Invariant Sets of Impulsive Differential Equations with Particularities in ω-Limit Set

Abstract

Sufficient conditions for the existence and asymptotic stability of the invariant sets of an impulsive system of differential equations defined in the direct product of a torus and an Euclidean space are obtained.

1. Introduction

The evolution of variety of processes in physics, chemistry, biology, and so forth, frequently undergoes short-term perturbations. It is sometimes convenient to neglect the duration of the perturbations and to consider these perturbations to be “instantaneous.” This leads to the necessity of studying the differential equations with discontinuous trajectories, the so-called impulsive differential equations. The fundamentals of the mathematical theory of impulsive differential equations are stated in [1–4]. The theory is developing intensively due to its applied value in simulations of the real world phenomena.

At the same time, this paper is closely related to the oscillation theory. In the middle of the 20th century, a sharp turn towards the investigations of the oscillating processes that were characterized as “almost exact” iterations within “almost the same” periods of time took place. Quasiperiodic oscillations were brought to the primary focus of investigations of the oscillation theory [5].

Quasiperiodic oscillations are a sufficiently complicated and sensitive object for investigating. The practical value of indicating such oscillations is unessential. Due to the instability of frequency basis, quasiperiodic oscillation collapses easily and may be transformed into periodic oscillation via small shift of the right-hand side of the system. This fact has led to search for more rough object than the quasiperiodic solution. Thus the minimal set that is covered by the trajectories of the quasiperiodic motions becomes the main object of investigations. As it is known, such set is a torus. The first profound assertions regarding the invariant toroidal manifolds were obtained by Bogoliubov et al. [6, 7]. Further results in this area were widely extended by many authors.

Consider the system of differential equations 𝑑𝑧(𝑧)𝑑𝑡=𝐹, (1.1) where the function 𝐹(𝑧) is defined in some subset 𝐷 of the (𝑚+𝑛)-dimensional Euclidean space 𝐸^𝑚+𝑛, continuous and satisfies a Lipschitz condition. Let 𝑀 be an invariant toroidal manifold of the system. While investigating the trajectories that begin in the neighborhood of the manifold 𝑀, it is convenient to make the change of variables from Euclidean coordinates (𝑧₁,…,𝑧_𝑚+𝑛) to so-called local coordinates 𝜑=(𝜑₁,…,𝜑_𝑚), 𝑥=(𝑥₁,…,𝑥_𝑛), where 𝜑 is a point on the surface of an 𝑚-dimensional torus 𝒯^𝑚 and 𝑥 is a point in an 𝑛-dimensional Euclidean space 𝐸^𝑛. The change of variables is performed in a way such that the equation, which defines the invariant manifold 𝑀, transforms into 𝑥=0, 𝜑∈𝒯^𝑚 in the new coordinates. In essence, the manifold 𝑥=0, 𝜑∈𝒯^𝑚 is the 𝑚-dimensional torus in the space 𝒯^𝑚×𝐸^𝑛. The character of stability of the invariant torus 𝑀 is closely linked with stability of the set 𝑥=0, 𝜑∈𝒯^𝑚: from stability, asymptotic stability, and instability of the manifold 𝑀, there follow the stability, asymptotic stability, and instability of the torus 𝑥=0, 𝜑∈𝒯^𝑚 correspondingly and vice versa. This is what determines the relevance and value of the investigation of conditions for the existence and stability of invariant sets of the systems of differential equations defined in 𝒯^𝑚×𝐸^𝑛. Theory of the existence and perturbation, properties of smoothness, and stability of invariant sets of systems defined in 𝒯^𝑚×𝐸^𝑛 are considered in [8].

2. Preliminaries

The main object of investigation of this paper is the system of differential equations, defined in the direct product of an 𝑚-dimensional torus 𝒯^𝑚 and an 𝑛-dimensional Euclidean space 𝐸^𝑛 that undergo impulsive perturbations at the moments when the phase point 𝜑 meets a given set in the phase space. Consider the system𝑑𝜑𝜑,𝑑𝑡=𝑎𝑑𝑥𝜑𝜑|𝑑𝑡=𝐴𝑥+𝑓,𝜑∉Γ,Δ𝑥_𝜑∈Γ=𝐵𝜑𝑥+𝑔𝜑, (2.1) where 𝜑=(𝜑₁,…,𝜑_𝑚)^𝑇∈𝒯^𝑚, 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝐸^𝑛, 𝑎(𝜑) is a continuous 2𝜋-periodic with respect to each of the components 𝜑_𝑣, 𝑣=1,…,𝑚 vector function that satisfies a Lipschitz condition𝑎𝜑^′′−𝑎𝜑^′≤𝐿𝜑^′′−𝜑^′ (2.2) for every 𝜑^′,𝜑^′′∈𝒯^𝑚. 𝐴(𝜑),𝐵(𝜑) are continuous 2𝜋-periodic with respect to each of the components 𝜑_𝑣, 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 𝜑_𝑣, 𝑣=1,…,𝑚 vector functions.

