proceedings of the american mathematical

society

Volume 86, Number 1, September 1982

ATTRACTORSWITH POSITIVELY LYAPUNOV STABLE TRAJECTORIES TADEUSZ NADZIEJA Abstract. We show that a finite dimensional attractor in locally connected metric space consisting of positively Lyapunov stable trajectories is a torus.

There appear in physics, chemistry and biology deterministic systems x —f(x) with an irregular, chaotic, turbulent time evolution. This phenomenon has received various mathematical interpretations [5,6,10,9,11]. According to the theory of Landau [6] the turbulent motion is asymptotically given by a quasi periodic function of time x(t) = /(«,/,..., w„f ), where/has period 1 in each argument separately and the frequences w,,..., w„ are not rationally related. The attractors in such systems are tori. Ruelle and Takens [11] argue that the occurrence of a strange attractor provides a mechanism for understanding turbulence. No definition of strange attractor is universally accepted. In [11] it is an attracting set which is compact, invariant and connected but is not a point, circle or a surface of any dimension. Ruelle [9,10] assumes that on a strange attractor all trajectories have sensitive dependence on initial data i.e. trajectories are Lyapunov unstable. In the investigation of systems with complex behavior, in particular turbulent flows, it seems reasonable to assume that the process takes place in some metric space and all the limit sets of trajectories lie in a single compact set. If a strange attractor occurs in a system, then the system is turbulent in the Ruelle-Takens sense. A natural question arises: What attractors may occur in systems without RuelleTakens turbulence? All the attractors in such systems consist of positively Lyapunov stable points. We will characterize these attractors in this note. Let (X, d) be a metric space. By cl A we denote the closure of a set A C X, Kr(p) is the open ball of radius r centered at p. A flow on I is a continuous map f: RX X -> X (R real numbers), satisfying the conditions: /(0, x) = x for x E X

and f(s + t, x) = f(t, f(s, x)) for x E X and t, s E R. The orbit of a point p E X is the set T(p) = [f(t, p): t E R) and the half positive trajectory is the set T+ (p) = [f(t, p): t > 0}. For any p E X the set u(a)(p) — [q E X: there is a sequence (i„) with tn -» + (-)oo

and limn^+xf(t„,

p) — q) is called its positive {negative)

limit

set [2]. A subset A C A'is invariant if f(t, A) C A for all t E R. Received by the editors March 31, 1981. 1980 Mathematics Subject Classification.Primary 34D20, 58F12. Key words and phrases. Attractor, turbulence, positively Lyapunov stable trajectory, almost periodic

orbit. ©1982 American Mathematical Society

0002-9939/81/0000- 1055/S02.25

87 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

88

TADEUSZ NADZIEJA

We shall say that an invariant, compact subset A C X is an attractor if: a. for every neighbourhood U of A, there is a neighbourhood V of A such that

f(t,V) C U for all positive t and o>(x) C A for every x E K, b. there exists a point p E /I such that cl T(p) —A. The domain of attraction D(A) of an attractor /I is the set of all points x E X such that co(x) C A. It is easy to see [2] that D(A) is open. A point q E X is positively Lyapunov stable [7] (l-stable) if for every e > 0 there is

5 > 0 such that if d(q, x) 0. If a point g£X

is /-stable then the point/(r, #) is /-stable for every t G R. Hence it is meaningful to say that the orbit of the point q E X is positively Lyapunov stable. For the remainder of this paper Ê will denote an attractor all of whose points are

/-stable. Lemma 1. There exists a point p E £ such that cl T+ (p) = £. Proof. By assumption, there exists a point p E £ such that cl T(p) = £. We will show that if g E a(p) then u(q) D T(p). Let x —f(tx, p) and pick e > 0. Since q El a(p) and p is /-stable, for every real K there exists t2< K and t2 < tx such that d(f(t2 + i, P), /(/, q)) < e for í > 0. Hence ¿/(x, /(/, — f2, çr)) < e, which proves our inclusion. a(p) being invariant and closed we thus have a(p) D w(p) and so co(p) D ?Xp). This together with the inclusion cl T+ (p) D w(p) implies cl T+ (p) = £.

A compact, invariant set A is called minimal if it contains no proper invariant, closed subset. It is easy to show [2] that a compact set A is minimal if and only if cl T(p) = A for every point p E A. Lemma 2. £ is a minimal set.

