Full text of DOI:10.1134/1.171737

Doklady Physics, Vol. 45, No. 4, 2000, pp. 183–185. Translated from Doklady Akademii Nauk, Vol. 371, No. 6, 2000, pp. 770–772.
Original Russian Text Copyright © 2000 by Druzhinina, Shestakov.
MECHANICS
On the Structure
of a Stable (in the Lyapunov Sense) Attractor
O. V. Druzhinina and A. A. Shestakov
Presented by Academician V.V. Rumyantsev July 7, 1999
Received July 14, 1999
An urgent problem in the Lyapunov stability theory
[1–3] is finding criteria that make it possible to distin-
guish simple-attracting sets (simple attractors) from
strange-attracting sets (strange attractors) [4–6]. There
is a principal distinction in the behavior of dynamic
systems with simple and strange attractors, because the
former systems are regular, while the latter ones are
random.
We will denote positive and negative limiting sets of
a point x Rn as ω(x) and α(x), respectively. An open
sphere of a radius δ with the center at the point y Rn
is denoted as Bδ(y).
In [8], the following statement on the uniform sta-
bility of the compact set M Rn is established.
Statement 1. Let M be a compact set stable in the
Lyapunov sense as t
+∞, and m1 M. Then, for
In this paper, it is shown that if the invariant set
each number ε > 0, there exists a number δ > 0, such
that
A
Rn of a dynamic system ϕ: Rn
Rn is compact,
attracting, stable in the Lyapunov sense, and there
exists a trajectory, whose closure is dense in A, then the
set A is a stable torus and, hence, the dynamic system
will be regular.
d(m1, m2) < δ d(ϕ(t, m1), ϕ(t, m2)) < ε
m2 M.
t
R+,
(1)
According to the assumption of Ruelle and Takens,
all trajectories of strange attractors are unstable in the
Lyapunov sense, and this was taken by these authors as
a starting point for explaining the turbulence phenome-
non. They proceed from the following turbulence defi-
nition: “… the motion of a liquid medium is turbulent
if this motion is described by the integral curve of a
vector field, which tends to an nonempty set A being
not an equilibrium state or a closed orbit” [6]. Theo-
rems proved by us elucidate the attractor structure in
various systems in the case of the absence of the turbu-
lence in the Ruelle–Takens sense.
Next, we formulate lemmas required in what fol-
lows.
Lemma 1. Let: (1) A be a compact set stable in the
Lyapunov sense as t +∞; (2) A be an attractor as
+∞, i.e., for arbitrary neighborhood U of the set A,
there exist a neighborhood V of the set A, such that
t
ϕ(t, V) V; (2)
R+ and ω(x)
U
t
A
x
(3) there exists a trajectory C(a), a A, which is dense
everywhere in the set A. Then, for an arbitrary λ > 0,
there exists a number δ(λ) > 0, such that
Following [7, 8], we refer to the invariant set M Rn
b1, b2
A
(3)
R.
of the dynamic system ϕ: Rn
Rn as stable in the
d(b1, b2) < δ(λ) d(ϕ(t, b1), ϕ(t, b2)) < λ
t
Lyapunov sense as t
+∞ provided that the condi-
tion
Lemma 1 is a direct corollary of Statement 1, as well
as of the definition of the attractor, and the presence in
the set A of the trajectory dense everywhere.
Corollary 1. For arbitrary two points b1, b2 A,
b1 ≠ b2, the relation
ε > 0 δ > 0, d(m, x) < δ d(ϕ(t, m), ϕ(t, x)) < ε
is satisfied at each point m of this set for all t R+,
where x M and d is the metric of the space Rn.
In other words, the stability of a set in the Lyapunov
sense implies that all points of this set are stable in the
Lyapunov sense. We note, that the equilibrium state,
periodic trajectory, and almost-periodic trajectory with
a compact closure are sets stable in the Lyapunov sense.
µ > 0, d(ϕ(t, b1), ϕ(t, b2)) > µ
takes place.
Lemma 2. Let the prerequisites of Lemma 1 be sat-
isfied. Then, the set A is the minimum set of almost peri-
odic (in the Bohr sense) trajectories.
t
R
(4)
State Russian Open Technical University
of Communications,
Proof. We prove initially, that
Chasovaya ul. 22/2, Moscow,
125808 Russia
a
A C+(a) = A.
(5)
1028-3358/00/4504-0183$20.00 © 2000 MAIK “Nauka/Interperiodica”

184
DRUZHININA, SHESTAKOV
By virtue of condition (3) of Lemma 1,
ble in the Lyapunov sense. By virtue of Statement 1, for
an arbitrary ε > 0, there exists δ(ε) > 0, such that
a
A C(a) = A.
a, b A,
R+.
