Localizing attractors via a generalized La Salle principle

P E R G A M O N

MATHEMATICAL AND COMPUTER MODELLING

Mathematical and Computer Modelling 36 (2002) 1085-1098 www.elsevier.com/locate/mcm

Loca l i z ing A t t r a c t o r s v ia a G e n e r a l i z e d La Sal le P r i n c i p l e

M . S A B A T I N I AND L . C . T R U S S O N I Dipart imento di Matematica, Universitg degli Studi di Trento

1-38050 Povo, (TN), I taly <sabatini><trussoni>@science. unitn, it

(Received and accepted May 2002)

A b s t r a c t - - W e are concerned with plane differential systems of the form x = P(x, y), N = Q(x~ y), with P, Q analytic. We propose a formal-numeric method to localize the attractors and the repellers of the system. Such a method consists of looking for a power series solution to a PDE of the type

OV OV P ~ + Q ~ = p(V), with p is an arbitrary analytic function. When p(V) -- p(V - V2), p > 0, the attractors are contained in the set V = 1, the repellers in the set V = 0. @ 2002 Elsevier Science Ltd. All rights reserved.

K e y w o r d s - - L i m i t cycles, Liapunov functions, Polynomial systems, Computer algebra, Rayleigh equation.

1. I N T R O D U C T I O N

We are concerned wi th a t t r a c t o r s of two-d imens iona l different ia l sys tems of the t y p e

= p ( z , y), y = Q(x, y), (1)

where P and Q are ana ly t i c funct ions defined in some open subse t of the plane. Such sys tems

arise in several different se t t ings , mode l ing p h e n o m e n a appea r ing in the s t u d y of physical , as well

as biological , chemical , and economica l sys tems [1-5]. The ma in p rob lem connec ted to the s t u d y

of such models consis ts of giving a comple te descr ip t ion of the behav iour of so lu t ions as t ~ oc. In

general , th is is not possible, due to the complex i ty of the equa t ions and the p h e n o m e n a involved.

The a im of the qua l i t a t i ve t heo ry is to give an a p p r o x i m a t e desc r ip t ion of the behav iou r of the

sys tem, by ident i fy ing su i tab le regions of the phase space, where the solu t ions behave in a s imi lar

way. This is the case, for ins tance, of the region of a t t r a c t i o n of an a s y m p t o t i c a l l y s t ab le solut ion.

The s imples t so lu t ions t h a t can exhib i t such a behav iour are equ i l ib r ium po in t s and l imi t cycles.

W h e n a sys tem has an a s y m p t o t i c a l l y s tab le cycle F, the sys t em ' s behav iou r is local ly d o m i n a t e d

by F: every o ther solut ion, close enough to F, even tua l ly a p p r o x i m a t e s it. Th is means t h a t the

phys ica l sys t em mode l l ed by (1) has a na tu r a l osci l la t ion mode t h a t is even tua l ly a p p r o x i m a t e d .

however one chooses the ini t ia l condi t ions , p rov ided t hey belong to the a t t r a c t i o n region of F.

Local iz ing l imi t cycles is not a s imple p rob lem. Even for s imple sys tems , as those ones ar is ing

from second-order scalar different ia l equat ions , several technica l difficulties have to be overcome

0895-7177/02/$ - see front matter @ 2002 Elsevier Science Ltd. All rights reserved. Typeset by AA/~-TF~X PII: S0895-7177(02)00260-1

1086 M. SABATINI AND L. G. TRUSSONI

in order to rigorously prove the existence of limit cycles. A classical example, the generalized Li6nard equation,

+ f(z).~ + g(z ) = 0, (2)

is still object of study, in spite of being associated to a differential system which is linear in one variable,

Jc = y, ~ = - g ( z ) - y f ( x ) . (3)

A typical approach consists of proving that the origin O is a repeller contained in a bounded at t ractor A. By Poincar6-Bendixson theory [6], if O is the unique equilibrium point contained in A. then A contains a limit cycle. The situation is more difficult if O is not the unique critical point in A. In this case, the solutions can be a t t racted to a 9eTzeralized limit cycle, which is a path consisting of finitely many critical points and connecting orbits. In both cases, one has only

a rough knowledge of the location of the (generalized) limit cycles. All this motivates the a t tempts to find approximation methods that allow" us to give an insight

into the structure of the limit sets of (1). Recently, after the appearance of formal-numerical packages that allow a combined symbolic and numerical manipulation, new approaches have been proposed. This was done in particular in order to look for limit cycles of polynomial systems. Ill !7], it was shown that the limit cycles of plane systems are contained in the zero-level set of an inverse integrating factor (IIF) that is a solution to the partial differential equation

1 2 = P - o ~ z + Q - ~ y = V. (4)

Under suitable hypotheses, an analytic I IF exists, so tha t it can be approximated by polynomials. The zero-level sets of such polynomials approach the zero-level set of the IIF, hence the limit

cycle of the system. In this way, one can approximate the limit cycles of (1) by means of algebraic curves. In [s,gJ, a different approach to find and count limit cycles was proposed. Such a method seems to be more appropriate in case of multiple limit cycles, but is restricted to the special class

of Li~nard equations with f ( - z ) = - f ( z ) and g(z ) = z. In [10], it was shown that , under mild additional hypotheses, an I IF vanishes on separatrices, so opening the possibility to approximate generalized limit cycles by zeroes of inverse integrating factors.

