Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems

Kehua Li; Changming Ding

doi:10.1515/math-2022-0478

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Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems

Kehua Li and Changming Ding

Published/Copyright: September 26, 2022

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$Open Mathematics$

From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this article, we deal with orbital stability and Zhukovskiǐ quasi-stability of periodic or recurrent orbits in an impulsive dynamical system defined in the n-dimensional Euclidean space R n . We show that for a periodic orbit of an impulsive system, its asymptotically orbital stability is equivalent to the asymptotically Zhukovskiǐ quasi-stability, and for a recurrent orbit, the orbital stability is equivalent to the Zhukovskiǐ quasi-stability.

Keywords: periodicity; recurrence; orbital stability; Zhukovskiǐ quasi-stability

MSC 2010: 37B25; 34D20

1 Introduction

Impulsive dynamical systems are a generalization of classical dynamical systems. They describe the evolution of systems where the continuous development of a process is interrupted by abrupt perturbations. The behavior of an impulsive system is much richer than that of the corresponding continuous dynamical system. In particular, the theory of impulsive dynamical systems represents a natural framework for the mathematical modeling of many real-world phenomena. Recently, the theory of impulsive dynamical systems has been intensively investigated. For the elementary results in this field, we refer readers to [1,2,3].

The research of impulsive semidynamical systems in a metric space was started by Kaul [4,5,6] and Rozhko [7,8]. Specifically, Rozhko dealt with a class of almost periodic motions in pulsed systems and the stability theory in terms of Lyapunov for impulsive systems. Later on, Kaul continued the study for impulsive semidynamical systems and established a list of important results about the structure of limit sets, periodicity and recurrence of an orbit, minimality and stabilities of closed subsets, etc. Also, Ciesielski presented many fundamental results in this field; for example, he applied his section theory of semidynamical systems to obtain the continuity of an impulsive time function [9,10,11]. Recently, in [12,13, 14,15], Bonotto and his research group developed a list of significant results on impulsive semidynamical systems, which include many counterparts of basic properties in classical dynamical systems. In addition, the authors of this article also established some interesting results on the limit sets and limit set maps [16,17], Lyapunov quasi-stability [18], and Zhukovskiǐ quasi-stability [19].

Poincaré (orbital) stability and Zhukovskiǐ stability are two different important stabilities of solutions of differential equations. Since Zhukovskiǐ stability admits a time lag, it is a more suitable concept for the study of impulsive dynamical systems. The Zhukovskiǐ quasi-stability in impulsive dynamical systems was first introduced in [18], where the author proved that the limit set of a uniformly asymptotically Zhukovskiǐ quasi-stability orbit is composed of a rest point or a periodic orbit. Clearly, the structure of limit sets can be determined by the stability of orbits. Furthermore, the inverse problem was considered in [19]. Actually, it was shown in [19] that (i) if the positive limit set of an orbit for a planar system is an asymptotically stable limit cycle, then it is a uniformly asymptotically Zhukovskiǐ quasi-stable orbit; (ii) if an orbit is not eventually periodic and its positive limit set is a periodic orbit, then it is an asymptotically Zhukovskiǐ quasi-stable orbit.

In this article, we deal with the recurrence, orbital stability, and Zhukovskiǐ quasi-stability of orbits in an impulsive dynamical system defined in R n . First, for a periodic orbit (or eventually periodic orbit), we show that the asymptotically orbital stability is equivalent to the asymptotically Zhukovskiǐ quasi-stability. Second, it is shown that for a recurrent orbit, the orbital stability is also equivalent to the Zhukovskiǐ quasi-stability.

2 Definitions and notations

We consider the system of differential equations

(1) X ′ = F ( X ) ,

where F : R n → R n . Obviously, the map F : R n → R n defines a vector field F of system (1) on R n . Assume that the vector field F defines a flow φ on R n ; i.e., φ : R n × R → R n is continuous such that φ ( p , 0 ) = p for all p ∈ R n and φ ( φ ( p , t ) , s ) = φ ( p , t + s ) for all p ∈ R n , t , s ∈ R . If A ⊂ R n and J ⊂ R , we write φ ( A × J ) = A ⋅ J , in particular, φ ( p , t ) = p ⋅ t . For a point p ∈ R n , the orbit of p is the set γ ( p ) = p ⋅ R . The positive and negative semi-orbits are the sets γ + ( p ) = p ⋅ R + and γ − ( p ) = p ⋅ R − , respectively.

