt, x0 Þ, x0 2 D, for a given t 2 I x0 .
International Journal of Systems Science ISSN 0020–7721 print/ISSN 1464–5319 online 2004 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/00207720410001714824
288
W. M. Haddad et al.
Definition 2.1: Consider the nonlinear dynamical
system G given by (1). The orbit sðt, x0 Þ 2 D Rn ,
t 2 I x0 , of G denotes the solution to (1) corresponding
to the initial condition xð0Þ ¼ x0 evaluated at time t.
The orbit sðt, x0 Þ, t 2 I x0 , of G is bounded if there exists
> 0 such that ksðt, x0 Þk < , t 2 I x0 .
This paper assumes that all solutions to (1) are
bounded over I x0 and hence by the Peano–Cauchy
theorem (Hale 1980, pp. 16, 17) can be extended to
infinity.
Definition 2.2: Consider the nonlinear dynamical
system G given by (1). A set M D is a positively invariant set for the dynamical system G if st ðMÞ M, for
all t 0, where st ðMÞ ¼ fst ðx0 Þ : x0 2 Mg. A set M D
is an invariant set for the dynamical system G if
st ðMÞ ¼ M for all t 2 R.
The following definition introduces two types of
stability of (1) with respect to a compact positively
invariant set.
Definition 2.3: Let D0 2 D be a compact positively
invariant set for the nonlinear dynamical system (1).
D0 is Lyapunov stable if for all > 0 there exists ¼
ðÞ > 0 such that if distðx0 , D0 Þ < , then distðsðt, x0 Þ,
D0 Þ < , t 0. D0 is asymptotically stable if D0 is
Lyapunov stable and there exists > 0 such that if
distðx0 , D0 Þ < , then distðsðt, x0 Þ, D0 Þ ! 0 as t ! 1.
Next, we introduce the notions of periodic solutions
and periodic orbits for (1).
Definition 2.4: A solution sðt, x0 Þ of (1) is periodic if
there exists a finite time T > 0 such that sðt þ T, x0 Þ ¼
sðt, x0 Þ for all t 0: A set O D is a periodic orbit
of (1) if O ¼ fx 2 D : x ¼ sðt, x0 Þ, 0 t Tg for some
periodic solution sðt, x0 Þ of (1).
Finally, the following definitions are used in the
paper.
Definition 2.5: Let Dc D. A function V : Dc ! R
is lower semicontinuous on Dc if for every sequence
fxn g1
such that limn!1 xn ¼ x, VðxÞ n¼0 2 Dc
lim inf n!1 Vðxn Þ.
the periodic orbit O ¼ fx 2 D : x ¼ sðt, pÞ, 0 t Tg.
Note that O is a compact invariant set. Furthermore,
as is standard in Poincaré’s method (Khalil 1996), we
assume that there exists a continuously differentiable
function X : D ! R such that the (n 1)-dimensional
hyperplane defined by H ¼ fx 2 D : X ðxÞ ¼ 0g contains
0
the point x ¼ p and X ð pÞ 6¼ 0. In addition, we assume
that the hyperplane H is not tangent to the periodic
orbit O at x ¼ p; that is, X 0 ð pÞf ð pÞ 6¼ 0. Next, define
the local section S H such that p 2 S, X 0 ðxÞ 6¼ 0,
x 2 S, and all orbits of (1) starting in S are not tangent
to H; that is, X 0 ðxÞf ðxÞ 6¼ 0, x 2 S. Note that an orbit
sðt, pÞ will intersect S at p in T seconds. Furthermore, let
U ¼ fx 2 S : there exists ðxÞ > 0 such that sððxÞ, xÞ
2 S and sðt, xÞ 2
= S, 0 < t < ðxÞg:
ð2Þ
Finally, define the Poincaré return map P : U ! S by
PðxÞ ¼ sððxÞ, xÞ,
x 2 U:
Figure 1 gives a visualization of the Poincaré return
map construction.
Next, define D1 ¼ fx 2 D : there exists ðxÞ > 0 such
that sððxÞ, xÞ 2 Sg. The existence of D1 is guaranteed
by continuous dependence of solutions of (1) on
initial data; moreover, for every x 2 O there exists
¼ ðxÞ > 0 such that B ðxÞ D1 and hence O D1 .