In this paper, we propose an alternative approach that consists of replacing equation (4) with

an equation of the form OV OV

where # is an arbi t rary analytic scalar function. We prove tha t if V is a solution to (5), then V is monotone along every solution of (1). As a consequence, the positive (negative) limit set of a positively (negatively) bounded solution is contained in the set {1) = 0} = {#(V) = 0}. This occurs without requiring that the monotonicity character be the same on all of the solutions.

V can be increasing along some solutions, and decreasing along other ones. This can be seen as a generalized version of La Salle invariance principle for Liapunov functions, which, in the original formulation, requires V to have the same monotonicity character on all of the solutions [3,6]. Under hypotheses similar to those assumed in [7], we prove that an analytic solution to (5) exists, so tha t it can be approximated by polynomials. Then we approximate the set {#(V) = 0} by means of suitable level sets of the approximating polynomials. As an example, we can choose #(s) p(s s2), 0 ¢ p E R, that leads to the PDE

_ _ _ _ o r ov (v v 2) (6) P o x + Q a y = p -

The set 1) = p ( V - V 2) = 0 consists of the points where V = 0 or V - 1. A more detailed analysis leads us to conclude tha t if p > 0, then {V = 0} contains the repellers of (1), while

Localizing Attractors 1087

{V = l} contains the attractors of (1). If Vd, d > 0, are the approximating polynomials. then

we can approximate both {V = 0}, and {V = l} by means of the algebraic curves {Vd = 0}

and {Vd = l}, respectively. This occurs both in the case of a limit cycltl, and in the case of a

generalized limit cycle, without the additional hypotheses required in [lo].

Our method seems to give better approximations at low degrees than the IIF method. In fact,

studying Van der Pol’s equation, we can give an eight-degree algebraic curve that gives as good an

approximation as the 32-degree algebraic curve found in [7], which is a substantial computational

improvement, since it reduces the number of coefficients potentially involved by a factor 10. The

presence of better approximations depends both on the choice of a different, PDE and on the

presence of a nonmultiplicative free parameter appearin g in our polynomials that can be used in

order to optimize the approximation.

We apply our method to study attractors of Van der Pol’s system, of a Rayleigh equation. non-

linear both in IC and in i. and of a perturbed Hamiltonian system with an attracting homoclinic

loop. We compare our results to those ones obtained via the IIF method.

We emphasize that our method is not bound to two-dimensional systems. The theory exposed

in the next section can be extended to finite-dimensional spaces. In a forthcoming paper. we

shall be concerned with applications to three-dimensional systems and t,heir attractors [ll].

Finally, we observe that, as in the case of [7]! our method is well fit for parametric approxima-

tions.

2. THE GENERALIZED INVARIANCE PRINCIPLE

Let

z = F(z), zEUCRn. FEC’(U,EP), (7)

be an autonomous differential system of class C’ in an open subset U of the n-dimensional

Euclidean space. Let us denote by r(t, Z) the unique solution of (7) satisfying ~(0, Z) = L, and

by I, c IF?, its maximal interval of existence. We shall write y(z) for the orbit {~(t, Z) : t E IZ}.

r+(z) for the positive semiorbit {y(t,z) : t E IZ, t > O}. and y-(z) for the negative semiorbit

{~(t.:) : t E I,, t 5 0). W e d o not require that the solutions of (7) be defined for all values of t.

We recall that positively bounded solutions are defined for all t 2 0. and negatively bounded

solutions are defined for all t < 0. If y(z) is a positively bounded orbit,. then its positive limit

set W(Z) is defined as follows:

W(Z) = n Y’(W)! WE?(Z)

where B is the topological closure in Ps” of a set B c IP. A point w belongs to d(z) if and only

if there exists a sequence t, --$ +oc such that lim,,, y(t,, Z) = PU. The negativtl limit set cr(-_)

is defined similarly, in the case of a negatively bounded orbit.

DEFINITION 1. We say that a function V E C’ (MT IR) 1s a Liapunov function for 17) on the open

set M c lRn if and only if, for every to such that y(to, 2) E M, t H V(y(t, z)) is monotone in a

neighbourhood of to.

If y(t, 2) is entirely contained in M, then V is monotone on all of y(t. 2). On the other hand, if

y(t, z) is not entirely contained in M, our definition ensures that V is monotone on the connected

components of y(t. z) that are contained in M.

We emphasize that we do not require that the character of monotonicity is the same for all

solutions. We admit the possibility to have solutions along which V increases, and solutions along

which V decreases. This is unusual, but fit for our applications.