Some aspects regarding existence and stability of invariant sets of systems similar to (2.1) were considered by different authors in [9–12].

We regard the point 𝜑=(𝜑₁,…,𝜑_𝑚)^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 𝜑_𝑣, 𝑣=1,…,𝑚 function.

The system of differential equations𝑑𝜑𝜑𝑑𝑡=𝑎 (2.3) defines a dynamical system on the torus 𝒯^𝑚. Denote by 𝜑_𝑡(𝜑) the solution of (2.3) that satisfies the initial condition 𝜑₀(𝜑)=𝜑. The Lipschitz condition (2.2) guarantees the existence and uniqueness of such solution. Moreover, the solutions 𝜑_𝑡(𝜑) satisfies a group property [8]𝜑_𝑡𝜑_𝜏𝜑=𝜑_𝑡+𝜏𝜑 (2.4) for all 𝑡,𝜏∈ℝ and 𝜑∈𝒯^𝑚.

Denote by 𝑡_𝑖(𝜑), 𝑖∈ℤ the solutions of the equation Φ(𝜑_𝑡(𝜑))=0 that are the moments of impulsive action in system (2.1). Let the function Φ(𝜑) be such that the solutions 𝑡=𝑡_𝑖(𝜑) exist, since otherwise, system (2.1) would not be an impulsive system. Assume that lim_𝑖→±∞𝑡_𝑖𝜑=±∞,lim_𝑇→±∞𝑖(𝑡,𝑡+𝑇)𝑇=𝑝<∞ (2.5) uniformly with respect to 𝑡∈ℝ, where 𝑖(𝑎,𝑏) is the number of the points 𝑡_𝑖(𝜑) in the interval (𝑎,𝑏). Hence, the moments of impulsive perturbations 𝑡_𝑖(𝜑) satisfy the equality [10, 11]𝑡_𝑖𝜑_−𝑡𝜑−𝑡_𝑖𝜑=𝑡. (2.6)

Together with system (2.1), we consider the linear system𝑑𝑥𝜑𝑑𝑡=𝐴_𝑡𝜑𝑥+𝑓𝜑_𝑡𝜑,𝑡≠𝑡_𝑖𝜑,Δ𝑥|_{𝑡=𝑡𝑖(𝜑)}=𝐵𝜑_{𝑡𝑖(𝜑)}𝜑𝑥+𝑔𝜑_{𝑡𝑖(𝜑)}𝜑 (2.7) that depends on 𝜑∈𝒯^𝑚 as a parameter. We obtain system (2.7) by substituting 𝜑_𝑡(𝜑) for 𝜑 in the second and third equations of system (2.1). By invariant set of system (2.1), we understand a set that is defined by a function 𝑢(𝜑), which has a period 2𝜋 with respect to each of the components 𝜑_𝑣, 𝑣=1,…,𝑚, such that the function 𝑥(𝑡,𝜑)=𝑢(𝜑_𝑡(𝜑)) is a solution of system (2.7) for every 𝜑∈𝒯^𝑚.

We call a point 𝜑^∗ an 𝜔-limit point of the trajectory 𝜑_𝑡(𝜑) if there exists a sequence {𝑡_𝑛}_𝑛∈ℕ in ℝ so that lim_𝑛→+∞𝑡_𝑛=+∞,lim_𝑛→+∞𝜑_𝑡𝑛𝜑=𝜑^∗. (2.8) The set of all 𝜔-limit points for a given trajectory 𝜑_𝑡(𝜑) is called 𝜔-limit set of the trajectory 𝜑_𝑡(𝜑) and denoted by Ω_𝜑.

Referring to system (2.7), the matrices 𝐴(𝜑_𝑡(𝜑)) and 𝐵(𝜑_𝑡(𝜑)), that influence the behavior of the solution 𝑥(𝑡,𝜑) of the system (2.7), depend not only on the functions 𝐴(𝜑) and 𝐵(𝜑) but also on the behavior of the trajectories 𝜑_𝑡(𝜑). Moreover, in [9], sufficient conditions for the existence and stability of invariant sets of a system similar to (2.1) were obtained in terms of a Lyapunov function 𝑉(𝜑,𝑥) that satisfies some conditions in the domain 𝑍={𝜑∈Ω,𝑥∈𝐽_ℎ}, where 𝐽_ℎ={𝑥∈𝐸^𝑛,‖𝑥‖≤ℎ,ℎ>0}, Ω=_{𝜑∈𝒯𝑚}Ω_𝜑. (2.9) Since the Lyapunov function has to satisfy some conditions not on the whole surface of the torus 𝒯^𝑚 but only in the 𝜔-limit set Ω, it is interesting to consider system (2.1) with specific properties in the domain Ω.

