Math Analysis Differential Equations and Applications ————————————— September 15–20, 2010. Sunny Beach, Bulgaria

On Existence and Stability of Invariant Sets of Discontinuous Dynamical Systems Mykola Perestyuk and Petro Feketa

Utilizing the concept of Green – Samoilenko function, sufficient conditions for the existence and asymptotic stability of invariant toroidal sets of impulsive system of differential equations defined in direct product of torus and Euclidean space are obtained. A certain class of discontinuous dynamical systems that has asymptotically stable invariant set are distinguished. AMS Subj. Classification: 34A37 Key Words: Impulsive differential equation, Invariant set, Asymptotic stability, Green– Samoilenko function.

1. Introduction The mathematical modeling of variety of processes in physics, chemistry, biology, etc. leads to necessity of studying the differential equations with solutions that are discontinuous functions or so called impulsive differential equations. The theory of impulsive differential equations developed intensively due to its applied value in simulation of real world phenomena. This article is concerned with investigation of the existence and stability of invariant toroidal sets of system of impulsive differential equations defined in direct product of m-dimensional torus T m and n-dimensional Euclidean space En. 2. Preliminaries The main object of investigation of this paper is the system of differential equations that undergo impulsive perturbations at the moments when the phase

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point meets the given set in the phase space. Consider the system dϕ = a(ϕ), dt dx (1) = A(ϕ)x + f (ϕ), ϕ 6∈ Γ, dt ∆x|ϕ∈Γ = B(ϕ)x+g(ϕ), where ϕ = (ϕ1 , . . . , ϕm )T ∈ T m , x = (x1 , . . . , xn )T ∈ E n , a(ϕ) is continuous 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m vector function that satisfies Lipschitz condition

′′

′ ′′ ′

(2)

a(ϕ ) − a(ϕ ) ≤ L ϕ − ϕ ′

′′

for every ϕ , ϕ ∈ T m . A(ϕ), B(ϕ) are continuous 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m square matrices; f (ϕ), g(ϕ) are continuous (piecewise continuous with first kind discontinuities in the set Γ) 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m vector functions. We regard the point ϕ = (ϕ1 , . . . , ϕm )T as a point of the m-dimensional torus T m so that the domain of the functions A(ϕ), B(ϕ), f (ϕ), g(ϕ) and a(ϕ) is the torus T m . We assume that the set Γ is a subset of the torus T m , which is a manifold of dimension m − 1 defined by the equation Φ(ϕ) = 0 for some continuous scalar 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m function. Denote by ϕt (ϕ) the solution of the first equation of system (1) that satisfies the initial condition ϕ0 (ϕ) = ϕ. Lipschitz condition (2) guarantees the existence and uniqueness of such solution. The system (1) characterized in the following way: between two successive meeting of the trajectory ϕt (ϕ) with the set Γ the motion proceeds along the trajectories of the system dx dϕ = a(ϕ), = A(ϕ)x + f (ϕ). dt dt At the moments of meeting of the phase point with the set Γ the point x(t) is instantly transferred to the point x(t) + B(ϕt (ϕ))x(t) + g(ϕt (ϕ)). Denote by ti (ϕ), i ∈ Z the solutions of equation Φ(ϕt (ϕ)) = 0 that are moments of impulsive action in system (1). Let function Φ(ϕ) be such that solutions t = ti (ϕ) exist since otherwise (1) would not be an impulsive system. Assume that lim ti (ϕ) = ±∞, i→±∞

(3) lim

T →±∞

i(t, t + T ) =p<∞ T

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uniformly with respect to t ∈ R, where i(a, b) is a number of the points ti (ϕ) in the interval (a, b). Together with system (1) we consider linear system (4)

dx = A(ϕt (ϕ))x + f (ϕt (ϕ)), t 6= ti (ϕ), dt ∆x|t=ti (ϕ) = B(ϕti (ϕ) (ϕ))x + g(ϕti (ϕ) (ϕ))