If the total derivative of P(Z) = [& V(y(t, z))]t=~ of V(y(t, 2)) w.r.t. the time t does not change

sign on -y(t. z)> whatever y(t: 2) intersecting M, then V is a Liapunov function for (7) on M. As

is well known, in order to compute P(Z) it is not necessary to know the solutions of (7)! since

V(Z) = T7V(z) F(Z), where denotes the scalar product in Iw”. Due to this. Liapunov functions


have become the main tool for the s tudy of the stabili ty and /o r a t t rac t iv i ty of solutions. In

general, theorems about stabili ty and Liapunov functions require some local sign properties both

on V and I/ [6]. This is not the case of La Salle inva~ance principle, which only requires a sign

condit ion on V.

We state and prove here a version of the invariance principle adap ted to the definition of

the Liapunov function tha t we have chosen. It is different from the version usually reported in

tex tbooks [3,6], but it is based on the same principle.

THEOREM 1. GENERALIZED LA SALLE INVARIANCE PRINCIPLE. Let M C U be open and

assume that 1/ c C~(M, R) is a Liapunov function for (7) on M. I f T(t, z) is positively bounded

and w(z) C M, then

The same holds for the a-l imit set of a negatively bounded orbit.

PROOF. If w C w(z) is a critical point of (7), then l)(w) = 0, and hence, we can assume

tha t w c w(z) and F(w) ~Z O. By definition, there exists a sequence tn E R, t~ ~ +oo,

such tha t w = l im~_.~ y(t~, z), where 7(t, z) is a positively bounded solution. Since V('~(t, z))

is mono tone - - say , decreas ing- -we have limt-~+o~ V('y( t , z ) ) = inft_>0 V ( 7 ( t , z ) ) = L. Hence, V(w) = L. Consider the solution s tar t ing at w. By s tandard arguments , one can prove tha t it is

defined for all t > 0. Then V(~/(t, w)) = l i m ~ o o V(7( t + t , , z)) = L. Hence, V(~/(t, w)) - L, so

tha t Ii~(y(t, w)) = 0. In particular, l )(w) = 0.

The proof for the a- l imit set proceeds in the same way.

The following corollary is an immedia te consequence of the above theorem. We recall tha t an

a t t rac tor is a set A having a ne ighbourhood UA such that , for every z E UA, 0 ¢ w(z) C A. The

definition of repeller is obtained similarly, by taking c~-limits. The next corollary is an immedia te

consequence of the theorem. We denote by OA the boundary of the set A.

COROLLARY 1. Let the hypotheses of the above theorem hold. I f A C UA C M is an attractor,

then

. : o l OA

In the case of plane systems, we can say something more. We recall tha t a limit cycle F is an

isolated cycle. In this case, all the solutions close enough to F either tend to F as t --* +oo, or

tend to F as t ~ - o c . It is possible tha t a limit cycle is a repeller on one side, and an a t t rac tor on the opposi te side. In other words, it can s imultaneously be an w-limit and an a-l imit . By

Poincar~-Bendixson theory, a t t rac tors of plane systems can also be critical points, or generalized

cycles: pa ths consisting of critical points and homoclinic or heteroclinic orbits [6].

COROLLARY 2. Let n = 2 and the hypotheses of the above theorem hold. I f F is a limit cycle or

a generalized limit cycle, and V is a Liapunov function defined in a neighbourhood of F, then

F C { w E M : l k ( w ) = 0 } .

For a scalar function # : R ~-* R and V : M -4 R, we set

z . v = {z E M : , ( V ( z ) ) = 0}. (8)

Let us now consider a special class of Liapunov functions, defined as solutions to the P D E

= (9)


THEOREM 2. Let # : R ~-* R be Lipschitzian and V be a solution of (5). Then, for every z E M such that 0 ¢ w(z) C M, we have

aJ(z) C Z , v . (10)

It is shnilar for the a-limit sets.

Let us first observe t h a t Z , v is invariant; t ha t is, solutions s ta r t ing at a point of Z , v do not

leave it. In fact, let 3'(t, z) be a solution of (7). The funct ion ~(t) = V(~/(t, z)) is a solut ion to the O D E fl = p(r/), which has uniqueness of solutions. I ts equi l ibr ium points are the zeroes of p.

H e n c e , if V(3'(0)) = 77(0) = So, wi th p(s0) = 0, then V(~/(t, z)) = rl(t) = so in all of its interval of existence.

By the cont inui ty of # and V, this implies t h a t l ) (z) = p (V) cannot change sign along a solution: if l?(z) > 0 ( '~(z) < 0), then V(y( t , z)) > 0 (~)(~(t, z)) < 0) in all of the interval of existence of 3@, z). Hence, V is a L iapunov funct ion for (7) on M.

T h e n the s t a t e m e n t comes f rom T h e o r e m 1.

By T h e o r e m 2, in order to find the a t t r ac to r s and repellers of (7), we can look for a solution of (5), and s t udy the zero-level sets of p (V) . In some cases, this gives us also some informat ion abou t the s tabi l i ty of the connected componen t s of {#(V) = 0}. Let us set Us(so) = {s ~ R :

s 0 - 6 < s < s 0 + 5 , s C s 0 } a n d V - l ( s 0 ) = { w E M : V ( w ) = s 0 } .