3. Main Result

Consider system (2.1) assuming that the matrices 𝐴(𝜑) and 𝐵(𝜑) are constant in the domain Ω: 𝐴𝜑_𝜑∈Ω=𝐵𝜑𝐴,_𝜑∈Ω=𝐵. (3.1) Therefore, for every 𝜑∈𝒯^𝑚lim_𝑡→+∞𝐴𝜑_𝑡𝜑=𝐴,lim_𝑡→+∞𝐵𝜑_𝑡𝜑=𝐵. (3.2) We will obtain sufficient conditions for the existence and asymptotic stability of an invariant set of the system (2.1) in terms of the eigenvalues of the matrices 𝐴 and 𝐵. Denote by 𝛾=max_{𝑗=1,…,𝑛}Re𝜆_𝑗𝐴,𝛼²=max_{𝑗=1,…,𝑛}𝜆_𝑗𝐸+𝐵^𝑇𝐸+𝐵. (3.3) Similar systems without impulsive perturbations have been considered in [13].

Theorem 3.1. Let the moments of impulsive perturbations {𝑡_𝑖(𝜑)} be such that uniformly with respect to 𝑡∈ℝ there exists a finite limit 𝑇lim_→∞𝑖𝑡,𝑡+𝑇𝑇=𝑝. (3.4) If the following inequality holds 𝛾+𝑝ln𝛼<0, (3.5) then system (2.1) has an asymptotically stable invariant set.