Proof. Pick q E £ and e > 0. By Lemma 1 there exists a point p, E £ such that cl T+ (px) = £ and d(f(t, px), f(t,q)) < e for t > 0. Since e is arbitrary we have

c\T+(q) = t. The point p G X will be called almost periodic if for every e > 0 there exists a relatively dense subset of numbers (t„) such that d(f(t, p), f(t + t„, p)) < e for all t £ R and eachrn. Remark 1. It was proved in [7, Theorem 8.04] that if £ is a compact set consisting of /-stable points then for every e > 0 there is a number 8 > 0 such that

d(f(t, p), f(t, q)) < e for all t ^ 0 ifp E B and d(p, q) < 5. Remark 2. It follows from Lemma 2 that if q E Z)(£) then w(q) = £. Using Theorem 5 of [3] and two last remarks we conclude that £ is a minimal set of almost periodic points. Remark 3. Corollary 8.09 of [7] implies that for any r > 0 there exists a number 8(r) > 0 such that for qx, q2 E £, d(qx, q2) < 8(r) implies d(f(t, qx), f(t, q2)) < r for t E R. An easy consequence of Remark 3 is: Remark 4. For any qx, q2 £ £, qx ¥= q2 there is a number a > 0 such that

d(f(t,qx), f(t, q2))> a for tE.R.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

■

attractors

with positively lyapunov stable trajectories

Lemma 3. The set D(t)

89

consists of l-s table points.

Proof. From Remark 1 it follows that for every e > 0 there exists 8(e) > 0 such

that d(f(t, p), f(t, q)) < e for all t > 0 if p £ £ and d(p, q) < 8(e). Let y E £>(£) and e > 0. We pick a number r > 0 such that Kr(y) C D(t) and for some T > 0 dist(/(i, x), £) < {-8(e/2) for x E Kr(y) and t > 7. The existence of such T follows from definition of the attractor [2]. By continuity of flow we may choose r < r such

that d(f(t, x), f(t, y)) < \8(e/2) for x E Kr(y) and t E [0, T]. It is easy to see that d(f(t, x), f(t, y)) < e for all í ^ 0 and x E K^y). For anypEfi

we define

»"(p)=

(x ££)(£):

Remark 4 implies that if p, ^p,

lim (£) = U {Ws(p): p E £). Proof. Let x £ i>(£) and /„ -» +00. Since w(x) C £ we may choose a subsequence tn -* +00 such that lim^+O0/(i„,x) = ze£. Consider the sequence z„ —f(-t„ , z). Due to z„ £ £ one can choose a convergent subsequence z„ . Suppose limA.,^+0Oz„, = p. From /-stability of the point p it follows that

lim^+oo */(/(/„ ,, p), /(*„,', x)) = 0. The last equality and /-stability of p imply x E Ws(p).

Lemma 5. // p £ £, ¿«e« /or every neighbourhood U of p, there exists a smaller neighbourhood Vsuch that x £ V and x £ Ws(q) imply q E U.

Proof. Let U be a neighbourhood of p E £. We may suppose that U is an open ball Kr(p) and Kr(p) C D(£). From /-stability of p it follows that there is a number

5 > 0 such that if x £ K£p)

then d(f(t, x), f(t, p)) < {-S(r) for t>0

(8(r)

corresponds to r as in Remark 3). Let x E K¡¡(p) and x E Ws(q). For sufficiently

large i"wehave d(f(t, q), f(t, x)) < \8(r), hence d(f(t, q), f(t, p)) < 8(r) which by definition of 8(r) implies í/(/(-í", f(t, q)), f(-t, f(i, p))) = d(q, p) < r. £ is a minimal set of almost periodic points hence we may equip £ with a structure of a compact topological group as in [7]. From now on we will suppose that covering dimension [4] of £ is finite. In [8] it was shown that a commutative connected finite dimensional compact group is locally homeomorphic to set B = G X «-cell where G is a compact 0-dimensional topological group. We have two possibilities, either G is a discrete or G is a perfect set. Since a perfect, 0-dimensional subset of a metric space is a Cantor set [4], £ is locally disc or a Cantor set X «-cell. It follows [8] that if £ is locally connected then £ is a torus. Theorem 1. A finite dimensional attractor £ contained in a locally connected metric space consisting of positively Lyapunov stable points is a torus.

Proof. We must show that £ is locally connected. If it is not so, every point p E £ has a neighbourhood U such that U n £ is the product of «-cell and a Cantor set. Let p E £ and F be a connected neighbourhood of p. By Lemma 5 we may suppose that V C U {Ws(q): qEtfMJ). Since U n £ is a product of a Cantor set and License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

90

TADEUSZ NADZIEJA

«-cell, U n £may be decomposed into a sum of two sets £,, £2 such that £,; n V ¥= 0, i = 1,2, and d(px, p2) > r > 0 for every p, E £, and p2 E £2. Now we put V¡ = U {Ws(q): q E £,} n V. We will show that Vt is open. Let 3; E Ws(q) n F and Î £ £,, From Lemma 3 it follows that there exists e > 0 such that if d(x, y) < e then d(f(t, x), f(t, y)) < 2-8(r) for t » 0 (8(r) corresponds to r as in Remark 3). If x e iE( v) n Vn Ws(p) then d(f(t, p),f(t,q))

society

Volume 86, Number 1, September 1982

ATTRACTORSWITH POSITIVELY LYAPUNOV STABLE TRAJECTORIES TADEUSZ NADZIEJA Abstract. We show that a finite dimensional attractor in locally connected metric space consisting of positively Lyapunov stable trajectories is a torus.