that depends on ϕ ∈ T m as a parameter. We obtain system (4) by substituting ϕt (ϕ) for ϕ in second and third equations of system (1). By invariant set of system (1) we understand a set that is defined by a piecewise continuous function u(ϕ), which has period 2π with respect to each of the components ϕv , v = 1, . . . , m, such that the function x(t, ϕ) = u(ϕt (ϕ)) is a solution of system (4) for every ϕ ∈ T m . The basic approach to investigate the invariant toroidal sets of system (1) is based on the concept of Green–Samoilenko function. Consider homogeneous system of differential equations dx = A(ϕt (ϕ))x, t 6= ti (ϕ), (5) dt ∆x|t=ti (ϕ) = B(ϕti (ϕ) (ϕ))x that depends on ϕ ∈ T m as a parameter. By Ωtτ (ϕ) we denote the fundamental matrix of system (5), which turns into an identity matrix at the point t = τ , i.e. Ωττ (ϕ) ≡ E. Such fundamental matrix will be called matriciant. Let C(ϕ) is continuous (piecewise continuous with first kind discontinuities in the set Γ) 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m square matrix. Let ( Ω0τ (ϕ)C(ϕτ (ϕ)), τ ≤ 0, (6) G0 (τ, ϕ) = Ω0τ (ϕ)(C(ϕτ (ϕ)) − E), τ > 0, where E is an identity matrix. Definition 1. Function G0 (τ, ϕ) is called Green – Samoilenko function of the system dϕ = a(ϕ), dt dx = A(ϕ)x, ϕ 6∈ Γ, dt ∆x|ϕ∈Γ = B(ϕ)x, if the following inequality holds Z +∞ kG0 (τ, ϕ)k dτ ≤ K < ∞. (7) −∞

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Then the invariant toroidal set of the system (1) may by presented in the form +∞ Z u(ϕ) = G0 (τ, ϕ)f (ϕτ (ϕ))dτ + −∞

X

G0 (ti (ϕ) + 0, ϕ)g(ϕti (ϕ) (ϕ)).

−∞
The integral and sum converge since the conditions (3) and (7) are fulfilled. Existence of Green–Samoilenko function G0 (τ, ϕ) and condition (3) fulfillment are sufficient for the existence of invariant toroidal set of system (1). 3. Main Results In this section we will distinguish a certain class of discontinuous dynamical systems that has Green–Samoilenko function. In [1], sufficient conditions for the existence and stability of invariant sets of system similar to (1) were obtained in terms of Lyapunov function V (ϕ, x) that satisfies some conditionsSin domain Z = {ϕ ∈ Ω x ∈ J¯h }, where J¯h = {x ∈ E n , kxk ≤ h, h > 0}, Ω = ϕ∈T m Ωϕ , Ωϕ is the ω-limit set of solution ϕt (ϕ). Since Lyapunov function has to satisfy some conditions not on the whole surface of torus T m but only in ω-limit set Ω, it is interesting to consider system (1) with specific properties in domain Ω. Therefore we consider system (1) assuming that the matrices A(ϕ) and B(ϕ) are constant in domain Ω: ˜ B(ϕ)|ϕ∈Ω = B. ˜ A(ϕ)|ϕ∈Ω = A, Hence, for every ϕ ∈ T m (8)

˜ lim A(ϕt (ϕ)) = A,

t→+∞

˜ lim B(ϕt (ϕ)) = B.

t→+∞

We will prove the existence of Green – Samoilenko function and obtain sufficient conditions for the existence and asymptotic stability of invariant set of system ˜ Denote (1) in terms of eigenvalues of matrices A˜ and B. ˜ α2 = max λj ((E + B) ˜ T (E + B)). ˜ γ = max Reλj (A), j=1,..n

j=1,..n

Similar systems without impulsive perturbations have been considered in [2]. Theorem 1. Let the moments of impulsive perturbations {ti (ϕ)} are such that uniformly with respect to t ∈ R there exists a finite limit i(t, t + T˜) = p. (9) lim T˜ T˜→∞ If the following inequality holds (10)

γ + p ln α < 0,

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then system (1) has asymptotically stable invariant toroidal set. P r o o f. Consider homogeneous system of differential equations dx = A(ϕt (ϕ))x, t 6= ti (ϕ), dt ∆x|t=ti (ϕ) = B(ϕti (ϕ) (ϕ))x