Proof. Consider a homogeneous system of differential equations𝑑𝑥𝜑𝑑𝑡=𝐴_𝑡𝜑𝑥,𝑡≠𝑡_𝑖𝜑,Δ𝑥|_{𝑡=𝑡𝑖(𝜑)}=𝐵𝜑_{𝑡𝑖(𝜑)}𝜑𝑥 (3.6) that depends on 𝜑∈𝒯^𝑚 as a parameter. By Ω^𝑡_𝜏(𝜑), we denote the fundamental matrix of system (3.6), which turns into an identity matrix at the point 𝑡=𝜏, that is, Ω^𝜏_𝜏(𝜑)≡𝐸. It can be readily verified [4] that Ω^𝑡_𝜏(𝜑) satisfies the equalities 𝜕Ω𝜕𝑡^𝑡𝜏𝜑=𝐴𝜑_𝑡𝜑Ω^𝑡_𝜏𝜑,Ω^𝑡_𝜏𝜑=Ω^𝑡_𝜏𝜑+2𝜋𝑒_𝑘,Ω^𝑡_𝑡+𝜏𝜑_−𝑡𝜑=Ω⁰_𝜏𝜑 (3.7) for all 𝑡,𝜏∈ℝ and 𝜑∈𝒯^𝑚. Rewrite system (3.6) in the form 𝑑𝑥=𝐴𝜑𝑑𝑡𝐴𝑥+_𝑡𝜑−𝐴𝑥,𝑡≠𝑡_𝑖𝜑,Δ𝑥|_{𝑡=𝑡𝑖(𝜑)}=𝐵𝑥+𝐵𝜑_{𝑡𝑖(𝜑)}𝜑−𝐵𝑥. (3.8) The fundamental matrix Ω^𝑡_𝜏(𝜑) of the system (3.6) may be represented in the following way [4]: Ω^𝑡_𝜏𝜑=𝑋^𝑡_𝜏𝜑+^𝑡_𝜏𝑋^𝑡_𝑠𝜑𝐴𝜑_𝑠𝜑−𝐴Ω^𝑡_𝑠𝜑𝑑𝑠+_{𝜏≤𝑡𝑖(𝜑)<𝑡}𝑋^𝑡_𝑡𝑖𝜑𝜑𝐵𝜑_{𝑡𝑖(𝜑)}𝜑−𝐵Ω^𝑡𝑖_𝜏^(𝜑)𝜑, (3.9) where 𝑋^𝑡_𝜏(𝜑) is the fundamental matrix of the homogeneous impulsive system with constant coefficients 𝑑𝑥=𝑑𝑡𝐴𝑥,𝑡≠𝑡_𝑖𝜑,Δ𝑥|_{𝑡=𝑡𝑖(𝜑)}=𝐵𝑥 (3.10) that depends on 𝜑∈𝒯^𝑚 as a parameter. Taking into account that the matrix 𝑋^𝑡_𝜏(𝜑) satisfies the estimate [14] 𝑋^𝑡_𝜏𝜑≤𝐾𝑒^{−𝜇(𝑡−𝜏)},𝑡≥𝜏 (3.11) for every 𝜑∈𝒯^𝑚 and some 𝐾≥1, where 𝛾+𝑝ln𝛼<−𝜇<0, we obtain Ω^𝑡_𝜏𝜑≤𝐾𝑒^{−𝜇(𝑡−𝜏)}+^𝑡_𝜏𝐾𝑒^{−𝜇(𝑡−𝑠)}𝐴𝜑_𝑠𝜑−𝐴Ω^𝑠_𝜏𝜑𝑑𝑠+_{𝜏≤𝑡𝑖(𝜑)<𝑡}𝐾𝑒^{−𝜇(𝑡−𝑡𝑖(𝜑))}𝐵𝜑_{𝑡𝑖(𝜑)}𝜑−𝐵Ω^𝑡𝑖_𝜏^(𝜑)𝜑. (3.12)
It follows from (3.2) that for arbitrary small 𝜀_𝐴 and 𝜀_𝐵, there exists a moment 𝑇 such that𝐴𝜑_𝑡𝜑−𝐴≤𝜀_𝐴,𝐵𝜑_𝑡𝜑−𝐵≤𝜀_𝐵 (3.13) for all 𝑡≥𝑇. Hence, multiplying (3.12) by 𝑒^{𝜇(𝑡−𝜏)}, utilizing (3.13), and weakening the inequality, we obtain 𝑒^{𝜇(𝑡−𝜏)}Ω^𝑡_𝜏𝜑≤𝐾+^𝑇_𝜏𝐾𝑒^{𝜇(𝑠−𝜏)}𝐴𝜑_𝑠𝜑−𝐴Ω^𝑠_𝜏𝜑+𝑑𝑠_𝑡𝜏𝐾𝜀_𝐴𝑒^{𝜇(𝑠−𝜏)}Ω^𝑠_𝜏𝜑𝑑𝑠+_{𝜏≤𝑡𝑖(𝜑)<𝑇}𝐾𝑒^{𝜇(𝑡𝑖(𝜑)−𝜏)}𝐵𝜑_{𝑡𝑖(𝜑)}𝜑−𝐵Ω^𝑡𝑖_𝜏^(𝜑)𝜑+_{𝜏≤𝑡𝑖(𝜑)<𝑡}𝐾𝜀_𝐵𝑒^{𝜇(𝑡𝑖(𝜑)−𝜏)}Ω^𝑡𝑖_𝜏^(𝜑)𝜑. (3.14) Using the Gronwall-Bellman inequality for piecewise continuous functions [4], we obtain the estimate for the fundamental matrix Ω^𝑡_𝜏(𝜑) of the system (3.6) Ω^𝑡_𝜏𝜑≤𝐾₁𝑒^{−(𝜇−𝐾𝜀𝐴−𝑝ln(1+𝐾𝜀𝐵))(𝑡−𝜏)}, (3.15) where 𝐾₁=𝐾+^𝑇_𝜏𝐾𝑒^{𝜇(𝑠−𝜏)}𝐴𝜑_𝑠𝜑−𝐴Ω^𝑠_𝜏𝜑𝑑𝑠+_{𝜏≤𝑡𝑖(𝜑)<𝑇}𝐾𝑒^{𝜇(𝑡𝑖(𝜑)−𝜏)}𝐵𝜑_{𝑡𝑖(𝜑)}𝜑−𝐵Ω^𝑡𝑖_𝜏^(𝜑)𝜑. (3.16) Choosing 𝜀_𝐴 and 𝜀_𝐵 so that 𝜇>𝐾𝜀_𝐴+𝑝ln(1+𝐾𝜀_𝐵), the following estimate holds Ω^𝑡_𝜏𝜑≤𝐾₁𝑒^{−𝛾1(𝑡−𝜏)} (3.17) for all 𝑡≥𝜏 and some 𝐾₁≥1, 𝛾₁>0.
Estimate (3.17) is a sufficient condition for the existence and asymptotic stability of an invariant set of system (2.1). Indeed, it is easy to verify that invariant set 𝑥=𝑢(𝜑) of the system (2.1) may be represented as𝑢𝜑=⁰_−∞Ω⁰_𝜏𝜑𝑓𝜑_𝜏𝜑𝑑𝜏+_{𝑡𝑖(𝜑)<0}Ω⁰_{𝑡𝑖(𝜑)}𝜑𝑔𝜑_{𝑡𝑖(𝜑)}𝜑. (3.18) The integral and the sum from (3.18) converge since inequality (3.17) holds and limit (3.4) exists. Utilizing the properties (3.7) of the matrix Ω^𝑡_𝜏(𝜑) (2.4), and (2.6), one can show that the function 𝑢(𝜑_𝑡(𝜑)) satisfies the equation 𝑑𝑥𝜑𝑑𝑡=𝐴_𝑡𝜑𝑥+𝑓𝜑_𝑡𝜑 (3.19) for 𝑡≠𝑡_𝑖(𝜑) and has discontinuities 𝐵(𝜑_{𝑡𝑖(𝜑)}(𝜑))𝑢(𝜑_{𝑡𝑖(𝜑)}(𝜑))+𝑔(𝜑_{𝑡𝑖(𝜑)}(𝜑)) at the points 𝑡=𝑡_𝑖(𝜑). It means that the function 𝑥(𝑡,𝜑)=𝑢(𝜑_𝑡(𝜑)) is a solution of the system (2.7). Hence, 𝑢(𝜑) defines the invariant set of system (2.1).
Let us prove the asymptotic stability of the invariant set. Let 𝑥=𝑥(𝑡,𝜑) be an arbitrary solutions of the system (2.7), and 𝑥^∗=𝑢(𝜑_𝑡(𝜑)) is the solution that belongs to the invariant set. The difference of these solutions admits the representation𝑥𝑡,𝜑−𝑢𝜑_𝑡𝜑=Ω^𝑡₀𝜑𝑥0,𝜑−𝑢𝜑. (3.20) Taking into account estimate (3.17), the following limit exists lim_𝑡→∞𝑥𝑡,𝜑−𝑢𝜑_𝑡𝜑=0. (3.21) It proves the asymptotic stability of the invariant set 𝑥=𝑢(𝜑).