Let M = { p ∈ U ∣ G ( p ) = 0 } be a simple smooth surface in an open subset U of R n , where G : U → R is a smooth function with G ′ ( p ) = ( ∂ G / ∂ x 1 , … , ∂ G / ∂ x n ) ≠ ( 0 , ⋯ , 0 ) for p = ( x 1 , x 2 , ⋯ , x n ) ∈ U . The surface M is said to be transversal to the vector field F if the inner product G ′ ( p ) ⋅ F ( p ) ≠ 0 for all p ∈ M , it is also called a contact-free surface [20]. Now, denote Ω = R n ⧹ M . Let I : M → Ω be a continuous function and N = I ( M ) . If p ∈ M , we shall denote I ( p ) by p + and say p jumps to p + . Meanwhile, M is said to be an impulsive set and I is called an impulsive function. For each p ∈ Ω , by M + ( p ) , we mean the set γ + ( p ) ∩ M . We can define a function ψ : Ω → R + ∪ { + ∞ } (the space of extended positive reals) by

ψ ( p ) = s , if p ⋅ s ∈ M and p ⋅ t ∉ M for t ∈ [ 0 , s ) , + ∞ , if M + ( p ) = ∅ .

In general, ψ : Ω → R + ∪ { + ∞ } is not continuous. Fortunately, some easy applicable conditions given by Ciesielski in [9] guarantee the continuity of ψ . Throughout this article, we always assume that ψ is a continuous function on Ω .

Now, we define an impulsive system ( Ω , φ ˜ ) by portraying the trajectory of each point in Ω . Let p ∈ Ω , the impulsive trajectory of p is an Ω -valued function φ ˜ p defined on a subset of R + . If M + ( p ) = ∅ , then ψ ( p ) = + ∞ , and we set φ ˜ p ( t ) = p ⋅ t for all t ∈ R + . If M + ( p ) ≠ ∅ , it is easy to see that there is a positive number t 0 such that p ⋅ t 0 = p 1 ∈ M and p ⋅ t ∉ M for all t ∈ [ 0 , t 0 ) . Thus, we define φ ˜ p ( t ) on [ 0 , t 0 ] by

φ ˜ p ( t ) = p ⋅ t , 0 ≤ t < t 0 , p 1 + , t = t 0 ,

where ψ ( p ) = t 0 and p 1 + = I ( p 1 ) ∈ Ω .

Since t 0 < + ∞ , we continue the process by starting with p 1 + . Similarly, if M + ( p 1 + ) = ∅ , i.e., ψ ( p 1 + ) = + ∞ , we define φ ˜ p ( t ) = p 1 + ⋅ ( t − t 0 ) for t 0 < t < + ∞ . Otherwise, let ψ ( p 1 + ) = t 1 , where p 1 + ⋅ t 1 = p 2 ∈ M , and p 1 + ⋅ t ∉ M for any t ∈ [ 0 , t 1 ) , then we define φ ˜ p ( t ) on [ t 0 , t 0 + t 1 ] by

φ ˜ p ( t ) = p 1 + ⋅ ( t − t 0 ) , t 0 ≤ t < t 0 + t 1 , p 2 + , t = t 0 + t 1 ,

where p 2 + = I ( p 2 ) ∈ Ω .

Thus, continuing inductively, the aforementioned process either ends after a finite number of steps, whenever M + ( p n + ) = ∅ for some n , or it continues infinitely, if M + ( p n + ) ≠ ∅ for n = 0 , 1 , 2 , … , and φ ˜ p is defined on the interval [ 0 , t p ) , where t p = ∑ i = 0 + ∞ t i . We call { t i } the impulsive intervals of φ ˜ p and { t p ( k ) = ∑ i = 0 k t i : k = 0 , 1 , 2 , … } the impulsive times of φ ˜ p . After setting each trajectory φ ˜ p for every point p ∈ Ω , we let φ ˜ ( p , t ) = φ ˜ p ( t ) for p ∈ Ω and t ∈ [ 0 , t p ) , then we obtain a discontinuous system ( Ω , φ ˜ ) satisfying the following properties:

φ ˜ ( p , 0 ) = p for all p ∈ Ω , and
φ ˜ ( φ ˜ ( p , s ) , t ) = φ ˜ ( p , s + t ) for all p ∈ Ω and s , t ∈ [ 0 , t p ) , such that s + t ∈ [ 0 , t p ) .

