# Invariant Sets of Impulsive Differential Equations with Particularities inω-Limit Set

Authors: Mykola Perestyuk; Petro Feketa
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 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(1.1)dzdt=F(z),
where the function F(z) is defined in some subset D of the (m+n)-dimensional Euclidean space Em+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,…,zm+n) to so-called local coordinates φ=(φ1,…,φm), x=(x1,…,xn), where φ is a point on the surface of an m-dimensional torus 𝒯m and x is a point in an n-dimensional Euclidean space En. The change of variables is performed in a way such that the equation, which defines the invariant manifold M, transforms into x=0, φ∈𝒯m in the new coordinates. In essence, the manifold x=0, φ∈𝒯m is the m-dimensional torus in the space 𝒯m×En. The character of stability of the invariant torus M is closely linked with stability of the set x=0, φ∈𝒯m: from stability, asymptotic stability, and instability of the manifold M, there follow the stability, asymptotic stability, and instability of the torus x=0, φ∈𝒯m 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 𝒯m×En. Theory of the existence and perturbation, properties of smoothness, and stability of invariant sets of systems defined in 𝒯m×En 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 anm-dimensional torus 𝒯m and an n-dimensional Euclidean space En that undergo impulsive perturbations at the moments when the phase point φ meets a given set in the phase space. Consider the system(2.1)dφdt=a(φ),dxdt=A(φ)x+f(φ),φ∉Γ,Δx|φ∈Γ=B(φ)x+g(φ),
where φ=(φ1,…,φm)T∈𝒯m, x=(x1,…,xn)T∈En, a(φ) is a continuous 2π-periodic with respect to each of the components φv, v=1,…,m vector function that satisfies a Lipschitz condition(2.2)‖a(φ′′)-a(φ′)‖≤L‖φ′′-φ′‖
for every φ′,φ′′∈𝒯m. 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.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 pointφ=(φ1,…,φm)T as a point of the m-dimensional torus 𝒯m so that the domain of the functions A(φ),B(φ),f(φ),g(φ), and a(φ) is the torus 𝒯m. We assume that the set Γ is a subset of the torus 𝒯m, 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.The system of differential equations(2.3)dφdt=a(φ)
defines a dynamical system on the torus 𝒯m. Denote by φt(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 φt(φ) satisfies a group property [8](2.4)φt(φτ(φ))=φt+τ(φ)
for all t,τ∈ℝ and φ∈𝒯m.Denote byti(φ), i∈ℤ the solutions of the equation Φ(φt(φ))=0 that are the moments of impulsive action in system (2.1). Let the function Φ(φ) be such that the solutions t=ti(φ) exist, since otherwise, system (2.1) would not be an impulsive system. Assume that (2.5)lim⁡i→±∞ti(φ)=±∞,lim⁡T→±∞i(t,t+T)T=p<∞
uniformly with respect to t∈ℝ, where i(a,b) is the number of the points ti(φ) in the interval (a,b). Hence, the moments of impulsive perturbations ti(φ) satisfy the equality [10, 11](2.6)ti(φ-t(φ))-ti(φ)=t.Together with system (2.1), we consider the linear system(2.7)dxdt=A(φt(φ))x+f(φt(φ)),t≠ti(φ),Δx|t=ti(φ)=B(φti(φ)(φ))x+g(φti(φ)(φ))
that depends on φ∈𝒯m as a parameter. We obtain system (2.7) by substituting φt(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 φv, v=1,…,m, such that the function x(t,φ)=u(φt(φ)) is a solution of system (2.7) for every φ∈𝒯m.We call a pointφ* an ω-limit point of the trajectory φt(φ) if there exists a sequence {tn}n∈ℕ in ℝ so that (2.8)lim⁡n→+∞tn=+∞,lim⁡n→+∞φtn(φ)=φ*.
