Localization of Compact Invariant Sets of Nonlinear Systems - wseas
Recent Advances in Applied and Theoretical Mathematics
Localization of Compact Invariant Sets of Nonlinear Systems ANATOLY N. KANATNIKOV Institute of System Analysis RAS pr. 60-letiya Oktyabrya, 9, 117312 Moscow RUSSIA [email protected]
ALEXANDER P. KRISHCHENKO Bauman Moscow State Technical University Department of Mathematical Modeling 2-aja Baumanskaja ul., 5, 105005 Moscow RUSSIA [email protected]
Abstract: The method of finding domains in the state space of a nonlinear system which contain all compact invariant sets is presented. We consider continuous-time and discrete-time dynamical systems and control systems which may contain disturbances. The results are based on the fact that any continuous function reaches maximum and minimum on a compact set. This method provides estimates of the areas with the chaotic behavior of trajectories. As examples, the Lorenz system and the H´enon system are considered. Key-Words: Localization, compact invariant sets, Lorenz system, H´enon system
1
2
Introduction
In this Section, when we talk about a localization we have in mind the following problem: find the set Ω ⊂ IRn (a localization set) that contains all compact invariant sets of the system (1). For any x0 ∈ IRn by x(t, x0 ), t ∈ IR we denote a solution of the system (1), with x(0, x0 ) = x0 . A set G ⊂ IRn is called invariant for (1) if for any x0 ∈ G we have: x(t, x0 ) ∈ G for all t ∈ IR. Important classes of compact invariant sets are equilibrium points, periodic orbits, heteroclinic orbits, homoclinic cycles and orbits. Let Q be a subset in IRn . We define a maximal compact invariant set of the system (1) contained in Q as a compact invariant set in Q containing any compact invariant set of the system (1) contained in Q. A maximal compact invariant set contained in Q may not exist.
The study of compact invariant sets is one of the important topics in the qualitative theory of ordinary differential equations closely related to analysis of a long-time behavior of a nonlinear system. Many researchers have been interested in the idea of finding some geometrical bounds for objects in the phase space such as attractors, periodical orbits and chaotic dynamics of a nonlinear autonomous system x˙ = f (x),
f ∈ C 1 (IRn ).
(1)
Some results were obtained by Lyapunov-type functions [2, 16–18, 20] and by the semipermeable surfaces [3]. Another approach was proposed in [10, 11] for solving the localization problem of all periodical solutions of the system (1). This method was developed in [1,12–15,19] in the case of compact invariant sets. Afterwards this localization method was extended to control systems and dynamical systems with disturbances [5, 8]. The case of discrete-time systems is considered in [5, 7, 9]. In all these cases the localization method is based on the fact that any continuous function reaches maximum and minimum on a compact set. In the case of chaotic systems it is important that the localization method does not use the numerical calculation of any trajectory. In this article we develop the main ideas of the localization method.
ISBN: 978-960-474-351-3
Compact Invariant Sets
2.1
Localization of Compact Invariant Sets
For a function ϕ ∈ C 1 (Q), we introduce the set Sϕ (Q) = {x ∈ Q : Lf ϕ = 0}, where Lf ϕ is a Lie derivative of the function ϕ(x) with respect to the vector field f (x) of the system (1) and define ϕinf (Q) =
inf
x∈Sϕ (Q)
ϕ(x), ϕsup (Q) =
sup ϕ(x). x∈Sϕ (Q)
Theorem 1 [12] For any φ(x) ∈ C ∞ (Rn ) all compact invariant sets of the system (1) located in Q are contained in the set defined by the formula Ωϕ (Q) = {x ∈ Q : ϕinf (Q) ≤ ϕ(x) ≤ ϕsup (Q)}
19
Recent Advances in Applied and Theoretical Mathematics
as well. If Ωϕ (Q) is a compact set then the system (1) has a maximal compact invariant set located in Ωϕ (Q).
For the localizing function p2 we notice that the set Sp2 (IR3 ) is defined by equality x22 = −bx23 + brx3 .
In Theorem 1 a smooth function defined on some subset Q of the state space is associated with a localization set. This localization set contains all compact invariant sets located in Q while this function itself is called localizing.
Following the calculations, we find p2 inf (IR3 ) = −
Theorem 2 Let ϕm (x), m = 0, 1, 2, . . . be a sequence of functions from C 1 (IRn ). The localization sets
p2 sup (IR3 ) = p∗ :=
r2 (b − 2)2 8(b − 1)
and we came to the localization set Ωp2 (IR3 ) = {
n
Ω0 = Ωϕ0 (IR ), Ωm = Ωϕm (Ωm−1 ), m > 0, contain any compact invariant set of the system (1) and Ω0 ⊃ Ω1 ⊃ . . . ⊃ Ωm ⊃ . . . .
x22 + x23 − rx3 ≤ p∗ }. 2
Let us consider the localizing function p3 . For this function we get that p3 inf (IR3 ) = −∞,
Proofs of these results may be realized in the same way like in [10, 11].