COROLLARY 3. Let the hypotheses of the above theorem hold, and So be an isolated zero of p. Let [; > 0 be such that so is the unique zero o f # in (so - 5, so + 5). We have

(1) i f ( s - so)ix(s) < 0 for all s C Us(so), then every compact connected component of V- ! (So) is asymptotically stable;

(2) i f ( s - so)ix(s) > 0 for ali s E Us(so) .4~.then every compact connected component of V - l ( s o ) is negatively asymptotically stable.

PROOF.

(1) It is sufficient to use (V - s0) 2 as a Liapunov funct ion for the set V- l ( so ) .

(2) I t is sufficient to use - ( V - so) 2 as a Liapunov funct ion for the set V l(s0).

T h e above corol lary allows us to dist inguish between a t t r ac to r s and repellers, provided one has a solution to a sui table equat ion of type (5). This is an advantage over the m e t h o d presented

in [7], in which a t t r ac to r s and repellers cannot be dis t inguished from each o ther by only s tudying the level sets of the I IF.

On the o ther hand, (4) is a linear PDE, while our me thod requires us to consider nonlinear

PDEs. This leads to heavier computa t ions , seemingly with be t te r app rox ima t ions at low degrees.

T h e procedure we follow is developed much in the same spirit as in [7]. In the following, we 0 ( 3 shall prove t h a t under mild hypotheses there exists an analyt ic solution to (5), V = ~ d = 0 Vd,

where the Vd are homogeneous polynomials of degree n. Equa t ion (5) forces the coefficients of Va to sat isfy a family of algebraic equat ions. We can solve recursively such equat ions , proceeding by increasing degree. In this way, we can de te rmine the polynomials Vd up to a given degree d, so obta in ing a t runca t ion of the solution V of (5). Then we consider a sui table level set of the

d t runca t ion ~d=O Vd, in order to get an approx imat ion of the desired level set of V.

In order to prove the existence of an analyt ic solution to (5), we first prove a pre l iminary lemma. Let us consider a scalar m a p ¢(s) = s + 0 ( s ) , defined in an open interval containing the origin.

LEMMA 1. Let ~(s) be analytic, with [~(s)l _< ks 2, k E R, in a neighbourhood ors = O. Then

( ~o ~ ~(s) d@ (11) /3(s) = sexp - s 2 + s ~ b ( s )

1090 M. SABATINI AND L. G. TRussoM

is analytic and locally invertible at O. Moreover,/3'(0) = 1, and

9'(s)- ~(s) ¢(s)'

for all s such that C(s) ¢ O.

P a o o F . If IO(s)l _< ks 2, then the integral in (11) converges, and hence, 3(s) is well defined in a ne ighbourhood of zero. An elementary computa t ion gives/3~(0) = 1. Moreover, we have

~b(s) dcr = exp ~;(s) da 3(s) = s e x p - s 2 + s ¢ ( s ) s 2 + s ~ ( s )

(12)

= exp da = exp dcr, s + ~ ( s )

tha t giv(~s J ( s ) = 9(s ) / ( ( s ) , whenever C(s) ¢ 0.

TUEOaEM 3. Let 0 be a critical point of (1), A1, A2 be the eigenvalues of the linear part of (1) at O. Assume that AIA2 ¢ 0 and the ratio A1/A2 is not rational. Assume that p(s) = p(s + ~(s))

is analytic, where p is either 2A1, or 2A2, or A1 + A2, and [¢(s)] < ks 2, k E R. Then equation (5) has an analytic solution in a neighbourhood of O.

PaOOF. If A1A2 ¢ 0 and A~/A2 is not rational, by the Poincar~ theorem [1, Section 24] system (1)

is analyt ical ly equivalent to its linear part. Let (u, v) = A(x, y) be the linearizing t ransformat ion, with inverse (x, y) = A - l ( u , v). In the new coordinates, the new sys tem appears as

i z = a u + b v , 9 = c u + d v , a ,b ,c , d E R . (13)

Let J be the coefficients matr ix of (13). Its eigenvalues are A1, A2. Let us consider a quadrat ic

homogeneous funct ion with a rb i t ra ry coefficients, W(x , y) = r, lX 2 +~2xy+ L'3y 2. Let us determine

its coefficients ~ so tha t W = ~cW, where W is the derivative w.r.t, the t ime of (13) and ~ is a

real constant . We obtain a linear system with coefficients matr ix

2b a + d - ~ 2c , (14)

0 b 2d -

whose de te rminant is (a + d - ~)(ec 2 - 2(a + d)~ + 4 ( a d - bc)) = ( t r ( J ) - ~)(~ - 2A1)(~ - 2A2).

Hence, there exists a nonzero W satisfying W = ~ W if and only if ~ assumes one of the three values 2A1,2A2, t r (J) = A1 + A2. Since p assumes one of such values, there exists a quadrat ic

function W such tha t W = pW.