There appear in physics, chemistry and biology deterministic systems x —f(x) with an irregular, chaotic, turbulent time evolution. This phenomenon has received various mathematical interpretations [5,6,10,9,11]. According to the theory of Landau [6] the turbulent motion is asymptotically given by a quasi periodic function of time x(t) = /(«,/,..., w„f ), where/has period 1 in each argument separately and the frequences w,,..., w„ are not rationally related. The attractors in such systems are tori. Ruelle and Takens [11] argue that the occurrence of a strange attractor provides a mechanism for understanding turbulence. No definition of strange attractor is universally accepted. In [11] it is an attracting set which is compact, invariant and connected but is not a point, circle or a surface of any dimension. Ruelle [9,10] assumes that on a strange attractor all trajectories have sensitive dependence on initial data i.e. trajectories are Lyapunov unstable. In the investigation of systems with complex behavior, in particular turbulent flows, it seems reasonable to assume that the process takes place in some metric space and all the limit sets of trajectories lie in a single compact set. If a strange attractor occurs in a system, then the system is turbulent in the Ruelle-Takens sense. A natural question arises: What attractors may occur in systems without RuelleTakens turbulence? All the attractors in such systems consist of positively Lyapunov stable points. We will characterize these attractors in this note. Let (X, d) be a metric space. By cl A we denote the closure of a set A C X, Kr(p) is the open ball of radius r centered at p. A flow on I is a continuous map f: RX X -> X (R real numbers), satisfying the conditions: /(0, x) = x for x E X

and f(s + t, x) = f(t, f(s, x)) for x E X and t, s E R. The orbit of a point p E X is the set T(p) = [f(t, p): t E R) and the half positive trajectory is the set T+ (p) = [f(t, p): t > 0}. For any p E X the set u(a)(p) — [q E X: there is a sequence (i„) with tn -» + (-)oo

and limn^+xf(t„,

p) — q) is called its positive {negative)

limit

set [2]. A subset A C A'is invariant if f(t, A) C A for all t E R. Received by the editors March 31, 1981. 1980 Mathematics Subject Classification.Primary 34D20, 58F12. Key words and phrases. Attractor, turbulence, positively Lyapunov stable trajectory, almost periodic

orbit. ©1982 American Mathematical Society

0002-9939/81/0000- 1055/S02.25

87 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

88

TADEUSZ NADZIEJA

We shall say that an invariant, compact subset A C X is an attractor if: a. for every neighbourhood U of A, there is a neighbourhood V of A such that

f(t,V) C U for all positive t and o>(x) C A for every x E K, b. there exists a point p E /I such that cl T(p) —A. The domain of attraction D(A) of an attractor /I is the set of all points x E X such that co(x) C A. It is easy to see [2] that D(A) is open. A point q E X is positively Lyapunov stable [7] (l-stable) if for every e > 0 there is

5 > 0 such that if d(q, x) 0. If a point g£X

is /-stable then the point/(r, #) is /-stable for every t G R. Hence it is meaningful to say that the orbit of the point q E X is positively Lyapunov stable. For the remainder of this paper Ê will denote an attractor all of whose points are

/-stable. Lemma 1. There exists a point p E £ such that cl T+ (p) = £. Proof. By assumption, there exists a point p E £ such that cl T(p) = £. We will show that if g E a(p) then u(q) D T(p). Let x —f(tx, p) and pick e > 0. Since q El a(p) and p is /-stable, for every real K there exists t2< K and t2 < tx such that d(f(t2 + i, P), /(/, q)) < e for í > 0. Hence ¿/(x, /(/, — f2, çr)) < e, which proves our inclusion. a(p) being invariant and closed we thus have a(p) D w(p) and so co(p) D ?Xp). This together with the inclusion cl T+ (p) D w(p) implies cl T+ (p) = £.

A compact, invariant set A is called minimal if it contains no proper invariant, closed subset. It is easy to show [2] that a compact set A is minimal if and only if cl T(p) = A for every point p E A. Lemma 2. £ is a minimal set.