4. Perturbation Theory

Let us show that small perturbations of the right-hand side of the system (2.1) do not ruin the invariant set. Let 𝐴₁(𝜑) and 𝐵₁(𝜑) be continuous 2𝜋-periodic with respect to each of the components 𝜑_𝑣, 𝑣=1,…,𝑚 square matrices. Consider the perturbed system𝑑𝜑𝜑,𝑑𝑡=𝑎𝑑𝑥=𝐴𝜑𝑑𝑡+𝐴₁𝜑𝜑|𝑥+𝑓,𝜑∉Γ,Δ𝑥_𝜑∈Γ=𝐵𝜑+𝐵₁𝜑𝑥+𝑔𝜑. (4.1)

Theorem 4.1. Let the moments of impulsive perturbations {𝑡_𝑖(𝜑)} be such that uniformly with respect to 𝑡∈ℝ, there exists a finite limit 𝑇lim_→∞𝑖𝑡,𝑡+𝑇𝑇=𝑝 (4.2) and the following inequality holds 𝛾+𝑝ln𝛼<0. (4.3) 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 𝜑_𝑣, 𝑣=1,…,𝑚 functions 𝐴₁(𝜑) and 𝐵₁(𝜑) such that max_{𝜑∈𝒯𝑚}𝐴₁𝜑≤𝑎₁,max_{𝜑∈𝒯𝑚}𝐵₁𝜑≤𝑏₁, (4.4) system (4.1) has an asymptotically stable invariant set.

Proof. The constants 𝑎₁ and 𝑏₁ exist since the matrices 𝐴₁(𝜑) and 𝐵₁(𝜑) are continuous functions defined in the torus 𝒯^𝑚, which is a compact manifold.
Consider the impulsive system that corresponds to system(4.1)𝑑𝑥𝜑𝑑𝑡=𝐴_𝑡𝜑𝑥+𝐴₁𝜑_𝑡𝜑𝑥,𝑡≠𝑡_𝑖𝜑,Δ𝑥|_{𝑡=𝑡𝑖(𝜑)}=𝐵𝜑_{𝑡𝑖(𝜑)}𝜑𝑥+𝐵₁𝜑_{𝑡𝑖(𝜑)}𝜑𝑥 (4.5) that depends on 𝜑∈𝒯^𝑚 as a parameter. The fundamental matrix Ψ^𝑡_𝜏(𝜑) of the system (4.5) may be represented in the following way Ψ^𝑡_𝜏𝜑=Ω^𝑡_𝜏𝜑+^𝑡_𝜏Ω^𝑡_𝑠𝜑𝐴₁𝜑_𝑠𝜑Ψ^𝑡_𝑠𝜑𝑑𝑠+_{𝜏≤𝑡𝑖(𝜑)<𝑡}Ω^𝑡_{𝑡𝑖(𝜑)}𝜑𝐵₁𝜑_{𝑡𝑖(𝜑)}𝜑Ψ^𝑡𝑖_𝜏^(𝜑)𝜑, (4.6) where Ω^𝑡_𝜏(𝜑) is the fundamental matrix of the system (3.6). Then taking estimate (3.17) into account, 𝑒^{𝛾1(𝑡−𝜏)}Ψ^𝑡_𝜏𝜑≤𝐾₁+^𝑡_𝜏𝐾₁𝑎₁𝑒^{𝛾1(𝑠−𝜏)}Ψ^𝑠_𝜏𝜑𝑑𝑠+_{𝜏≤𝑡𝑖(𝜑)<𝑡}𝐾₁𝑏₁𝑒^{𝛾1(𝑡𝑖(𝜑)−𝜏)}Ψ^𝑡𝑖_𝜏^(𝜑)𝜑. (4.7) Using the Gronwall-Bellman inequality for piecewise continuous functions, we obtain the estimate for the fundamental matrix Ψ^𝑡_𝜏(𝜑) of the system (4.5) Ψ^𝑡_𝜏𝜑≤𝐾₁𝑒^{−(𝛾1−𝐾1𝑎1−𝑝ln(1+𝐾1𝑏1))(𝑡−𝜏)}. (4.8) Let the constants 𝑎₁ and 𝑏₁ be such that 𝛾₁>𝐾₁𝑎₁+𝑝ln(1+𝐾₁𝑏₁). Hence, the matrix Ψ^𝑡_𝜏(𝜑) satisfies the estimate Ψ^𝑡_𝜏𝜑≤𝐾₂𝑒^{−𝛾2(𝑡−𝜏)} (4.9) for all 𝑡≥𝜏 and some 𝐾₂≥1, 𝛾₂>0. As in Theorem 3.1, from estimate (4.9), we conclude that the system (4.1) has an asymptotically stable invariant set 𝑥=𝑢(𝜑), which admits the representation 𝑢𝜑=⁰_−∞Ψ⁰_𝜏𝜑𝑓𝜑_𝜏𝜑𝑑𝜏+_{𝑡𝑖(𝜑)<0}Ψ⁰_{𝑡𝑖(𝜑)}𝜑𝑔𝜑_{𝑡𝑖(𝜑)}𝜑. (4.10)