We call ( Ω , φ ˜ ) , with φ ˜ as defined earlier, an impulsive dynamical system associated with ( Ω , φ ) . Also for simplicity of exposition, we denote φ ˜ ( p , t ) by p ∗ t . Thus, (ii) reads ( p ∗ s ) ∗ t = p ∗ ( s + t ) . Similarly, if A ⊂ Ω and J ⊂ R + , we denote A ∗ J = { p ∗ t ∣ p ∈ A and t ∈ J } . In particular, if J = { t } , we let A ∗ t = A ∗ { t } = φ ˜ t ( A ) . For p ∈ Ω the mapping φ ˜ p : R + → Ω defined by t → p ∗ t and for a t ∈ R + the mapping φ ˜ t : Ω → Ω defined by p → p ∗ t may not be continuous. However, φ ˜ p is continuous from the right hand for any p ∈ Ω .

For an impulsive dynamical system ( Ω , φ ˜ ) , the trajectories that are of interest are those with an infinite number of discontinuities and with [ 0 , + ∞ ) as the interval of definition. Following Kaul in [4], the trajectories are called infinite impulsive trajectories. Furthermore, for an impulsive dynamical system, Ciesielski used time reparametrization to obtain an isomorphic system whose impulsive trajectories are global, i.e., the resulting dynamics is defined for all positive times, [11]. Hence, from now on, we always assume t p = + ∞ for any p ∈ Ω .

In the following, for a point p ∈ Ω , let B δ ( p ) = { q ∈ Ω ∣ d ( p , q ) < δ } be the open ball in Ω with center p and radius δ > 0 , where d is the Euclidean metric on R n , and the closed ball B ¯ δ ( p ) = { q ∈ Ω ∣ d ( p , q ) ≤ δ } . In addition, for S ⊂ Ω , the r -neighborhood of S in Ω is denoted by U ( S , r ) = { q ∈ Ω ∣ d ( q , S ) < r } for r > 0 , where d ( q , S ) = inf { d ( q , p ) ∣ p ∈ S } . Here, with no confusion, we also use d for the distance between a point and a set. The orbit of p in ( Ω , φ ˜ ) is the set γ ˜ ( p ) = p ∗ R . The positive and negative semi-orbits of p are the sets γ ˜ + ( p ) = p ∗ R + and γ ˜ + ( p ) = p ∗ R − , respectively. A subset S of Ω is said to be positively invariant if γ ˜ + ( p ) ⊂ S for any p ∈ S ; furthermore, it is said to be invariant if it is positively invariant, and for any p ∈ S and t ∈ R + , there exist a q ∈ S such that q ∗ t = p .

Now, we introduce several definitions that will be used in the sequel.

Definition 2.1

Let p ∈ Ω . The positive semi-orbit γ ˜ + ( p ) = p ∗ R + is said to be orbitally stable if, given an ε > 0 , there exists a δ = δ ( ε ) > 0 such that for any q ∈ B δ ( p ) , then we have that q ∗ R + ⊂ U ( p ∗ R + , ε ) . Moreover, if there is a η > 0 such that if q ∈ B η ( p ) implies d ( q ∗ t , p ∗ R + ) → 0 as t → + ∞ , then the positive semi-orbit γ ˜ + ( p ) is asymptotically orbitally stable.

Next, we give the following concept of φ ˜ -recurrence in impulsive dynamical systems, which was introduced first in [21].

Definition 2.2

A point p ∈ Ω is said to be φ ˜ -recurrent if for every ε > 0 , there exists a T = T ( ε ) > 0 , such that for any t , s ∈ R + , the interval [ 0 , T ] contains a real number τ > 0 such that

d ( p ∗ t , p ∗ ( s + τ ) ) < ε .