2.2
r2 , 2
p3 sup (IR3 ) = p∗ :=
M 2 (b − 2σ)2 r 8σ(2σ − M )(σ − b)
and
The Lorenz system
The Lorenz system
Ωp3 (IR3 ) = {x21 −
x˙ 1 = −σx1 + σx2 , x˙ 2 = rx1 − x2 − x1 x3 , x˙ 3 = x1 x2 − bx3
2σ − M 2 (x2 + x23 ) − M x3 ≤ p∗ }. r
Hence we found the M -parametric family of localization sets ωM = Ωp3 (IR3 ). The intersection of the localization sets is a localization set. Therefore all compact invariant sets of the Lorenz system are contained in the sets ω = ∩M
Localization of Compact Invariant Sets of Nonlinear Systems ANATOLY N. KANATNIKOV Institute of System Analysis RAS pr. 60-letiya Oktyabrya, 9, 117312 Moscow RUSSIA [email protected]
ALEXANDER P. KRISHCHENKO Bauman Moscow State Technical University Department of Mathematical Modeling 2-aja Baumanskaja ul., 5, 105005 Moscow RUSSIA [email protected]
Abstract: The method of finding domains in the state space of a nonlinear system which contain all compact invariant sets is presented. We consider continuous-time and discrete-time dynamical systems and control systems which may contain disturbances. The results are based on the fact that any continuous function reaches maximum and minimum on a compact set. This method provides estimates of the areas with the chaotic behavior of trajectories. As examples, the Lorenz system and the H´enon system are considered. Key-Words: Localization, compact invariant sets, Lorenz system, H´enon system
1
2
Introduction
In this Section, when we talk about a localization we have in mind the following problem: find the set Ω ⊂ IRn (a localization set) that contains all compact invariant sets of the system (1). For any x0 ∈ IRn by x(t, x0 ), t ∈ IR we denote a solution of the system (1), with x(0, x0 ) = x0 . A set G ⊂ IRn is called invariant for (1) if for any x0 ∈ G we have: x(t, x0 ) ∈ G for all t ∈ IR. Important classes of compact invariant sets are equilibrium points, periodic orbits, heteroclinic orbits, homoclinic cycles and orbits. Let Q be a subset in IRn . We define a maximal compact invariant set of the system (1) contained in Q as a compact invariant set in Q containing any compact invariant set of the system (1) contained in Q. A maximal compact invariant set contained in Q may not exist.
The study of compact invariant sets is one of the important topics in the qualitative theory of ordinary differential equations closely related to analysis of a long-time behavior of a nonlinear system. Many researchers have been interested in the idea of finding some geometrical bounds for objects in the phase space such as attractors, periodical orbits and chaotic dynamics of a nonlinear autonomous system x˙ = f (x),
f ∈ C 1 (IRn ).
(1)
Some results were obtained by Lyapunov-type functions [2, 16–18, 20] and by the semipermeable surfaces [3]. Another approach was proposed in [10, 11] for solving the localization problem of all periodical solutions of the system (1). This method was developed in [1,12–15,19] in the case of compact invariant sets. Afterwards this localization method was extended to control systems and dynamical systems with disturbances [5, 8]. The case of discrete-time systems is considered in [5, 7, 9]. In all these cases the localization method is based on the fact that any continuous function reaches maximum and minimum on a compact set. In the case of chaotic systems it is important that the localization method does not use the numerical calculation of any trajectory. In this article we develop the main ideas of the localization method.
ISBN: 978-960-474-351-3
Compact Invariant Sets
2.1
Localization of Compact Invariant Sets
For a function ϕ ∈ C 1 (Q), we introduce the set Sϕ (Q) = {x ∈ Q : Lf ϕ = 0}, where Lf ϕ is a Lie derivative of the function ϕ(x) with respect to the vector field f (x) of the system (1) and define ϕinf (Q) =
inf
x∈Sϕ (Q)
ϕ(x), ϕsup (Q) =
sup ϕ(x). x∈Sϕ (Q)
Theorem 1 [12] For any φ(x) ∈ C ∞ (Rn ) all compact invariant sets of the system (1) located in Q are contained in the set defined by the formula Ωϕ (Q) = {x ∈ Q : ϕinf (Q) ≤ ϕ(x) ≤ ϕsup (Q)}
19
Recent Advances in Applied and Theoretical Mathematics
as well. If Ωϕ (Q) is a compact set then the system (1) has a maximal compact invariant set located in Ωϕ (Q).
For the localizing function p2 we notice that the set Sp2 (IR3 ) is defined by equality x22 = −bx23 + brx3 .
In Theorem 1 a smooth function defined on some subset Q of the state space is associated with a localization set. This localization set contains all compact invariant sets located in Q while this function itself is called localizing.
Following the calculations, we find p2 inf (IR3 ) = −
Theorem 2 Let ϕm (x), m = 0, 1, 2, . . . be a sequence of functions from C 1 (IRn ). The localization sets
p2 sup (IR3 ) = p∗ :=
r2 (b − 2)2 8(b − 1)
and we came to the localization set Ωp2 (IR3 ) = {
n
Ω0 = Ωϕ0 (IR ), Ωm = Ωϕm (Ωm−1 ), m > 0, contain any compact invariant set of the system (1) and Ω0 ⊃ Ω1 ⊃ . . . ⊃ Ωm ⊃ . . . .
x22 + x23 − rx3 ≤ p∗ }. 2
Let us consider the localizing function p3 . For this function we get that p3 inf (IR3 ) = −∞,
Proofs of these results may be realized in the same way like in [10, 11].
2.2
r2 , 2
p3 sup (IR3 ) = p∗ :=
M 2 (b − 2σ)2 r 8σ(2σ − M )(σ − b)
and
The Lorenz system
The Lorenz system
Ωp3 (IR3 ) = {x21 −
x˙ 1 = −σx1 + σx2 , x˙ 2 = rx1 − x2 − x1 x3 , x˙ 3 = x1 x2 − bx3
2σ − M 2 (x2 + x23 ) − M x3 ≤ p∗ }. r
Hence we found the M -parametric family of localization sets ωM = Ωp3 (IR3 ). The intersection of the localization sets is a localization set. Therefore all compact invariant sets of the Lorenz system are contained in the sets ω = ∩M