Consider the nonlinear system of differential equations with impulsive perturbations of the form𝑑𝜑𝜑,𝑑𝑡=𝑎𝑑𝑥|𝑑𝑡=𝐹𝜑,𝑥,𝜑∉Γ,Δ𝑥_𝜑∈Γ=𝐼𝜑,𝑥, (4.11) where 𝜑∈𝒯^𝑚, 𝑥∈𝐽_ℎ, 𝑎(𝜑) is a continuous 2𝜋-periodic with respect to each of the components 𝜑_𝑣, 𝑣=1,…,𝑚 vector function and satisfies Lipschitz conditions (2.2); 𝐹(𝜑,𝑥) and 𝐼(𝜑,𝑥) are continuous 2𝜋-periodic with respect to each of the components 𝜑_𝑣, 𝑣=1,…,𝑚 functions that have continuous partial derivatives with respect to 𝑥 up to the second order inclusively. Taking these assumptions into account, system (4.11) may be rewritten in the following form:𝑑𝜑𝜑,𝑑𝑡=𝑎𝑑𝑥𝑑𝑡=𝐴₀𝜑𝑥+𝐴₁𝜑|𝜑,𝑥𝑥+𝑓,𝜑∉Γ,Δ𝑥_𝜑∈Γ=𝐵₀𝜑𝑥+𝐵₁𝜑,𝑥𝑥+𝑔𝜑, (4.12) where 𝐴𝜑,𝑥=¹₀𝜕𝐹𝜑,𝜏𝑥𝜕()=𝜏𝑥𝑑𝜏,𝐵𝜑,𝑥¹₀𝜕𝐼𝜑,𝜏𝑥𝜕(𝜏𝑥)𝑑𝜏, (4.13)

𝐴₀(𝜑)=𝐴(𝜑,0), 𝐴₁(𝜑,𝑥)=𝐴(𝜑,𝑥)−𝐴(𝜑,0), 𝐵₀(𝜑)=𝐵(𝜑,0), 𝐵₁(𝜑,𝑥)=𝐵(𝜑,𝑥)−𝐵(𝜑,0), 𝑓(𝜑)=𝐹(𝜑,0), and 𝑔(𝜑)=𝐼(𝜑,0). We assume that the matrices 𝐴₀(𝜑) and 𝐵₀(𝜑) are constant in the domain Ω: 𝐴₀(𝜑)_𝜑∈Ω=𝐵𝐴,₀(𝜑)_𝜑∈Ω=𝐵 (4.14) and the inequality 𝛾+𝑝ln𝛼<0 holds.

We will construct the invariant set of system (4.12) using an iteration method proposed in [8]. As initial invariant set 𝑀₀, we consider the set 𝑥=0, as 𝑀_𝑘—the invariant set of the system𝑑𝜑𝜑,𝑑𝑡=𝑎𝑑𝑥𝑑𝑡=𝐴₀𝜑𝑥+𝐴₁𝜑,𝑢_𝑘−1𝜑𝜑|𝑥+𝑓,𝜑∉Γ,Δ𝑥_𝜑∈Γ=𝐵₀𝜑𝑥+𝐵₁𝜑,𝑢_𝑘−1𝜑𝑥+𝑔𝜑, (4.15) where 𝑥=𝑢_𝑘−1(𝜑) is the invariant set on (𝑘−1)-step.