(11)

that depends on ϕ ∈ T m as a parameter. As in Section 2, by Ωtτ (ϕ) we denote the matriciant of system (11). Rewrite system (11) in the form dx ˜ + (A(ϕt (ϕ)) − A)x, ˜ = Ax t 6= ti (ϕ), dt ˜ + (B(ϕt (ϕ) (ϕ)) − B)x. ˜ ∆x|t=ti (ϕ) = Bx i Thus, the matriciant Ωtτ (ϕ) of system (11) may be represented in the following way [6] Z t ˜ ts (ϕ)ds+ Xst (ϕ)(A(ϕs (ϕ)) − A)Ω Ωtτ (ϕ) = Xτt (ϕ) + τ X t ˜ τti (ϕ) (ϕ), Xti (ϕ) (ϕ)(B(ϕti (ϕ) (ϕ)) − B)Ω + τ ≤ti (ϕ)
where Xτt (ϕ) is the matriciant of homogeneous impulsive system with constant coefficients dx ˜ t 6= ti (ϕ), = Ax, dt ˜ ∆x|t=ti (ϕ) = Bx that depends on ϕ ∈ T m as a parameter. Taking into account that matriciant Xτt (ϕ) satisfies the estimate [3]

t

Xτ (ϕ) ≤ Ke−µ(t−τ ) , t ≥ τ

for every ϕ ∈ T m and some K ≥ 1, where γ + p ln α < −µ < 0, we have Z t

t



Ωτ (ϕ) ≤ Ke−µ(t−τ ) + Ke−µ(t−s) A(ϕs (ϕ)) − A˜ kΩsτ (ϕ)k ds+ τ



X

−µ(t−ti (ϕ)) ˜ + Ke

Ωτti (ϕ) (ϕ) .

B(ϕti (ϕ) (ϕ)) − B τ ≤ti (ϕ)
From (8) it follows that for an arbitrary small εA and εB there exists moment T such that





˜

≤ εB

A(ϕt (ϕ)) − A˜ ≤ εA , B(ϕt (ϕ)) − B

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for all t ≥ T . Hence,

+

Z

eµ(t−τ ) Ωtτ (ϕ) ≤ K + t

τ

Z

T τ

KεA eµ(s−τ ) kΩsτ (ϕ)k ds +



Keµ(s−τ ) A(ϕs (ϕ)) − A˜ kΩsτ (ϕ)k ds+ X

τ ≤ti (ϕ)


× Ωτti (ϕ) (ϕ) +

X

τ ≤ti (ϕ)


˜ Keµ(ti (ϕ)−τ ) B(ϕti (ϕ) (ϕ)) − B

×



KεB eµ(ti (ϕ)−τ ) Ωτti (ϕ) (ϕ) .

Using the Gronwall–Bellman inequality for piecewise continuous functions [5, 6], we have

t

Ωτ (ϕ) ≤ K1 e−(µ−KεA −p ln(1+KεB ))(t−τ ) , where

T

Keµ(s−τ ) A(ϕs (ϕ)) − A˜ kΩsτ (ϕ)k ds+ K1 = K + τ



X

˜ Keµ(ti (ϕ)−τ ) B(ϕti (ϕ) (ϕ)) − B +

Ωτti (ϕ) (ϕ) . Z

τ ≤ti (ϕ)
Choosing εA and εB so that µ > KεA + p ln(1 + KεB ) the following estimate holds

t

Ωτ (ϕ) ≤ K1 e−γ1 (t−τ ) (12)

for all t ≥ τ and some K1 ≥ 1, γ1 > 0. Estimate (12) is sufficient condition for the existence and asymptotic stability of invariant set of system (1). Indeed, there exists Green–Samoilenko function G0 (τ, ϕ) of the form ( Ω0τ (ϕ), τ ≤ 0, (13) G0 (τ, ϕ) = 0, τ > 0.