We assume that if b α(a), then ω(b)
Indeed, let x = ϕ(a, t1), and we choose a number ε > 0.
Since the point b α(a) is stable in the Lyapunov
C(a).
d(a, b) < δ d(ϕ(t, a), ϕ(t, b)) < ε
Let ε > 0 and y Π(A). We choose the number
λ > 0, such that Bλ(y) Π(A) and the inequality
t
sense, then, for an arbitrary number c R, there exist
t2 < c and t1 > t2, such that d(ϕ(t2 + t, a), ϕ(t, b)) > ε
1
2
ε
2
-- --
r(ϕ(t, x), A) < δ
, x Bλ (y), t > τ
t
R+ and d(x, ϕ(t1 – t2, b)) > ε. This proves the inclu-
sion. Since α(a) is invariant and closed, then α(a)
ω(a) and, thus, ω(a) C(a). Since C+(a) ω(a), then
C+(a) = A. Thus, statement (5) is established.
is satisfied for a certain τ > 0, where r is the distance
from the point to the set.
The existence of the number τ follows from the def-
inition of the attractor. Due to the continuity of the
dynamic system, we can choose λ1 < λ, such that the
following expressions hold:
We now show that the set A is the minimum one. To
do this, we choose b A and ε > 0. Then, by virtue of
arguments proved above, there exists the point a1 A,
1
2
ε
2
such that C+(a1) = A and d(ϕ(t, a1), ϕ(t, b)) < ε
R+. Since ε is arbitrary, then C+(b) = A.
t
-- --
d(ϕ(t, x), ϕ(t, y)) < δ
,
x
Bλ (y),
t
[0, τ].
1
Because the point b is arbitrary, each half-trajectory
of the set A is dense everywhere in A, and hence, the
set A consists of almost periodic trajectories. Thus,
Lemma 2 is proved.
Hence,
d(ϕ(t, x), ϕ(t, y)) < ε (t, x) R+ × Bλ (y).
1
Thus, Lemma 3 is proved.
Let A be an attractor as t
tion set as Π(A), such that
+∞. Denote the attrac-
Let A be a set for which conditions (1)–(3) of
Lemma 1 are satisfied. We show that if a A then, for
each neighborhood N1 of the point a, there exists the
neighborhood N2 N1, such that
Π(A) = {x Rn: ω(x) A}.
In addition, for an arbitrary point a A, we define
x
(N2 ∩ K(b))
b
N1.
(7)
Actually, let N1 be the neighborhood of a point
A. We may take that N1 is an open sphere Bλ(a) with
the radius λ and Bλ(a) Π(A). From the property of
stability in the Lyapunov sense of the point a, it follows
that
K(a)
a
={x Π(A): d((ϕ(t, x), ϕ(t, a))
0, (t
+∞))}.
It is easy to show that for different points a1 and a2,
the sets K(a1) and K(a2) are nonintersecting and
δ1 > 0
Π(A) =
{K(a): a A}.
(6)
(8)
R+.
1
2
--
Bδ (a) d(ϕ(t, x), ϕ(t, a)) < δ(λ)
x
t
We now establish equality (6). Indeed, let x Π(A)
1
and τn +∞. Since ω(x) A, then we can choose a
subsequence sn +∞, such that ϕ(sn, x) A as
The choice of the number δ(λ) is performed according
to (3). Let x Bδ (a) ∩ K(b). For large values of s, we
y
1
sn
+∞. We define the sequence ys = ϕ(sn, y). Since
n
1
2
ys
A, then we can choose the convergent subse-
--
have d(ϕ(s, b), ϕ(s, x)) < δ(λ). Consequently,
n
quence ys , such that ys
a as k
point a is stable in the Lyapunov sense. Therefore,
d(ϕ(sk, a) ϕ(sk, x)) 0 as k +∞. Consequently,
+∞. The
k
k
d(ϕ(–s, ϕ(s, b)), ϕ(–s, ϕ(s, a))) = d(b, a) < λ.
Implication (7) follows from the inequality obtained
and condition (3) of Lemma 1.
from the stability of the point a, it follows that x K(a).
Thereby, the validity of relation (6) is proved.
The following theorem also takes place:
Lemma 3. The attraction set Π(A) is open and sta-
Theorem 1. Let the invariant set A
Rn of the
ble in the Lyapunov sense as t
+∞.
dynamic system ϕ: Rn
Rn be compact, attracting as
Proof. The property of openness for the set Π(A) is
established in [8]. We will prove that the set Π(A) is sta-
t
+∞, stable in the Lyapunov sense as t
+∞,
and there exist in A a trajectory everywhere dense.