Using Theorem 4.1, the invariant set 𝑥=𝑢_𝑘(𝜑), 𝑘=1,2,… may be represented as𝑢_𝑘𝜑=⁰_−∞Ψ⁰_𝜏𝜑,𝑘𝑓𝜑_𝜏𝜑𝑑𝜏+_{𝑡𝑖(𝜑)<0}Ψ⁰_{𝑡𝑖(𝜑)+0}𝜑,𝑘𝑔𝜑_{𝑡𝑖(𝜑)}𝜑, (4.16) where Ψ^𝑡_𝜏(𝜑,𝑘) is the fundamental matrix of the homogeneous system 𝑑𝑥=𝐴𝑑𝑡⁰𝜑_𝑡𝜑+𝐴₁𝜑_𝑡𝜑,𝑢_𝑘−1𝜑_𝑡𝜑𝑥,𝑡≠𝑡_𝑖𝜑,Δ𝑥|_{𝑡=𝑡𝑖(𝜑)}=𝐵₀𝜑_𝑡𝜑+𝐵₁𝜑_𝑡𝜑,𝑢_𝑘−1𝜑_𝑡𝜑𝑥 (4.17) that depends on 𝜑∈𝒯^𝑚 as a parameter and satisfies the estimateΨ^𝑡_𝜏𝜑,𝑘≤𝐾₂𝑒^{−𝛾2(𝑡−𝜏)} (4.18) for all 𝑡≥𝜏 and some 𝐾₂≥1, 𝛾₂>0 only if max_{𝜑∈𝒯𝑚}𝐴₁𝜑,𝑢_𝑘−1𝜑≤𝑎₁,max_{𝜑∈𝒯𝑚}𝐵₁𝜑,𝑢_𝑘−1𝜑≤𝑏₁. (4.19) Let us prove that the invariant sets 𝑥=𝑢_𝑘(𝜑) belong to the domain 𝐽_ℎ. Denote bymax_{𝜑∈𝒯𝑚}𝑓𝜑≤𝑀_𝑓,max_{𝜑∈𝒯𝑚}𝑔𝜑≤𝑀_𝑔. (4.20) Since the torus 𝒯^𝑚 is a compact manifold, such constants 𝑀_𝑓 and 𝑀_𝑔 exist. Analogously to [4], using the representation (4.16) and estimate (4.18), we obtain that𝑢_𝑘𝜑≤𝐾₂𝛾₂𝑀_𝑓+𝐾₂1−𝑒^{−𝛾2𝜃1}𝑀_𝑔, (4.21) where 𝜃₁ is a minimum gap between moments of impulsive actions. Condition (3.4) guarantees that such constant 𝜃₁ exists. Assume that the constants 𝐾₂ and 𝛾₂ are such that ‖𝑢(𝜑)‖≤ℎ.

Let us obtain the conditions for the convergence of the sequence {𝑢_𝑘(𝜑)}. For this purpose, we estimate the difference 𝑤_𝑘+1(𝜑)=𝑢_𝑘+1(𝜑)−𝑢_𝑘(𝜑) and take into account that the functions 𝑢_𝑘(𝜑_𝑡(𝜑)) satisfy the relations 𝑑𝑢𝑑𝑡^𝑘𝜑_𝑡𝜑=𝐴₀𝜑_𝑡𝜑+𝐴₁𝜑_𝑡𝜑,𝑢_𝑘−1𝜑_𝑡𝜑×𝑢_𝑘𝜑_𝑡𝜑+𝑓𝜑_𝑡𝜑,𝑡≠𝑡_𝑖𝜑,Δ𝑢_𝑘𝜑_𝑡𝜑_{𝑡=𝑡𝑖(𝜑)}=𝐵₀𝜑_𝑡𝜑+𝐵₁𝜑_𝑡𝜑,𝑢_𝑘−1𝜑_𝑡𝜑×𝑢_𝑘𝜑_𝑡𝜑+𝑔𝜑_𝑡𝜑 (4.22) for all 𝜑∈𝒯^𝑚, 𝑘=1,2,…. Hence, the difference 𝑤_𝑘+1(𝜑)=𝑢_𝑘+1(𝜑)−𝑢_𝑘(𝜑) is the invariant set of the linear impulsive system 𝑑𝜑𝜑,𝑑𝑡=𝑎𝑑𝑥=𝐴𝑑𝑡⁰𝜑+𝐴₁𝜑,𝑢_𝑘𝜑𝑥+𝐴₁𝜑,𝑢_𝑘𝜑−𝐴₁𝜑,𝑢_𝑘−1𝜑𝑢_𝑘𝜑|,𝜑∉Γ,Δ𝑥_𝜑∈Γ=𝐵₀𝜑+𝐵₁𝜑,𝑢_𝑘𝜑𝑥+𝐵₁𝜑,𝑢_𝑘𝜑−𝐵₁𝜑,𝑢_𝑘−1𝜑𝑢_𝑘𝜑. (4.23) Then, taking (4.21) into account, max_{𝜑∈𝒯𝑚}𝑢_𝑘+1𝜑−𝑢_𝑘𝜑≤𝐾₂𝛾₂𝐴₁𝜑,𝑢_𝑘𝜑−𝐴₁𝜑,𝑢_𝑘−1𝜑𝑢_𝑘𝜑+𝐾₂1−𝑒^−𝛾2𝜃𝐵₁𝜑,𝑢_𝑘𝜑−𝐵₁𝜑,𝑢_𝑘−1𝜑𝑢_𝑘𝜑. (4.24) Let the functions 𝐴₁(𝜑,𝑥) and 𝐵₁(𝜑,𝑥) satisfy the Lipschitz condition with constants 𝐿_𝐴 and 𝐿_𝐵 correspondingly. Then max_{𝜑∈𝒯𝑚}𝑢_𝑘+1𝜑−𝑢_𝑘𝜑≤𝐾₂𝛾₂𝐿_𝐴ℎ𝑢_𝑘𝜑−𝑢_𝑘−1𝜑+𝐾₂1−𝑒^−𝛾2𝜃𝐿_𝐵ℎ𝑢_𝑘𝜑−𝑢_𝑘−1𝜑=𝐾₂ℎ𝛾₂𝐿_𝐴+𝐾₂ℎ1−𝑒^−𝛾2𝜃𝐿_𝐵𝑢_𝑘𝜑−𝑢_𝑘−1𝜑. (4.25) Assuming that the constants 𝐿_𝐴 and 𝐿_𝐵 are so small that 𝐾₂ℎ𝛾₂𝐿_𝐴+𝐾₂ℎ1−𝑒^−𝛾2𝜃𝐿_𝐵<1, (4.26) we conclude that the sequence {𝑢_𝑘(𝜑)} converges uniformly with respect to 𝜑∈𝒯^𝑚 and lim_𝑘→∞𝑢_𝑘𝜑=𝑢𝜑. (4.27) Thus, the invariant set 𝑥=𝑢(𝜑) admits the representation 𝑢𝜑=⁰_−∞Ψ⁰_𝜏𝜑𝑓𝜑_𝜏𝜑𝑑𝜏+_{𝑡𝑖(𝜑)<0}Ψ⁰_{𝑡𝑖(𝜑)}𝜑𝑔𝜑_{𝑡𝑖(𝜑)}𝜑, (4.28) where Ψ^𝑡_𝜏(𝜑) is the fundamental matrix of the homogeneous system 𝑑𝑥=𝐴𝜑𝑑𝑡^𝑡𝜑+𝐴₁𝜑_𝑡𝜑,𝑢𝜑_𝑡𝜑𝑥,𝑡≠𝑡_𝑖𝜑,Δ𝑥|_{𝑡=𝑡𝑖(𝜑)}=𝐵𝜑_𝑡𝜑+𝐵𝜑_𝑡𝜑,𝑢𝜑_𝑡𝜑𝑥 (4.29) that depends on 𝜑∈𝒯^𝑚 as a parameter and satisfies the estimation Ψ^𝑡_𝜏𝜑≤𝐾₂𝑒^{−𝛾2(𝑡−𝜏)} (4.30) for all 𝑡≥𝜏 and some 𝐾₂≥1, 𝛾₂>0. The following assertion has been proved.