The set of all ω-limit points for a given trajectory φt(φ) is called ω-limit set of the trajectory φt(φ) and denoted by Ωφ.Referring to system (2.7), the matrices A(φt(φ)) and B(φt(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 φt(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∈En,∥x∥≤h,h>0}, (2.9)Ω=⋃φ∈TmΩφ.
Since the Lyapunov function has to satisfy some conditions not on the whole surface of the torus 𝒯m 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: (3.1)A(φ)|φ∈Ω=Ã,B(φ)|φ∈Ω=B̃.
Therefore, for every φ∈𝒯m(3.2)lim⁡t→+∞A(φt(φ))=Ã,lim⁡t→+∞B(φt(φ))=B̃.
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 Ã and B̃. Denote by (3.3)γ=max⁡j=1,…,nReλj(Ã),α2=max⁡j=1,…,nλj((E+B̃)T(E+B̃)).
Similar systems without impulsive perturbations have been considered in [13].Theorem 3.1.
Let the moments of impulsive perturbations{ti(φ)} be such that uniformly with respect to t∈ℝ there exists a finite limit
(3.4)lim⁡T̃→∞i(t,t+T̃)T̃=p.
If the following inequality holds
(3.5)γ+pln⁡α<0,
then system (2.1) has an asymptotically stable invariant set.Proof.
Consider a homogeneous system of differential equations(3.6)dxdt=A(φt(φ))x,t≠ti(φ),Δx|t=ti(φ)=B(φti(φ)(φ))x
that depends on φ∈𝒯m as a parameter. By Ωτt(φ), we denote the fundamental matrix of system (3.6), which turns into an identity matrix at the point t=τ, that is, Ωττ(φ)≡E. It can be readily verified [4] that Ωτt(φ) satisfies the equalities
(3.7)∂∂tΩτt(φ)=A(φt(φ))Ωτt(φ),Ωτt(φ)=Ωτt(φ+2πek),Ωt+τt(φ-t(φ))=Ωτ0(φ)
for all t,τ∈ℝ and φ∈𝒯m. Rewrite system (3.6) in the form
(3.8)dxdt=Ãx+(A(φt(φ))-Ã)x,t≠ti(φ),Δx|t=ti(φ)=B̃x+(B(φti(φ)(φ))-B̃)x.
The fundamental matrix Ωτt(φ) of the system (3.6) may be represented in the following way [4]:
(3.9)Ωτt(φ)=Xτt(φ)+∫τtXst(φ)(A(φs(φ))-Ã)Ωst(φ)ds+∑τ≤ti(φ)<tXti(φ)t(φ)(B(φti(φ)(φ))-B̃)Ωτti(φ)(φ),
where Xτt(φ) is the fundamental matrix of the homogeneous impulsive system with constant coefficients
(3.10)dxdt=Ãx,t≠ti(φ),Δx|t=ti(φ)=B̃x
that depends on φ∈𝒯m as a parameter. Taking into account that the matrix Xτt(φ) satisfies the estimate [14]
(3.11)‖Xτt(φ)‖≤Ke-μ(t-τ),t≥τ
for every φ∈𝒯m and some K≥1, where γ+plnα<-μ<0, we obtain
(3.12)‖Ωτt(φ)‖≤Ke-μ(t-τ)+∫τtKe-μ(t-s)‖A(φs(φ))-Ã‖‖Ωτs(φ)‖ds+∑τ≤ti(φ)<tKe-μ(t-ti(φ))‖B(φti(φ)(φ))-B̃‖‖Ωτti(φ)(φ)‖.
It follows from (3.2) that for arbitrary small ɛA and ɛB, there exists a moment T such that(3.13)‖A(φt(φ))-Ã‖≤ɛA,‖B(φt(φ))-B̃‖≤ɛB
for all t≥T. Hence, multiplying (3.12) by eμ(t-τ), utilizing (3.13), and weakening the inequality, we obtain
(3.14)eμ(t-τ)‖Ωτt(φ)‖≤K+∫τTKeμ(s-τ)‖A(φs(φ))-Ã‖‖Ωτs(φ)‖ds+∫τtKɛAeμ(s-τ)‖Ωτs(φ)‖ds+∑τ≤ti(φ)<TKeμ(ti(φ)-τ)‖B(φti(φ)(φ))-B̃‖‖Ωτti(φ)(φ)‖+∑τ≤ti(φ)<tKɛBeμ(ti(φ)-τ)‖Ωτti(φ)(φ)‖.