Now consider the function W1 = A - I ( W ) . W1 is an analyt ic function defined in a neighbour-

hood of the origin, such tha t l)dl = pW1, where l?dl denotes the derivative w.r.t, the t ime of (1).

Let us set ((s) = (1/p)p(s). We have ((s) = s + ¢(s ) , with I%s)l _< ks 2. Let us define 3(s) as

in the previous lemma, and set V = 3-1(W1) . We claim tha t V is the required solution to (5). In fact, (5) is trivially satisfied at O. For

(x, y) ¢; (0, 0), consider tha t we have W1 = / 3 ( V ) and

,o/~(V) : / o W 1 : l/g 1 ~ - /~ ' (V) l~ - ~ ( V ) y (15) -Vv5 '

where all the derivatives are made w.r.t, the t ime of (1). Since O is an isolated zero of V, and 0 is an isolated zero of ~(s), we have 1) = p£(V) = p(V) in a ne ighbourhood of O.

REMARK 1. The conditions on p are sufficient, but not necessary, as it can be shown by examples. Moreover, the above theorem not only gives the existence of an analytic solution, but also proves


tha t a solution V s ta r t ing with nonzero quadrat ic par t exists. In fact, if O is a (negative)

a t t ractor , then every solution obtained by the procedure of Theorem 3 is definite in sign in a

ne ighbourhood of O. Such a solution is not unique, since in the proof of thd above theorem we

could replace the funct ion W with TW, ~- E IR. For every ~- E R, we would get a different analytic

solution V(~-) to (5).

We emphasize tha t the above theorem does not hold only in the case of an a t t rac t ing critical

point O, but whenever the eigenvalues satisfy the given conditions. For instance, this is the case of the Hamil tonian system

: - x , 9 : Y. (16)

The eigenvalues of the linear par t at O are t l = 1 and 12 = - 1 . If we choose p = 2 t l = 2, then

the funct ion W(x, y) of the proof of Theorem 3 is y2. If we choose p = 2£2 = - 2 , then we have

W(x, y) = x 2. If we choose p = 11 + A2 = 0, then we have W(x, y) = .rg, which is a first integral.

Hamil tonian systems have no at t ractors , and hence, in this case the set {I%'(:r, y) = 0} does not

contain any.

If O is (negatively) asymptot ica l ly stable, then we can say something more about the definition

domain of an analyt ic solution of (5).

COROLLARY 4. Let 0 be (negatively) asymptotically stable. Then V can be extended to ali of the region of (negative) attraction of O.

PRooY. If V is defined at a point of a solution y(t, x, y), then it can be analyt ical ly extended to

all of ^/(t, x, y) as the unique solution of the scalar O D E ~t V(7(t, x, g)) = / ~ ( V ( ? ( t , x, :~j))). Such

a solution depends analyt ical ly on ( t , z , y ) . By Theorem 3, an analyt ic solution V exists in a

ne ighbourhood Uo of O. Let Ao be the region of a t t rac t ion of O. Since for every (z, Y) c Ao, the orbit "y(t, x, y) meets [70 for some value of t, V can be extended to all of Ao.

In some cases, V exists on all of R 2, as for the following system:

~ ? : y - x ( x 2 + y 2 - 1 ) , ~ ) : - x - g ( x 2 + y 2 - 1), (17)

which has a negatively asymptot ica l ly stable point at O. Its region of uegadve a t t rac t ion Ao is the circle {x 2 + y2 < 1}, whose boundary OAo is a limit cycle. Choosing p = 2, the function

V(x, y) = x 2 + y2 satisfies equat ion (6), is defined on the whole plane, vanishes at O, and assumes

the value 1 on the limit cycle. This is consistent with Theorem 2, since in this case the set, the set

{~) = 0} = {p(V - V 2) = 0} splits into the two subsets {V = 0} and {V = 1}. From Theorem 2 we know tha t {V = 0} contains the repellers of (17), while {V = 1} contains its a t t ractors .

In general, equat ion (6) is our best choice, because it the simplest one in the class of nonlinear

equations. W h e n we consider second-order scalar ODEs with a repelling critical point O and an

a t t rac t ing limit cycle, V will be zero at O and 1 on the limit cycle. If OAo is a generalized cycle F.

V will be zero at O and 1 on F. Anyway, in general we cannot expect tha t V extends out of Ao, since p(V - V 2) satisfy the hypotheses of Corol lary 3, so tha t every componen t of {V = 1} has to

be an a t t rac tor . This means tha t if OAo is semistable, then V cannot be extended beyond OAo. In this case, a different P D E should be chosen. For instance, a P D E suitable for semistable sets

is

17 : p V ( V - 1) 2. (18)

In this paper, we shall not consider this case.

3. T H E A L G O R I T H M

We will now describe the procedure we apply in order to cons t ruc t the approximat ing polyno-

mials Vd. It consists of two main steps, the first one algebraic, the second one numerical.


We assume P and Q to be analytic, and we replace them with their Taylor expansion up to a convenient degree,

so that we actually work with polynomials. We assume the origin to be a repeller, and the system (1) to satisfy the hypotheses of Theorem 3.