Theorem 4.2. Let the matrices 𝐴₀(𝜑) and 𝐵₀(𝜑) be constant in the domain Ω:𝐴₀(𝜑)_𝜑∈Ω=𝐵𝐴,₀(𝜑)_𝜑∈Ω=𝐵, (4.31) uniformly with respect to 𝑡∈ℝ, there exists a finite limit 𝑇lim_→∞𝑖𝑡,𝑡+𝑇𝑇=𝑝 (4.32) and the following inequality holds 𝛾+𝑝ln𝛼<0, (4.33) where 𝛾=max_{𝑗=1,…,𝑛}Re𝜆_𝑗𝐴,𝛼²=max_{𝑗=1,…,𝑛}𝜆_𝑗𝐸+𝐵^𝑇𝐸+𝐵. (4.34) Then there exist sufficiently small constants 𝑎₁ and 𝑏₁ and sufficiently small Lipschitz constants 𝐿_𝐴 and 𝐿_𝐵 such that for any continuous 2𝜋-periodic with respect to each of the components 𝜑_𝑣, 𝑣=1,…,𝑚 matrices 𝐹(𝜑,𝑥) and 𝐼(𝜑,𝑥), which have continuous partial derivatives with respect to 𝑥 up to the second order inclusively, such that max_{𝜑∈𝒯𝑚,𝑥∈𝐽ℎ}𝐴₁𝜑,𝑥≤𝑎₁,max_{𝜑∈𝒯𝑚,𝑥∈𝐽ℎ}𝐵₁𝜑,𝑥≤𝑏₁ (4.35) and for any 𝑥^′,𝑥^′′∈𝐽_ℎ𝐴₁𝜑,𝑥^′−𝐴₁𝜑,𝑥^′′≤𝐿_𝐴𝑥^′−𝑥^′′,𝐵₁𝜑,𝑥^′−𝐵₁𝜑,𝑥^′′≤𝐿_𝐵𝑥^′−𝑥^′′, (4.36) system (4.11) has an asymptotically stable invariant set.

5. Conclusion

In summary, we have obtained sufficient conditions for the existence and asymptotic stability of invariant sets of a linear impulsive system of differential equations defined in 𝒯^𝑚×𝐸^𝑛 that has specific properties in the 𝜔-limit set Ω of the trajectories 𝜑_𝑡(𝜑). We have proved that it is sufficient to impose some restrictions on system (2.1) only in the domain Ω to guarantee the existence and asymptotic stability of the invariant set.