We obtain function (13) by substituting an arbitrary matrix C(ϕ) by an identity matrix E. Condition (7) is fulfilled since the inequality (12) holds. Hence, invariant toroidal set x = u(ϕ) of system (1) may be represented as Z 0 X Ω0τ (ϕ)f (ϕτ (ϕ))dτ + Ω0ti (ϕ)+0 (ϕ)g(ϕti (ϕ) (ϕ)). (14) u(ϕ) = −∞

ti (ϕ)<0

Integral and sum from (14) converge since the inequality (12) holds and the limit (9) exists. Let us prove the asymptotic stability of the invariant set. Let x = x(t, ϕ) is an arbitrary solution of system (4) and x∗ = u(ϕt (ϕ)) is the solution that

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belongs to the invariant set. The difference of these solutions admits the representation x(t, ϕ) − u(ϕt (ϕ)) = Ωt0 (ϕ)(x(0, ϕ) − u(ϕ)). Taking into account the estimate (12), the following limit exists lim kx(t, ϕ) − u(ϕt (ϕ))k = 0.

t→∞

It proves asymptotic stability of invariant set x = u(ϕ). Let us prove that small perturbations of right-hand side of system (1) do not ruin the invariant set. Let A1 (ϕ) and B1 (ϕ) are continuous 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m square matrices. Consider perturbed system dϕ = a(ϕ), dt (15)

dx = (A(ϕ)+A1 (ϕ))x + f (ϕ), ϕ 6∈ Γ, dt ∆x|ϕ∈Γ = (B(ϕ) + B1 (ϕ))x + g(ϕ).

Theorem 2. Let the moments of impulsive perturbations {ti (ϕ)} are such that uniformly with respect to t ∈ R there exists a finite limit i(t, t + T˜) =p T˜ T˜→∞ lim

and the following inequality holds γ + p ln α < 0. Then there exist sufficiently small constants a1 > 0 and b1 > 0 such that for any continuous 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m functions A1 (ϕ) and B1 (ϕ) such that max kA1 (ϕ)k ≤ a1 , maxm kB1 (ϕ)k ≤ b1

ϕ∈T m

ϕ∈T

system (15) has asymptotically stable invariant toroidal set. P r o o f. Constants a1 and b1 exist since the matrices A1 (ϕ) and B1 (ϕ) are continuous functions defined in torus T m , which is a compact manifold. Consider impulsive system that depends on ϕ ∈ T m as a parameter that corresponds to system (15) (16)

dx = A(ϕt (ϕ))x + A1 (ϕt (ϕ))x, t 6= ti (ϕ), dt ∆x|t=ti (ϕ) = B(ϕti (ϕ) (ϕ))x + B1 (ϕti (ϕ) (ϕ))x.

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Matriciant Ψtτ (ϕ) of system (16) may be represented in the following way Z t t t Ωts (ϕ)A1 (ϕs (ϕ))Ψts (ϕ)ds+ Ψτ (ϕ) = Ωτ (ϕ) + τ X + Ωtti (ϕ) (ϕ)B1 (ϕti (ϕ) (ϕ))Ψτti (ϕ) (ϕ). τ ≤ti (ϕ)
Taking into account the estimate (12), we have Z t

γ1 (t−τ ) t K1 a1 eγ1 (s−τ ) kΨsτ (ϕ)k ds+ e Ψτ (ϕ) ≤ K1 + τ

X

γ1 (ti (ϕ)−τ ) ti (ϕ) K1 b1 e +

Ψτ (ϕ) . τ ≤ti (ϕ)
Using the Gronwall–Bellman inequality for piecewise continuous functions, we obtain

t

Ψτ (ϕ) ≤ K1 e−(γ1 −K1 a1 −p ln(1+K1 b1 ))(t−τ ) . Let a1 and b1 are such that γ1 > K1 a1 + p ln(1 + K1 b1 ). Then the matriciant Ψtτ (ϕ) satisfies the estimate

t

Ψτ (ϕ) ≤ K2 e−γ2 (t−τ ) (17) for all t ≥ τ and some K2 ≥ 1, γ2 > 0. As in theorem 1, utilizing the estimate (17), we conclude that system (15) has asymptotically stable invariant set x = u(ϕ), which admits the representation Z 0 X Ψ0τ (ϕ)f (ϕτ (ϕ))dτ + Ψ0ti (ϕ)+0 (ϕ)g(ϕti (ϕ) (ϕ)). u(ϕ) = −∞

ti (ϕ)<0

Consider the nonlinear system of differential equations with impulsive perturbations of the form dϕ dx = a(ϕ), = F (ϕ, x), ϕ 6∈ Γ, (18) dt dt ∆x|ϕ∈Γ = I(ϕ, x), where ϕ ∈ T m , x ∈ J¯h , a(ϕ) is continuous 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m vector function and satisfies Lipschitz conditions (2); F (ϕ, x) and I(ϕ, x) are continuous 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m functions that have up to second continuous partial derivative by x. Taking these assumptions into account, system (18) may be rewritten in the following form dϕ dx = a(ϕ), = A0 (ϕ)x + A1 (ϕ, x)x + f (ϕ), ϕ 6∈ Γ, dt dt ∆x|ϕ∈Γ = B0 (ϕ)x + B1 (ϕ, x)x + g(ϕ),