We work on equation (6), that we recall here,

= ( v -

We (:an integrate it by separation of variables, so to have

vo V(7(t , zo)) = Vo + (1 - Vo)e-pt' (20)

where 7(t, zo) is a solution to (1) and V0 = V(zo). This gives a different proof of the fact that , if p > 0, then V(zo) ¢ 0 implies limt~_oo V(7(t , z0)) = 0, l i m t - ~ V(7(t , z0)) = 1, when such limits exist. The existence of such limits depends obviously on the existence of V. By Theorem 3, if

we choose p among the values 2),~, 2A2, ~ + ~2, then (6) has an analytic solution V = ~ - = 0 Va, where the V~ are homogeneous polynomials of degree d. Equation (6) imposes algebraic conditions on the coefficients of P, Q, Vd. We want to determine the unknown coefficients of Va start ing

from the given coefficients of P and Q. Let us consider a fixed degree d. For the sake of simplicity, we write Z for Va,

Z = E z i , j x i y j (21) 2<i,j<d

Z has no zero and first degree terms because if the origin is a repeller, then our V has to satisfy V(0,0) = 0 and V(x , y ) ¢ 0 near the origin (see Remark 1). Let us now consider the new

polynomial

• o z o z ( z z 2) w ,Sy =2-p(z-z2)=P +ON-p - i+j<_2d

(22)

Notice that the degree of W(d, x, y) is greater than d. The coefficients wi, j of W can be regarded as functions of the coefficients of Z, P, and Q. Since Z - Vd is the Taylor expansion of V up to degree d, the terms of W, up to degree d, are identically zero. Hence, we can determine the

coefficients of Z by solving the system of algebraic equations,

S = {w~,j =0}~d, O < i , j , i + j <_ d. (23)

Let. us consider the equation wi,j = 0, for H = i + j < d,

E [(h + 1)Zh+l,kP/,m + (k + 1)Zh,k+lqt,m] -- pzi,j -- p E h+l=i h+l=i

k+m=j k + m = j

Zh,kZl,,~ = 0. (24)

Since z0,0 = 0, this equation is linear in Zh,k, with h + k = H. It is quadratic in Zh,k, with h + k < H. Moreover, since P0,0 = q0,0 = 0, it does not contain zh,k, with h + k > H. This allows us to solve the system S by "blocks" SH = {w~ a = O}~+j=H<_d, in a recursive way. We first

solve $2, finding z2,0, z1,1, z0,2; then we solve $3, in which z2,o, z1,1, zo,2 are no longer unknowns,

but coefficients, and so on, by increasing degree.


Let us consider the block $2, whose coefficient matr ix can be obta ined from (14) by sett ing

a = pl,0, b = P0,1, c = qL0, d = q0,1. Under the hypotheses of Theorem 3, such a linear system

has Rank 2, so we can always determine a triplet (f2,0, 51,1, ~0n) ¢ (0, 0, 0) such tha t

Z = T52,0 x2 + 721,1xy + 720,2Y 2 + .- . ,

where 7 is a free parameter .

Also observe tha t in (24) the coefficients of the unknowns zh,k, h + k = H, are constants

(depending on the coefficients of P and Q), while the known term is made of a sum of products

of t hezh ,k , h + k < H, with coefficients of P and Q or other zh',k', h ' + k ' < H. This ensures

t ha t the zh,k, h + k = H are determined as polynomials of the kind 72h,k(T, . . . ) , where 7 is

the parameter coming from the resolution of $2. Consequently, such a ~- divides the whole

polynomial Z, t ha t from now on will be denoted by Z(T). In all the cases we examined, the de terminant of SH, for H > 2, has been found to be different from zero. This is consistent with

the form of (20), which depends on a single parameter .

This concludes the algebraic par t of the procedure. Now we have a polynomial Vd(7) -- Z(7) of degree d, depending on a parameter 7. For every 7, it represents a t runca t ion of an analytic

solution V(T) of the P D E (6) (see Remark 1), and the value of the parameter T determines which

solution, among the existing ones, we are approximating. We want to find, for any degree d,

the value of 7 tha t gives the best approximation. This task is achieved by es t imat ing Wd(7) = ~/'d(7) -- p ( V d ( T ) -- Vf(T)), t ha t gives a measure of how much Vd is not a solution to (5). Let us

consider the set where IWd(~r)l < ~,

f/d(~-) = { (x ,y) E R 2 s.t. Wd(T)(x,y) C [ -~ ,~]} . (25)

f~d(7) is the "good" set, from our point of view. We are interested in taking 7 such tha t i~d(r) is

as large as possible, but we are not actual ly interested in considering those parts of ~ d ( T ) which

are too far f rom the a t t ractor . Since a rough est imate of the region where the a t t rac tor lies is

possible, based on a classical phase-plane analysis [2], we may work in a suitable bounded region B

containing O. Additionally, in the region of negative a t t rac t ion of O, we have 0 _< V(x, y) < 1,

so we may consider only the set Vd(x, y) E [0, 1 + rl], where r~ > 0, which takes account of possible

over-oscillations of Vd. We do not consider possible under-oscillations of Vd, because Vd will be

taken positive definite at O, and increasing along solutions. Hence, we consider also the set