Using the Gronwall-Bellman inequality for piecewise continuous functions [4], we obtain the estimate for the fundamental matrix Ωτt(φ) of the system (3.6)
(3.15)‖Ωτt(φ)‖≤K1e-(μ-KɛA-pln⁡(1+KɛB))(t-τ),
where
(3.16)K1=K+∫τTKeμ(s-τ)‖A(φs(φ))-Ã‖‖Ωτs(φ)‖ds+∑τ≤ti(φ)<TKeμ(ti(φ)-τ)‖B(φti(φ)(φ))-B̃‖‖Ωτti(φ)(φ)‖.
Choosing ɛA and ɛB so that μ>KɛA+pln(1+KɛB), the following estimate holds
(3.17)‖Ωτt(φ)‖≤K1e-γ1(t-τ)
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 as(3.18)u(φ)=∫-∞0Ωτ0(φ)f(φτ(φ))dτ+∑ti(φ)<0Ωti(φ)0(φ)g(φti(φ)(φ)).
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 Ωτt(φ) (2.4), and (2.6), one can show that the function u(φt(φ)) satisfies the equation
(3.19)dxdt=A(φt(φ))x+f(φt(φ))
for t≠ti(φ) and has discontinuities B(φti(φ)(φ))u(φti(φ)(φ))+g(φti(φ)(φ)) at the points t=ti(φ). It means that the function x(t,φ)=u(φt(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. Letx=x(t,φ) be an arbitrary solutions of the system (2.7), and x*=u(φt(φ)) is the solution that belongs to the invariant set. The difference of these solutions admits the representation(3.20)x(t,φ)-u(φt(φ))=Ω0t(φ)(x(0,φ)-u(φ)).
Taking into account estimate (3.17), the following limit exists
(3.21)lim⁡t→∞‖x(t,φ)-u(φt(φ))‖=0.
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 φv, v=1,…,m square matrices. Consider the perturbed system(4.1)dφdt=a(φ),dxdt=(A(φ)+A1(φ))x+f(φ),φ∉Γ,Δx|φ∈Γ=(B(φ)+B1(φ))x+g(φ).Theorem 4.1.
Let the moments of impulsive perturbations{ti(φ)} be such that uniformly with respect to t∈ℝ, there exists a finite limit
(4.2)lim⁡T̃→∞i(t,t+T̃)T̃=p
and the following inequality holds
(4.3)γ+pln⁡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
(4.4)max⁡φ∈Tm‖A1(φ)‖≤a1,max⁡φ∈Tm‖B1(φ)‖≤b1,
system (4.1) has an asymptotically stable invariant set.Proof.
The constantsa1 and b1 exist since the matrices A1(φ) and B1(φ) are continuous functions defined in the torus 𝒯m, which is a compact manifold.
Consider the impulsive system that corresponds to system(4.1)(4.5)dxdt=A(φt(φ))x+A1(φt(φ))x,t≠ti(φ),Δx|t=ti(φ)=B(φti(φ)(φ))x+B1(φti(φ)(φ))x
that depends on φ∈𝒯m as a parameter. The fundamental matrix Ψτt(φ) of the system (4.5) may be represented in the following way
(4.6)Ψτt(φ)=Ωτt(φ)+∫τtΩst(φ)A1(φs(φ))Ψst(φ)ds+∑τ≤ti(φ)<tΩti(φ)t(φ)B1(φti(φ)(φ))Ψτti(φ)(φ),
where Ωτt(φ) is the fundamental matrix of the system (3.6). Then taking estimate (3.17) into account,
(4.7)eγ1(t-τ)‖Ψτt(φ)‖≤K1+∫τtK1a1eγ1(s-τ)‖Ψτs(φ)‖ds+∑τ≤ti(φ)<tK1b1eγ1(ti(φ)-τ)‖Ψτti(φ)(φ)‖.