Similarly, define D ¼ fx 2 D1 : sððxÞ, xÞ 2 S g and
U ¼ fx 2 S : sððxÞ,
xÞ 2 S g, where S ¼ B ð pÞ \ S,
> 0, and O D D1 . The function : D1 ! Rþ
defines the time required for the solution sðt, xÞ, x 2
D1 , to return to the local section S. Note that
ðxÞ > 0, x 2 U. The following lemma shows that ðÞ
is continuous at x 2 D1 nH.
Lemma 3.1: Consider the nonlinear dynamical system
(1). Assume that the point p 2 D1 generates the periodic
orbit O ¼ fx 2 D1 : x ¼ sðt, pÞ, 0 t Tg, where sðt, pÞ,
H
S
P (x)
Definition 2.6: A function V : D ! R is positive definite on DnD0 , where D0 D, if VðxÞ ¼ 0, x 2 D0 , and
VðxÞ > 0, x 2 DnD0 .
x
p
U
3. Unification between Lyapunov’s second
method and Poincaré’s theorem
This section provides a Lyapunov function proof of
Poincaré’s theorem. To proceed, it is assumed that for
the point p 2 D, the dynamical system (1) has a periodic
solution sðt, pÞ, t 0, with period T > 0 that generates
ð3Þ
Figure 1.
Visualization of the Poincaré return map.
289
Poincare´’s theorem via Lyapunov’s second method
t 0, is the periodic solution with period T ð pÞ. Then
the function : D1 ! Rþ is continuous on D1 nH.
Proof: The proof is similar to the proof of Lemma 3
of Grizzle et al. (2001) and hence is omitted.
œ
Finally, define the discrete-time dynamical system
given by:
zðk þ 1Þ ¼ PðzðkÞÞ,
zð0Þ 2 U,
k 2 N:
ð4Þ
Clearly x ¼ p is a fixed point of (4) since T ¼ ð pÞ and
hence p ¼ Pð pÞ. The following theorem is a direct
application of the standard discrete-time Lyapunov
stability theorem for general dynamical systems to the
dynamical system (4).
Theorem 3.1: The equilibrium solution zðkÞ p to (4) is
Lyapunov (respectively, asymptotically) stable if and
only if there exist a scalar > 0 and a lower semicontinuous (respectively, continuous) function V : S ! R such
that VðÞ is continuous at x ¼ p, Vð pÞ ¼ 0, VðxÞ > 0,
x 2 S , x 6¼ p, and VðPðxÞÞ VðxÞ 0, x 2 U (respectively, VðPðxÞÞ VðxÞ < 0, x 2 U , x 6¼ p).
The following theorem gives sufficient conditions
for Lyapunov and asymptotic stability of a positively
invariant set with respect to (1) in terms of a lower
semicontinuous Lyapunov function. For a proof of
this result, see Leonessa et al. (2000, 2001).
Theorem 3.2 (Leonessa et al. 2000, 2001): Consider the
nonlinear dynamical system (1). Assume that the point
p 2 D generates the periodic orbit O ¼ fx 2 D : x ¼
sðt, pÞ, 0 t Tg, where sðt, pÞ, t 0, is a periodic
solu
tion with period T ð pÞ such that O D0 D.
Furthermore, assume that there exists a lower semicontinuous, positive-definite (on D0 nO) function V : D0 ! R
such that VðÞ is continuous on O and
Vðsðt, xÞÞ Vðsð, xÞÞ,
0 t,
x 2 D0 :
ð5Þ
n ¼ 0, 1, . . . ,
Vðsðt, xÞÞ Vðsð, xÞÞ,
ð6Þ
then O is asymptotically stable.
The following theorem provides a converse to
Theorem 3.2. For this result, define the notation Or ¼
fx 2 D1 : distðx, OÞ < rg, r > 0, to denote an r open
neighbourhood of O.
Theorem 3.3: Consider the nonlinear dynamical system (1). If the periodic orbit O generated by the point
p 2 D is Lyapunov stable, then there exists a lower
0 t,
x 2 D0 :
ð7Þ
If the periodic orbit O generated by the point p 2 D is
asymptotically stable, then there exists a continuous, positive-definite (on D0 nO) function V : D0 ! R such that
inequality (7) is strictly satisfied.