% ( r ) = { (x ,y) • R2 s.t. 0 < Vd(7)(x,y) < 1 + ~} , (26)

t ry ing to choose 7 so to maximize #(fld("r) C3 ~d(7)), in relation to # (~d(7 ) ) , where # is the

s tandard Lebesgue measure. The value 7min will be found minimizing the following function:

E d = 1 - n (27)

Since we work on the bounded set B, we actual ly perform our est imates on the sets ~d = /IJd C'i B ,

~=~d = ~ d FI B. The value Wmi, will be chosen among a discrete set of values, as t ha t one minimizing the quant i ty

d(7) = 1 n ad) (2s)

# ( ~ d N ~d) and # (~d) are evaluated on a s tandard rectangular mesh on a suitable B = [a, b] x

[a', b']. For every 7 we have obvious ly / )d --< 1. For all the systems we consider in next section, the

quant i ty /~d takes values close to 1 for very small values of 7, then reaches a local min imum 7,~,

then increases again, approaching what seems to be an asympto t ic value. In the next section, we produce plots of the level 1 curves approximat ing the a t t rac tors of the

system considered, and compare them both to R u n g e - K u t t a approximat ions , and to the algebraic approximat ions obta ined via the I IF method.


4. E X A M P L E S

In this section, we apply the procedure developed in previous sections to approximate the

attractors of' three distinct plane differential systems. We start with Van der Pol system.

5: = y, '~) = - x + !1 (1 - z2) , (29)

which is a sort of benchmark.

In Figure la, we show the eight-degree approximation obtained using our method. In dotted

lines we plot the limit cycle, obtained via a Runge-Kutta method (starting point at (x0,Y0) =

(0.1,0.1), time from 20 to 27), while in solid lines we plot the level 1 set of our polynomial Vs. In

Figure lb, we show the 32-degree approximation obtained using the IIF method. In dotted lines

we plot the same limit cycle of the previous figure obtained via the Runge-Kutta method, while

in solid lines we plot the level 0 set of the 32-degree polynomial approximating the IIF.

,/ , / " y

I ~ 1

' - 1 0

-3

\ 3

-3 \

(a) (b)

Figure 1.

The figure highlights the fact that we obtain a very precise approximation at low degree, which

is important because low degree polynomials are simpler and faster to handle, and save a large

amount of memory. In Figure 2, we plot the error function /)s, with B = [-3, 3] x [--3, 3], and parameters 77 = 0,

e = 0.1. The measures of sets involved in the calculation have been obtained by evaluating the

truth functions at the crosspoints of a 26 x 26 points mesh. The plot appears rough because of

the roughness of the mesh, but the minimum of the error function is quite evident.

In Figures 3a and 3b, we consider the system

11 9 y3 (30) 2 = y , ~ ) = - - - x + x 2 - x 3 + y - 2

obtained from a Rayleigh equation not reducible to a Li~nard equation by differentiating with

respect to the time. Such an equation, different from the Van der Pol system, is nonlinear both

in x and in 5:. In Figure 3a, we plot in dotted lines a Runge-Kutta solution of the system (starting

Localizing Att ractors 1095

0.54

0.52

0.5

0.48

0.46

0.44

0.42

0.4

0.38

0.36

0.34

0,32

0.3

0.28

0.26

0.24

0,22

0.2

0.18

0.16

1.2 , J

1 / /

0.8

" YO.6

0.4

0.2

~ . 6 ~ . 4 1 r~ .2 ' 10

-0 .2

-0.4

- 0 . 6 "

~ -0.8

-1

-1.2

\ \

' , - , , ! , , - i . . . . . . . . L - , i , , i • , , - , - , , ' - - r l " ' , - - H , . . . . . . . . . I ' , . . . . = . . . . . I ~

0.02 0.06 0.1 0.14 0.18 0,22 0.26 0.3 0.34 0.38 tau

Figure 2.

/

' . [

0 : 2 r 0:4 x

/

/

0.2

i=-0.6 ~ .4 -0.2 0 -0 .2

-0.4 =, i,

: -0 .6

; -0.8

0.8 q YO.6

0.4

0.2

-1.2

0.4 0i6 x

s

(a)

Figure 3.

(6)


point (x, !1) = (0.1, 0.1), t ime f rom 15 to 18), and in solid lines the approx ima t ion at degree eight obta ined via our procedure. In Figure 3b, we plot the same R u n g e - K u t t a solut ion and in solid lines the I I F approx ima t ion at degree 20.