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where A(ϕ, x) = B(ϕ, x) =

Z

1 0

Z

1 0

∂F (ϕ, τ x) dτ, ∂(τ x)

A0 (ϕ) = A(ϕ, 0),

A1 (ϕ, x) = A(ϕ, x) − A(ϕ, 0) ,

∂I(ϕ, τ x) dτ , ∂(τ x)

B0 (ϕ) = B(ϕ, 0),

B1 (ϕ, x) = B(ϕ, x)−B(ϕ, 0) ,

f (ϕ) = F (ϕ, 0) ,

g(ϕ) = I(ϕ, 0) .

Utilizing the iteration method proposed in [4] and the previous theorems one can prove the following assertion Theorem 3. Let matrices A0 (ϕ) and B0 (ϕ) are constant in domain Ω: ˜ B0 (ϕ)|ϕ∈Ω = B, ˜ A0 (ϕ)|ϕ∈Ω = A, uniformly with respect to t ∈ R there exists a finite limit i(t, t + T˜) =p T˜ T˜→∞ lim

and the following inequality holds γ + p ln α < 0. Then there exist sufficiently small constants a1 and b1 and sufficiently small Lipschitz constants LA and LB such that for any continuous 2π-periodic with respect to each of the components ϕv , v = 1, . . . , m that have up to second continuous partial derivative by x matrices F (ϕ, x) and I(ϕ, x) such that max

ϕ∈T m ,x∈J¯h

kA1 (ϕ, x)k ≤ a1 ,

max

ϕ∈T m ,x∈J¯h

kB1 (ϕ, x)k ≤ b1

′ ′′ and for any x , x ∈ J¯h

′

′′ ′ ′′

A1 (ϕ, x ) − A1 (ϕ, x ) ≤ LA x − x ,

′

′′ ′ ′′

B1 (ϕ, x ) − B1 (ϕ, x ) ≤ LB x − x

system (18) has asymptotically stable invariant toroidal set. Conclusion

We have obtained sufficient conditions for the existence and asymptotic stability of invariant toroidal sets of impulsive system of differential equations defined in direct product of torus T m and Euclidean space E n that has specific properties in ω-limit set Ω of the trajectories ϕt (ϕ).

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[2]

[3]

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References D o u d z y a n y S . I . , P e r e s t y u k M . O . ”On stability of trivial invariant torus of a certain class of systems with impulsive perturbations”, Ukrainian Mathematical Journal, vol. 50, no. 3, 1998, pp. 338-349. P e r e s t y u k M . O . , B a l o h a S . I . ”Existence of invariant torus of a certain class of systems of differential equations”, Nonlinear Oscillations, vol. 11, no. 4, 2008, pp. 520-529. S a m o i l e n k o A . M . , P e r e s t y u k M . O . ”On stability of solutions of systems with impulsive perturbations”, Differential Equations, vol. XVII, no. 11, 1981, pp. 1995-2001 [in Russian]. S a m o i l e n k o A . M . Elements of mathematical theory of multifrequency oscillations, Kluwer Acad. Publ., Dordrecht, 1991. S a m o i l e n k o A . M . , P e r e s t y u k N . A . Impulsive differential equations, Vyscha Shkola, Kyiv, 1987, [in Russian]. S a m o i l e n k o A . M . , P e r e s t y u k N . A . Impulsive differential equations, World Scientific, Singapore, 1995.

Mykola Perestyuk Faculty of Mechanics and Mathematics Kyiv Taras Shevchenko National University Volodymyrska Str. 64, Kyiv 01033, Ukraine E-MAIL: [email protected] Petro Feketa Faculty of Mechanics and Mathematics Kyiv Taras Shevchenko National University Volodymyrska Str. 64, Kyiv 01033, Ukraine E-MAIL: [email protected]