Proof: Let > 0. Since the periodic orbit O is
Lyapunov stable it follows that there exists ¼
ðÞ > 0 such that if x0 2 O , then sðt, x0 Þ 2 O , t 0.
Now, let D0 ¼ fy 2 O : there exists t 0 and x0 2 O
such that y ¼ sðt, x0 Þg. Note that D0 O , D0 is
positively invariant set, and O D0 . Hence, O D0 .
Next, define VðxÞ ¼ supt0 distðsðt, xÞ, OÞ, x 2 D0 , and
since D0 is positively invariant and bounded it follows
that VðÞ is well defined on D0 . Now, since O is invariant, x 2 O implies VðxÞ ¼ 0, x 2 O. Furthermore,
VðxÞ distðsð0, xÞ, OÞ > 0, x 2 D0 , x 2
= O. Next, since
f ðÞ in (1) is such that for every x 2 D0 , sðt, xÞ, t 0, is
the unique solution to (1), it follows that sðt, xÞ ¼
sðt , sð, xÞÞ, 0 t. Hence, for every t, 0, such
that t ,
Vðsð, xÞÞ ¼ sup distðsð, sð, xÞÞ, OÞ
0
¼ sup distðsð þ , xÞ, OÞ
0
sup distðsð þ , xÞ, OÞ
t
¼ sup distðsð ðt Þ, sðt, xÞÞ, OÞ
t
¼ sup distðsð, sðt, xÞÞ, OÞ
0
¼ Vðsðt, xÞÞ,
Then O is Lyapunov stable. If, in addition, for all
= O, there exists an increasing unbounded
x 2 D0 , x 2
sequence ftn g1
n¼0 , with t0 ¼ 0, such that
Vðsðtnþ1 , xÞÞ < Vðsðtn , xÞÞ,
semicontinuous, positive-definite (on D0 nO) function
V : D0 ! R, where D0 D, such that O D0 , VðÞ is
continuous on O, and
ð8Þ
which proves (7). Next, let pO 2 O. Since the periodic
orbit O is Lyapunov stable it follows that for every
^ > 0 there exists ^ ¼ ^ðÞ > 0 such that if x0 2 B^ð pO Þ,
then sðt, x0 Þ 2 O^=2 , t 0, which implies that Vðx0 Þ ¼
supt0 distðsðt, x0 Þ, OÞ ^=2. Hence, for every point
pO 2 O and ^ > 0 there exists ^ ¼ ^ð^Þ such that if
x0 2 B^ð pO Þ, then Vðx0 Þ < ^ establishing that VðÞ is
continuous on O. Finally, to show that VðÞ is lower
semicontinuous everywhere on D0 , let x 2 D0 , let
^ > 0, and note that since VðxÞ ¼ supt0 distðsðt, xÞ, OÞ,
there exists T ¼ Tðx, ^Þ > 0 such that VðxÞ
distðsðT, xÞ, OÞ < ^. Now, consider a sequence fxi g1
i¼1 2
D0 such that xi ! x as i ! 1. Next, since sðt, Þ is
continuous for every t 0 and distð, OÞ : D0 ! R
290
W. M. Haddad et al.
is continuous, it follows that distðsðT, xÞ, OÞ ¼
limi!1 distðsðT, xi Þ, OÞ which implies that:
VðxÞ < distðsðT, xÞ, OÞ þ ^
¼ lim distðsðT, xi Þ, OÞ þ ^
i!1
lim inf sup distðsðt, xi Þ, OÞ þ ^
i!1
t0
¼ lim inf Vðxi Þ þ ^,
ð9Þ
i!1
which, since ^ > 0 is arbitrary, further implies that
VðxÞ lim inf i!1 Vðxi Þ. Thus, since fxi g1
i¼1 is an arbitrary sequence converging to x, it follows that VðÞ is
lower semicontinuous on D0 .
The existence of a continuous, positive-definite
(on D0 nO) function V : D0 ! R in the case where O is
asymptotically stable follows using similar arguments
as above with
ð1
VðxÞ ¼
0
sup distðsðt, sð, xÞÞ, OÞe d,
x 2 DA ,
t0
ð10Þ
where DA is a domain of attraction of O. For details
see Bhatia and Szegö (1970, Theorem 2.2, p. 66).