Again, our p rocedure supplies a more precise approx ima t ion at a lower degree, since the I IF

approx ima t ion does not seem to produce a closed curve. We have examined the I1F approx imat ion on a larger por t ion of the p l a n e - - w e do not repor t it since it would be difficult to read it - a n d

still the two branches appear open. In Figure 4, we plot the error function, with B = [ -3 , 3] x [ -3 , 31, and pa rame te r s r/ = 0,

e = 0.1, and again the plot is quite rough, but the m i n i m u m is evident. In Figure 5, we have the approx imat ion of an homoclinic orbit , belonging to the pe r tu rbed

Hami l ton ian sys tem

0.4

0.3

0.2

0.1

0 0.2 0.4 0.6 . . . . 0 ' 8 ~ tau

Figure 4.

-0.), :

Figure 5.

Localizing Attractors 109;

:\

I

0.38

0.36

0.34

0.32

0.3

0.28

0.26

0.24

0.22

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

7

0 1 2 3 4 5 6 7 tau

Figure 6.

j: = -2y, $ = 2x - 3X2 -y x2+yZ-23-; ( )

(31)

In solid lines, we plot our approximation at degree 10, while in dotted lines a Runge-Kutta

solution is shown (starting point (~0, yc) = (O.l,O.l), time from 5 to 10). In Figure 6, we plot

the error function Eic, with B = [-0.5, l] x [-0.5,0.5], and parameters n = 0, E = 0.1. Here,

we do not plot the IIF approximation at degree 10, because it is virtually coinciding with our

curve. In this case, the two methods are equally efficient. The attractor is the algebraic curve

x2 + y” _ x3 - 4/27 = 0, which is the level set of the Hamiltonian x2 + y2 - r3 passing through

the point (2/3,0). This example suggests that probably both methods could be improved, since

approximating an algebraic curve should be easier than approximating a possibly nonalgebraic

limit cycle. On the other hand, it is possible that the approximation is slow because one is trying

to approximate a curve with a singularity at (2/3,0) by means of nonsingular curves.

We point out the fact that in all the examined cases, the heaviest calculation was the one

related to the error function. Using low degree polynomials and a rough mesh, our procedure is

fast enough to be performed on a standard PC. In all the plots, the minimizing parameter to be

substituted in the approximation has been deduced from the graphs, showing that the procedure

achieves good precision with rough calculations.

The calculations have been performed using the Maple program from Waterloo Research Inc.,

both on a Compaq DS20 with Digital Unix and on a standard Intel Pentium II PC.

5. CONCLUSIONS

We propose a new method for the formal-numerical approximation of attractors of autonomous

ODES. Such a method is based on the construction and truncation of suitable Liapunov func-

tions V, implicitly defined as solutions to nonlinear PDEs. The abstract framework is that of

the La Salle invariance principle, valid in arbitrary finite-dimensional spaces. In this paper, it is


applied to s tudy two-dimensional systems, looking for limit cycles and generalized limit cycles. In order to localize the at tractors of polynomial plane systems, we use the simple PDE

ov ov (v v 2) - .

We give polynomial approximations of the limit cycle of Van der Pol's equation and of a fully nonlinear Rayleigh equation. We also consider a perturbed Hamiltonian system with a homoclinic

attractor. The polynomial truncations we find give a bet ter approximation at low degrees than

those given in I71. The approximations we find seem to converge to the limit sets of the systems studied, even in the case of homoclinic or heteroclinic orbits. Our method can as well be applied to parametr ic systems, generating parametric truncations and approximations.

R E F E R E N C E S

1. V.I. Arnold, Geometrical methods in the theory of ordinary differential equations, Grund. Math. Wiss. 250, Springer-Verlag, New York, (1983).

2. L. Cesari, Asymptotic behaviour and stability problems in ordinary differential equations, Ergeb. Math. Crcnzgeb. 16, Springer-Verlag, Heidelberg, (1963).

3. N Rouche, P. Habets and M. Laloy, Stability theory by Liapunov direct method, Er9eb. Math. Grenzgeb. 16, Springer-Verlag, Heidelberg, (1977).

4. S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, (1994). 5. L. Edelstein-Keshet, Mathematical Models in Biology, Random House, New York, (1988). 6. H. Amann, Ordinary differential equations, In de Gruyter Studies zn Mathematics, Volume 13, Waiter de

Gruyter, Berlin, (1990). 7. H. Giacomini and M. Viano, Determination of limit cycles for two-dimensional dynamical systems, Phys.

Rev. E 5 2 (1A), 222 228, (1995). 8. H. Giacomini and S. Neukirch, Number of limit cycles of the Li6nard equation, Phys. Rev. E 56 (4), 3809-

3813. (1997). 9. J. Llibre. L. Pizarro and E. Ponce, Limit cycles of polynomial Li6nard systems. Comment on: "Number of

limit cycles of the Li~nard equation" by H. Giacomini and S. Neukireh, Phys. Rev. E 58 (4), 5185 5187, (1998).

10. L.R. Berrone and H. Giacomini, On the vanishing set of inverse integrating factors, Qual. Th. Dyn. Systems 1 (2), 211 230, (2000).

11. M. Sabatini and L.G. Trussoni, Polynomial approximations to at tractors of three-dimensional systems, (in preparation).

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Localizing attractors via a generalized La Salle principle