Using the Gronwall-Bellman inequality for piecewise continuous functions, we obtain the estimate for the fundamental matrix Ψτt(φ) of the system (4.5)
(4.8)‖Ψτt(φ)‖≤K1e-(γ1-K1a1-pln⁡(1+K1b1))(t-τ).
Let the constants a1 and b1 be such that γ1>K1a1+pln(1+K1b1). Hence, the matrix Ψτt(φ) satisfies the estimate
(4.9)‖Ψτt(φ)‖≤K2e-γ2(t-τ)
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
(4.10)u(φ)=∫-∞0Ψτ0(φ)f(φτ(φ))dτ+∑ti(φ)<0Ψti(φ)0(φ)g(φti(φ)(φ)).Consider the nonlinear system of differential equations with impulsive perturbations of the form(4.11)dφdt=a(φ),dxdt=F(φ,x),φ∉Γ,Δx|φ∈Γ=I(φ,x),
where φ∈𝒯m, x∈J¯h, a(φ) is a continuous 2π-periodic with respect to each of the components φv, 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 φv, 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:(4.12)dφdt=a(φ),dxdt=A0(φ)x+A1(φ,x)x+f(φ),φ∉Γ,Δx|φ∈Γ=B0(φ)x+B1(φ,x)x+g(φ),
where (4.13)A(φ,x)=∫01∂F(φ,τx)∂(τx)dτ,B(φ,x)=∫01∂I(φ,τx)∂(τx)dτ,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: (4.14)A0(φ)|φ∈Ω=Ã,B0(φ)|φ∈Ω=B̃
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 Mk—the invariant set of the system(4.15)dφdt=a(φ),dxdt=A0(φ)x+A1(φ,uk-1(φ))x+f(φ),φ∉Γ,Δx|φ∈Γ=B0(φ)x+B1(φ,uk-1(φ))x+g(φ),
where x=uk-1(φ) is the invariant set on (k-1)-step.Using Theorem4.1, the invariant set x=uk(φ), k=1,2,… may be represented as(4.16)uk(φ)=∫-∞0Ψτ0(φ,k)f(φτ(φ))dτ+∑ti(φ)<0Ψti(φ)+00(φ,k)g(φti(φ)(φ)),
where Ψτt(φ,k) is the fundamental matrix of the homogeneous system (4.17)dxdt=(A0(φt(φ))+A1(φt(φ),uk-1(φt(φ))))x,t≠ti(φ),Δx|t=ti(φ)=(B0(φt(φ))+B1(φt(φ),uk-1(φt(φ))))x
that depends on φ∈𝒯m as a parameter and satisfies the estimate(4.18)‖Ψτt(φ,k)‖≤K2e-γ2(t-τ)
for all t≥τ and some K2≥1, γ2>0 only if (4.19)max⁡φ∈Tm‖A1(φ,uk-1(φ))‖≤a1,max⁡φ∈Tm‖B1(φ,uk-1(φ))‖≤b1.
Let us prove that the invariant sets x=uk(φ) belong to the domain J¯h. Denote by(4.20)max⁡φ∈Tm‖f(φ)‖≤Mf,max⁡φ∈Tm‖g(φ)‖≤Mg.