œ
The next theorem presents the main result of the
paper.
Theorem 3.4: Consider the nonlinear dynamical system
(1) with the Poincare´ map defined by (3). Assume that the
point p 2 D generates the periodic orbit O ¼ fx 2 D : x ¼
sðt, pÞ, 0 t Tg, where sðt, pÞ, t 0, is the periodic solution with period T ð pÞ. Then the following statements
hold:
(1) p 2 D is a Lyapunov stable fixed point of (4) if and
only if the periodic orbit O generated by p is
Lyapunov stable.
(2) p 2 D is an asymptotically stable fixed point of (4) if
and only if the periodic orbit O generated by p is
asymptotically stable.
Proof: (1) To show necessity, assume that x ¼ p is a
Lyapunov stable fixed point of (4). Then it follows from
Theorem 3.1 that for sufficiently small > 0 there exists
a lower semicontinuous function Vd : S ! R such that
Vd ðÞ is continuous at x ¼ p, Vd ð pÞ ¼ 0, Vd ðxÞ > 0,
x 2 S , x 6¼ p, and Vd ðPðxÞÞ Vd ðxÞ 0, x 2 U .
Next, define a function V : D ! R such that VðxÞ ¼
Vd ðsððxÞ, xÞÞ, x 2 D . It follows from the definition of
ðÞ, Lemma 3.1, and the joint continuity of solutions
of (1) that VðÞ is a lower semicontinuous function on
D and
VðxÞ ¼ 0, x 2 O, VðxÞ > 0, x 2 D nO, where
O D . Alternatively, it follows from the Lyapunov
stability of the fixed point x ¼ p of (4) that for > 0
such that B ð pÞ \ S U , there exists ¼ ðÞ > 0 such
that zðkÞ 2 B ð pÞ \ S, k 2 N , for all zð0Þ 2 B ð pÞ \ S,
where zðkÞ 2 N satisfies (4). Now, define D ¼ fx 2 D :
sððxÞ, xÞ 2 B ð pÞ \ Sg and O D . Hence, Vðsðt, xÞÞ Vðsð, xÞÞ, 0 t, for any x 2 D D . Now, to
show that VðÞ is continuous on O let pO 2 O be
such that pO 6¼ p and consider any arbitrary sequence
fxn g1
n¼0 2 D such that limn!1 xn ¼ pO . Then, since ðÞ
is continuous on D1 nH, it follows that limn!1 ðxn Þ ¼
ð pO Þ and, by joint continuity of solutions of (1),
sððxn Þ, xn Þ ! sðð pO Þ, pO Þ ¼ p as n ! 1. Now, since
Vd ðÞ is continuous at x ¼ p, it follows that limn!1
Vðxn Þ ¼ limn!1 Vd ðsððxn Þ, xn ÞÞ ¼ Vd ð pÞ ¼ 0
which,
since fxn g1
is
arbitrary,
implies
continuity
of
VðÞ
at
n¼0
any point pO 2 O, pO 6¼ p. Next, we show the continuity of VðÞ at x ¼ p. Note that VðÞ is not necessarily
continuous at every point of S but x ¼ p. Consider any
arbitrary sequence fxn g1
n¼0 2 D such that limn!1 xn ¼
p. For this sequence we have one of the following
three cases: either limn!1 ðxn Þ ¼ 0, limn!1 ðxn Þ ¼ T,
1
or there exist subsequences fxnk g1
k¼0 and fxnm gm¼0 such
1
1
1
that fxnk gk¼0 [ fxnm gm¼0 ¼ fxn gn¼0 , xnk ! p, ðxnk Þ ! 0,
as k ! 1, and xnm ! p, ðxnm Þ ! T, as m ! 1. We
assume the latter case, since the analysis for the first
two cases follows immediately from the arguments for
pO 2 O, pO 6¼ p, presented above. The characterization
of both subsequences and joint continuity of solutions
of (1) yield sððxnk Þ, xnk Þ ! sð0, pÞ ¼ p and sððxnm Þ,
xnm Þ ! sðT, pÞ ¼ p, as k ! 1, and m ! 1, respectively. Now, since Vd ðÞ is continuous at x ¼ p, it follows that limk!1 Vðxnk Þ ¼ limk!1 Vd ðsððxnk Þ, xnk ÞÞ ¼
Vd ð pÞ ¼ 0 and limm!1 Vðxnm Þ ¼ limm!1 Vd ðsððxnm Þ,
xnm ÞÞ ¼ Vd ð pÞ ¼ 0, and thus limn!1 Vðxn Þ ¼ Vð pÞ ¼ 0,
which implies that VðÞ is continuous at x ¼ p and
hence VðÞ is continuous on O. Finally, since all the
assumptions of Theorem 3.2 hold, the periodic orbit O
is Lyapunov stable.