Proof. Pick q E £ and e > 0. By Lemma 1 there exists a point p, E £ such that cl T+ (px) = £ and d(f(t, px), f(t,q)) < e for t > 0. Since e is arbitrary we have

c\T+(q) = t. The point p G X will be called almost periodic if for every e > 0 there exists a relatively dense subset of numbers (t„) such that d(f(t, p), f(t + t„, p)) < e for all t £ R and eachrn. Remark 1. It was proved in [7, Theorem 8.04] that if £ is a compact set consisting of /-stable points then for every e > 0 there is a number 8 > 0 such that

d(f(t, p), f(t, q)) < e for all t ^ 0 ifp E B and d(p, q) < 5. Remark 2. It follows from Lemma 2 that if q E Z)(£) then w(q) = £. Using Theorem 5 of [3] and two last remarks we conclude that £ is a minimal set of almost periodic points. Remark 3. Corollary 8.09 of [7] implies that for any r > 0 there exists a number 8(r) > 0 such that for qx, q2 E £, d(qx, q2) < 8(r) implies d(f(t, qx), f(t, q2)) < r for t E R. An easy consequence of Remark 3 is: Remark 4. For any qx, q2 £ £, qx ¥= q2 there is a number a > 0 such that

d(f(t,qx), f(t, q2))> a for tE.R.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

■

attractors

with positively lyapunov stable trajectories

Lemma 3. The set D(t)

89

consists of l-s table points.

Proof. From Remark 1 it follows that for every e > 0 there exists 8(e) > 0 such

that d(f(t, p), f(t, q)) < e for all t > 0 if p £ £ and d(p, q) < 8(e). Let y E £>(£) and e > 0. We pick a number r > 0 such that Kr(y) C D(t) and for some T > 0 dist(/(i, x), £) < {-8(e/2) for x E Kr(y) and t > 7. The existence of such T follows from definition of the attractor [2]. By continuity of flow we may choose r < r such

that d(f(t, x), f(t, y)) < \8(e/2) for x E Kr(y) and t E [0, T]. It is easy to see that d(f(t, x), f(t, y)) < e for all í ^ 0 and x E K^y). For anypEfi

we define

»"(p)=

(x ££)(£):

Remark 4 implies that if p, ^p,

lim (£) = U {Ws(p): p E £). Proof. Let x £ i>(£) and /„ -» +00. Since w(x) C £ we may choose a subsequence tn -* +00 such that lim^+O0/(i„,x) = ze£. Consider the sequence z„ —f(-t„ , z). Due to z„ £ £ one can choose a convergent subsequence z„ . Suppose limA.,^+0Oz„, = p. From /-stability of the point p it follows that

lim^+oo */(/(/„ ,, p), /(*„,', x)) = 0. The last equality and /-stability of p imply x E Ws(p).

Lemma 5. // p £ £, ¿«e« /or every neighbourhood U of p, there exists a smaller neighbourhood Vsuch that x £ V and x £ Ws(q) imply q E U.

Proof. Let U be a neighbourhood of p E £. We may suppose that U is an open ball Kr(p) and Kr(p) C D(£). From /-stability of p it follows that there is a number

5 > 0 such that if x £ K£p)

then d(f(t, x), f(t, p)) < {-S(r) for t>0

(8(r)

corresponds to r as in Remark 3). Let x E K¡¡(p) and x E Ws(q). For sufficiently

large i"wehave d(f(t, q), f(t, x)) < \8(r), hence d(f(t, q), f(t, p)) < 8(r) which by definition of 8(r) implies í/(/(-í", f(t, q)), f(-t, f(i, p))) = d(q, p) < r. £ is a minimal set of almost periodic points hence we may equip £ with a structure of a compact topological group as in [7]. From now on we will suppose that covering dimension [4] of £ is finite. In [8] it was shown that a commutative connected finite dimensional compact group is locally homeomorphic to set B = G X «-cell where G is a compact 0-dimensional topological group. We have two possibilities, either G is a discrete or G is a perfect set. Since a perfect, 0-dimensional subset of a metric space is a Cantor set [4], £ is locally disc or a Cantor set X «-cell. It follows [8] that if £ is locally connected then £ is a torus. Theorem 1. A finite dimensional attractor £ contained in a locally connected metric space consisting of positively Lyapunov stable points is a torus.

Proof. We must show that £ is locally connected. If it is not so, every point p E £ has a neighbourhood U such that U n £ is the product of «-cell and a Cantor set. Let p E £ and F be a connected neighbourhood of p. By Lemma 5 we may suppose that V C U {Ws(q): qEtfMJ). Since U n £ is a product of a Cantor set and License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

90

TADEUSZ NADZIEJA

«-cell, U n £may be decomposed into a sum of two sets £,, £2 such that £,; n V ¥= 0, i = 1,2, and d(px, p2) > r > 0 for every p, E £, and p2 E £2. Now we put V¡ = U {Ws(q): q E £,} n V. We will show that Vt is open. Let 3; E Ws(q) n F and Î £ £,, From Lemma 3 it follows that there exists e > 0 such that if d(x, y) < e then d(f(t, x), f(t, y)) < 2-8(r) for t » 0 (8(r) corresponds to r as in Remark 3). If x e iE( v) n Vn Ws(p) then d(f(t, p),f(t,q))