DOKLADY PHYSICS Vol. 45 No. 4
2000

ON THE STRUCTURE OF A STABLE (IN THE LYAPUNOV SENSE) ATTRACTOR
185
Then, the set A is a torus. In particular, if A is a hyper- it follows that x Ui, U1 ∩ U2 = л, U1 U2 = N2, and
bolic set, then it will be either an equilibrium state or a
closed trajectory.
the set N2 is a combination of two open nonempty sets,
which contradicts to the connectedness of the set N2.
The contradiction obtained proves the local connected-
ness of the set A. Therefore, the set A is a torus. The the-
orem is proved.
Theorem 1 is generalized to the case of a connected
metric space X and a finite-dimensional attractor A X.
Namely, the following theorem takes place.
Proof. By lemmas 1 and 2, the set A is a minimum
set of almost-periodic trajectories. We impart to A the
structure of a compact topologic group, which is
always possible [8]. The set is compact, and, conse-
quently, by the Pontryagin theorem [9], the commuta-
tive connected finite-dimensional topologic group is
locally homeomorphic to the set being the Cartesian
product Γ1 × Γ2. Here, Γ1 is the compact zero-dimen-
sional topological group and Γ2 is the n-dimensional set
homeomorphic to the sphere |x| < 1. The set Γ1 is dis-
crete or perfect. A perfect zero-dimensional set from Rn
is known from [8] to be the Cantor set. Therefore, A is
a local disk or a product of the Cantor set by an
n-dimensional element. It follows from the Pontryagin
theorem that if A is connected locally, then A is the Car-
tesian product of n circumferences, i.e., A is an
n-dimensional torus T n.
Theorem 2. Let X be a locally connected metric
space. Let the invariant set A X of a dynamic system ϕ:
X
X be finite-dimensional, attracting as t
+∞,
stable in the Lyapunov sense as t
+∞, and a trajec-
tory dense everywhere there exist in A. Then, the set A
is a topological torus. In particular, if A is hyperbolic
set, then it will be either an equilibrium state, or a
closed trajectory.
The proof of Theorem 2 is performed in the same
manner as that of Theorem 1.
Comments to Theorem 2. It was established by
V. V. Nemytskiœ and V. V. Stepanov [8], that for any
compact metric group G, there exists a dynamic sys-
tem, such that G is a minimal set of almost-periodic tra-
jectories in this dynamic system. Therefore, any such a
We now establish that A possesses the property of a
local connectedness. To do this, we assume the con-
trary. Then, each point a A has a neighborhood N1,
such that A ∩ N1 is a product of a n-dimensional ele-
ment and a Cantor set. Let N2 be the connected neigh- group can be a stable attractor, and hence, the condition
of Theorem 2 stating that the set A is contained in a cer-
tain closed locally connected metric space cannot be
relaxed.
borhood of a point a A. Since for any neighborhood
N1, there exist a neighborhood N2 N1, such that
x
N2, x K(b)
then we can assume that
N2 {K(b): b A ∩ N1}.
b
N1,
ACKNOWLEDGMENTS
The authors are grateful toAcademicianV.V. Rumy-
antsev for his attention to this work.
Due to the property of the intersection A ∩ N1, A can be
decomposed into a sum of two sets Ai (i = 1, 2), such
that
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(Gostekhizdat, Moscow, Leningrad, 1950).
Ai ∩ N2 =
,
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d(a1, a2) > c > 0 ai Ai (i = 1, 2).
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We now assume that
4. G. Alland and E. S. Thomas, J. Differ. Equat. 15, 158
Ui =
{K(b): b Ai} ∩ N2.
(1974).
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Π(A) is stable in the Lyapunov sense. Hence, we have
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ε > 0,
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1
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--
d(x, y) < ε d(ϕ(t, x), ϕ(t, y)) < δ(λ),
2
where δ(λ) is a number corresponding to the number
λ > 0 chosen in the same manner as in the item (3) of
Lemma 1. If x Bε(y) ∩ N2 ∩ K(a), then for a suffi-
ciently large t, we have d(ϕ(t, a), ϕ(t, b)) < δ(λ). Hence,
d(a, b) < λ. From the last inequality and statement (7),
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(Nauka, Moscow, 1988), Vol. 3.
Translated by V. Devitsyn
DOKLADY PHYSICS Vol. 45
No. 4
2000