To show sufficiency, assume that the periodic orbit O
generated by the point p 2 D is Lyapunov stable. Then it
follows from the Theorem 3.3 that there exists a lower
semicontinuous, positive-definite (on D0 nO) function
V : D0 ! R such that (7) is satisfied. Now, for sufficiently small > 0, construct a function Vd : S ! R
such that Vd ðxÞ ¼ VðxÞ, x 2 S . Thus, in this case the
sufficient conditions of Theorem 3.1 are satisfied for
Vd ðÞ which implies that the point x ¼ p is a Lyapunov
stable fixed point of (4).
(2) To show necessity, assume that x ¼ p is an asymptotically stable fixed point of (4). Then it follows
from Theorem 3.1 that there exists a continuous function Vd : S ! R such that Vd ð pÞ ¼ 0, Vd ðxÞ > 0,
x 2 S , x 6¼ p, and Vd ðPðxÞÞ Vd ðxÞ < 0, x 2 U , x 6¼ p.
Next, as in (i), construct the lower semicontinuous
function V : D ! R such that VðxÞ ¼ Vd ðsððxÞ, xÞÞ,
x 2 D ,
VðxÞ ¼ 0, x 2 O,
VðxÞ > 0, x 2 D , x 2
= O,
291
Poincare´’s theorem via Lyapunov’s second method
Vðsðt, xÞÞ Vðsð, xÞÞ, 0 t, for any x 2 D D ,
and VðÞ is continuous on O. Furthermore, for any
x 2 D define an increasing unbounded sequence
ftn g1
n¼0 such that t0 ¼ 0 and tk ¼ ðzðk 1ÞÞ, k ¼ 1,
2, . . ., where zðÞ satisfies (4). Then it follows from the
definition of the function VðÞ that Vðsðtnþ1 , xÞÞ <
Vðsðtn , xÞÞ, n 2 N , establishing that all the conditions
of the Theorem 3.2 hold, and hence the periodic orbit
O is asymptotically stable.
To show sufficiency, assume that the periodic orbit O
generated by the point p 2 D is asymptotically stable.
Then it follows from the Theorem 3.3 that there
exists a continuous, positive-definite (on D0 nO) function
V : D0 ! R such that (7) is strictly satisfied. Now,
for sufficiently small > 0, construct a function
Vd : S ! R such that Vd ðxÞ ¼ VðxÞ, x 2 S . Thus, the
sufficient conditions of Theorem 3.1 are satisfied for
Vd ðÞ which implies that the point x ¼ p is an asymptotically stable fixed point of (4).