Since the torus 𝒯m is a compact manifold, such constants Mf and Mg exist. Analogously to [4], using the representation (4.16) and estimate (4.18), we obtain that(4.21)‖uk(φ)‖≤K2γ2Mf+K21-e-γ2θ1Mg,
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{uk(φ)}. For this purpose, we estimate the difference wk+1(φ)=uk+1(φ)-uk(φ) and take into account that the functions uk(φt(φ)) satisfy the relations (4.22)ddtuk(φt(φ))=(A0(φt(φ))+A1(φt(φ),uk-1(φt(φ))))×uk(φt(φ))+f(φt(φ)),t≠ti(φ),Δuk(φt(φ))|t=ti(φ)=(B0(φt(φ))+B1(φt(φ),uk-1(φt(φ))))×uk(φt(φ))+g(φt(φ))
for all φ∈𝒯m, k=1,2,…. Hence, the difference wk+1(φ)=uk+1(φ)-uk(φ) is the invariant set of the linear impulsive system (4.23)dφdt=a(φ),dxdt=(A0(φ)+A1(φ,uk(φ)))x+(A1(φ,uk(φ))-A1(φ,uk-1(φ)))uk(φ),φ∉Γ,Δx|φ∈Γ=(B0(φ)+B1(φ,uk(φ)))x+(B1(φ,uk(φ))-B1(φ,uk-1(φ)))uk(φ).
Then, taking (4.21) into account, (4.24)max⁡φ∈Tm‖uk+1(φ)-uk(φ)‖≤K2γ2‖A1(φ,uk(φ))-A1(φ,uk-1(φ))‖‖uk(φ)‖+K21-e-γ2θ‖B1(φ,uk(φ))-B1(φ,uk-1(φ))‖‖uk(φ)‖.
Let the functions A1(φ,x) and B1(φ,x) satisfy the Lipschitz condition with constants LA and LB correspondingly. Then (4.25)max⁡φ∈Tm‖uk+1(φ)-uk(φ)‖≤K2γ2LAh‖uk(φ)-uk-1(φ)‖+K21-e-γ2θLBh‖uk(φ)-uk-1(φ)‖=(K2hγ2LA+K2h1-e-γ2θLB)‖uk(φ)-uk-1(φ)‖.
Assuming that the constants LA and LB are so small that (4.26)K2hγ2LA+K2h1-e-γ2θLB<1,
we conclude that the sequence {uk(φ)} converges uniformly with respect to φ∈𝒯m and (4.27)lim⁡k→∞uk(φ)=u(φ).
Thus, the invariant set x=u(φ) admits the representation (4.28)u(φ)=∫-∞0Ψτ0(φ)f(φτ(φ))dτ+∑ti(φ)<0Ψti(φ)0(φ)g(φti(φ)(φ)),
where Ψτt(φ) is the fundamental matrix of the homogeneous system (4.29)dxdt=(A(φt(φ))+A1(φt(φ),u(φt(φ))))x,t≠ti(φ),Δx|t=ti(φ)=(B(φt(φ))+B(φt(φ),u(φt(φ))))x
that depends on φ∈𝒯m as a parameter and satisfies the estimation (4.30)‖Ψτt(φ)‖≤K2e-γ2(t-τ)
for all t≥τ and some K2≥1, γ2>0. The following assertion has been proved.Theorem 4.2.
Let the matricesA0(φ) and B0(φ) be constant in the domain Ω:(4.31)A0(φ)|φ∈Ω=Ã,B0(φ)|φ∈Ω=B̃,
uniformly with respect to t∈ℝ, there exists a finite limit
(4.32)lim⁡T̃→∞i(t,t+T̃)T̃=p
and the following inequality holds
(4.33)γ+pln⁡α<0,
where
(4.34)γ=max⁡j=1,…,nReλj(Ã),α2=max⁡j=1,…,nλj((E+B̃)T(E+B̃)).
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 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
(4.35)max⁡φ∈Tm,x∈J¯h‖A1(φ,x)‖≤a1,max⁡φ∈Tm,x∈J¯h‖B1(φ,x)‖≤b1
and for any x′,x′′∈J¯h(4.36)‖A1(φ,x′)-A1(φ,x′′)‖≤LA‖x′-x′′‖,‖B1(φ,x′)-B1(φ,x′′)‖≤LB‖x′-x′′‖,
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 in𝒯m×En that has specific properties in the ω-limit set Ω of the trajectories φt(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.



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