œ
Theorem 3.4 is a re-statement of the classical Poincaré
theorem. However, in proving necessary and sufficient
conditions for Lyapunov and asymptotic stability of
the periodic orbit O, we constructed explicit Lyapunov
functions in the proof of Theorem 3.4. Specifically,
in order to show necessity of Poincaré’s theorem via
Lyapunov’s second method, we constructed the lower
semicontinuous (respectively, continuous), positive
definite (on D nO) Lyapunov function:
Theorem 3.4 presents necessary and sufficient conditions for Lyapunov and asymptotic stability of a
periodic orbit of the nonlinear dynamical system (1)
based on the stability properties of a fixed point of the
n-dimensional discrete-time dynamical system (4) involving the Poincaré map (3). Next, using a Lyapunov
function proof, we present a classical corollary to
Poincaré’s theorem that allows us to analyse the
stability of periodic orbits by replacing the nth-order
nonlinear dynamical system by an (n 1)th-order discrete-time system. To present this result assume, without
loss of generality, that ð@X ðxÞ=@xn Þ 6¼ 0, x 2 S , where
x ¼ ½x1 , . . . , xn T and > 0 is sufficiently small. Then it
follows from the implicit function theorem (Khalil
1996) that xn ¼ gðx1 , . . . , xn1 Þ, where gðÞ is a con
tinuously differentiable function at xr ¼ ½x1 , . . . , xn1 T
such that ½xTr , gðxr Þ
T 2 S . Note that in this case
P : U ! S in (4) is given by PðxÞ ¼ ½P1 ðxÞ, . . . , Pn ðxÞ
T ,
where
Pn ðxr , gðxr ÞÞ ¼ gðP1 ðxr , gðxr ÞÞ, . . . , Pn1 ðxr , gðxr ÞÞÞ: ð14Þ
Hence, we can reduce the n-dimensional discrete-time
system (4) to the (n 1)-dimensional discrete-time
system given by:
zr ðk þ 1Þ ¼ P r ðzr ðkÞÞ,
k 2 N,
ð15Þ
where zr 2 Rn1 , ½zTr ðÞ, gðzr ðÞÞ
T 2 S , and
VðxÞ ¼ Vd ðsððxÞ, xÞÞ,
x 2 D ,
ð11Þ
where the existence of the lower semicontinuous (respectively, continuous), positive definite (on S n p) function
Vd : S ! R is guaranteed by the Lyapunov (respectively, asymptotic) stability of a fixed point p 2 D of (4).
Alternatively, in the proof of sufficiency, Lyapunov
(respectively, asymptotic) stability of the periodic orbit
O implies the existence of the lower semicontinuous
(respectively, continuous), positive definite (on D0 nO)
Lyapunov function given by:
VðxÞ ¼ sup distðsðt, xÞ, OÞ,
x 2 D0 ,
ð12Þ
t0
and
ð1
VðxÞ ¼
0
sup distðsðt, sð, xÞÞ, OÞe d,
x 2 DA ,
t0
ð13Þ
respectively. Using the Lyapunov function Vd ðxÞ ¼
VðxÞ, x 2 S , we showed the stability of a fixed point
p 2 D of (4).
2
3
P1 ðxr , gðxr ÞÞ
7
6
..
7:
P r ðxr Þ ¼ 6
.
4
5
Pn1 ðxr , gðxr ÞÞ
Note that it follows from (14) and (16)
p ¼ ½ pTr , gð pr Þ
T 2 S is a fixed point of (4) if and
if pr is a fixed point of (15). To present the
result define S r ¼ fxr 2 Rn1 : ½xTr , gðxr Þ
T 2 S g
U r ¼ fxr 2 S r : ½xTr , gðxr Þ
T 2 U g.
ð16Þ
that
only
next
and
Corollary 3.1: Consider the nonlinear dynamical system
(1) with the Poincare´ return map defined by (3). Assume
that ð@X ðxÞ=@xn Þ 6¼ 0, x 2 S , and the point p 2 S gener
ates the periodic orbit O ¼ fx 2 D : x ¼ sðt, pÞ, 0 t Tg, where sðt, pÞ, t 0, is the periodic solution with
the period T ¼ ð pÞ such that sðð pÞ, pÞ ¼ p. Then the following statements hold:
(1) For p ¼ ½ pTr , gð pr Þ
T 2 S , pr is a Lyapunov stable
fixed point of (15) if and only if the periodic orbit O
is Lyapunov stable.
(2) For p ¼ ½ pTr , gð pr Þ
T 2 S , pr is an asymptotically
stable fixed point of (15) if and only if the periodic
orbit O is asymptotically stable.
292
W. M. Haddad et al.
Proof: (1) To show necessity, assume that pr 2 S r is a
Lyapunov stable fixed point of (15). Then it follows
from Theorem 3.1 that there exists a lower semicontinuous function Vr : S r ! R such that Vr ðÞ is continuous
at pr , Vr ð pr Þ ¼ 0, Vr ðxr Þ > 0, xr 6¼ pr , xr 2 S r , and
Vr ðP r ðxr ÞÞ Vr ðxr Þ 0, xr 2 U r . Define V : S ! R
such that VðxÞ ¼ Vr ðxr Þ, xr 2 S r . To show that VðÞ is
continuous at p 2 S , consider an arbitrary sequence
fxk g1
k¼1 such that xk 2 S and xk ! p as k ! 1.
Then, xrk ! pr as k ! 1 and, since Vr ðÞ is continuous
at pr , limk!1 Vðxk Þ ¼ limk!1 Vr ðxrk Þ ¼ Vr ð pr Þ ¼ Vð pÞ.
Hence, VðÞ is continuous at p 2 S . Similarly,
for the sequence defined above VðxÞ ¼ Vr ðxr Þ lim inf k!1 Vr ðxrk Þ ¼ lim inf k!1 Vðxk Þ, x 2 S , which
implies that VðÞ is lower semicontinuous. Next, note
that Vð pÞ ¼ Vr ð pr Þ ¼ 0 and suppose, ad absurdum,
there exists x 6¼ p such that x 2 S and VðxÞ ¼ 0.
Then, Vr ðxr Þ ¼ 0 and xr ¼ pr , which implies that xn ¼
gð pr Þ and x ¼ p, leading to a contradiction. Hence,
VðxÞ > 0, x 6¼ p, x 2 S . Next, note that
a contradiction. Hence, Vr ðxr Þ > 0, xr 6¼ pr , xr 2 S r .
Finally, using (14), it follows that
Vr ðP r ðxr ÞÞ Vr ðxr Þ ¼ VðP r ðxr Þ, gðP r ðxr ÞÞÞ Vðxr , gðxr ÞÞ
¼ VðP r ðxr Þ, Pn ðxr , gðxr ÞÞÞ VðxÞ
¼ VðPðxÞÞ VðxÞ
0,
xr 2 U r ,
ð18Þ
and hence by Theorem 3.1 the point pr 2 S r is a
Lyapunov stable fixed point of (15).
(2) The proof is analogous to (1) and hence is
omitted.
œ
4.
Conclusion
This paper presented a new proof of Poincaré’s theorem using Lyapunov’s second method. Specifically,
using stability of set notions, it constructed lower semicontinuous Lyapunov functions to provide a Lyapunov
function proof of the classical Poincaré theorem.
VðPðxÞÞ VðxÞ ¼ Vr ðP1 ðxÞ, . . . , Pn1 ðxÞÞ Vr ðxr Þ
¼ Vr ðP1 ðxr , gðxr ÞÞ, . . . , Pn1 ðxr , gðxr ÞÞÞ
Vr ðxr Þ
¼ Vr ðP r ðxr ÞÞ Vr ðxr Þ
0,
x 2 U ,
ð17Þ
and hence by Theorem 3.1 the point p 2 S is a
Lyapunov stable fixed point of (4). Finally, Lyapunov
stability of the periodic orbit O follows from
Theorem 3.4.
To show sufficiency, assume that the periodic orbit O
is Lyapunov stable. Then, it follows from Theorem 3.4
that the point p 2 S is a Lyapunov stable fixed point
of (4). Hence, it follows from Theorem 3.1 that there
exists a lower semicontinuous function V : S ! R
such that VðÞ is continuous at p 2 S , Vð pÞ ¼ 0,
VðxÞ > 0, x 6¼ p, x 2 S ,
and
VðPðxÞÞ VðxÞ 0,
x 2 U . Next, define Vr : S r ! R such that Vr ðxr Þ ¼
Vðxr , gðxr ÞÞ. The proofs of continuity of Vr ðÞ at
pr 2 S r and lower semicontinuity of Vr ðÞ follow
similarly as in the proof of necessity. Next, note
that Vr ð pr Þ ¼ Vð pr , gð pr ÞÞ ¼ Vð pÞ ¼ 0 and suppose,
ad absurdum, there exists xr 6¼ pr such that xr 2 S r
and Vr ðxr Þ ¼ 0. Then, Vðxr , gðxr ÞÞ ¼ 0 and x ¼ ½xTr ,
gðxr Þ
T ¼ p, which implies that xr ¼ pr , leading to
Acknowledgements
Research was supported in part by the National
Science Foundation under Grant ECS-9496249 and by
the Air Force Office of Scientific Research under
Grant F49620-001-0095.
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