A positive semi-orbit γ ˜ + ( p ) is said to be φ ˜ -recurrent if p is φ ˜ -recurrent.

Obviously, p ∈ Ω is φ ˜ -recurrent means for any ε > 0 ; there exists a T = T ( ε ) > 0 such that γ ˜ + ( p ) ⊂ U ( p ∗ [ s , s + T ] , ε ) holds for every s ≥ 0 .

The idea of a time reparametrization is useful in our discussion about Zhukovskiǐ quasi-stabilities, see [16,18].

Definition 2.3

A time reparametrization is a homeomorphism ρ from R + onto R + with ρ ( 0 ) = 0 . Furthermore, for a σ > 0 , by a time σ -reparametrization, we mean a homeomorphism ρ from R + onto R + with ρ ( 0 ) = 0 such that ∣ ρ ( t ) − t ∣ < σ for all t ≥ 0 .

Now, we recall the concepts of Zhukovskiǐ quasi-stabilities, which were first introduced for impulsive dynamical systems in [16].

Definition 2.4

Let p ∈ Ω . The positive semi-orbit γ ˜ + ( p ) = p ∗ R + is Zhukovskiǐ quasi-stable provided that given any ε > 0 , there exists a δ = δ ( p , ε ) > 0 such that if q ∈ B δ ( p ) , then one can find a time reparametrization ρ q such that d ( p ∗ t , q ∗ ρ q ( t ) ) < ε holds for all t ≥ 0 . Moreover, if there is a λ > 0 such that if q ∈ B λ ( p ) , then d ( p ∗ t , q ∗ ρ q ( t ) ) → 0 as t → + ∞ , and then the orbit p ∗ R + is said to be asymptotically Zhukovskiǐ quasi-stable.

Furthermore, the property of Zhukovskiǐ quasi-stability can be strengthened to that of uniformly asymptotically Zhukovskiǐ quasi-stability as follows:

Definition 2.5

The positive semi-orbit γ ˜ + ( p ) = p ∗ R + of p ∈ Ω is uniformly asymptotically Zhukovskiǐ quasi-stable provided that given any ε > 0 ; there exists a δ > 0 such that for each s > 0 and q ∈ B δ ( p ∗ s ) , one can find a time reparametrization ρ q such that d ( p ∗ ( s + t ) , q ∗ ρ q ( t ) ) < ε holds for all t ≥ 0 , and also, d ( p ∗ ( s + t ) , q ∗ ρ q ( t ) ) → 0 as t → + ∞ .

The definition of an impulsive periodic orbit was first presented by Kaul in [4] as follows:

Definition 2.6

Let p ∈ Ω . The positive semi-orbit γ ˜ + ( p ) = p ∗ R + is said to be (impulsive) periodic of period τ and order k if γ ˜ + ( p ) has k components and τ is the least positive number such that p ∗ τ = p . Thus the point p is called an (impulsive) periodic point of order k .

A periodic orbit of an impulsive dynamical system ( Ω , φ ˜ ) is an invariant closed set in Ω , and it is not connected if k ≠ 1 . If γ ˜ + ( p ) is not a periodic orbit, but there exists a t > 0 such that γ ˜ + ( p ∗ t ) is a periodic orbit, then γ ˜ + ( p ) is said to be eventually periodic. Clearly, a periodic orbit is eventually periodic, but easy examples can be constructed to show that the converse may not be true.

The continuous dependence on the initial conditions is a fundamental property in the theory of dynamical systems. Fortunately, in [16], the author establishes a counterpart of the continuous dependence for impulsive dynamical systems, which is crucial for the study of impulsive dynamical systems. It is called Quasi-Continuous Dependence and is presented as follows.

Quasi-continuous dependence: Let ( Ω , φ ˜ ) be an impulsive dynamical system and p ∈ Ω . For any ε > 0 , σ > 0 , and a positive number τ , there exists a δ > 0 such that if q ∈ B δ ( p ) , then the inequality d ( p ∗ t , q ∗ ρ q ( t ) ) < ε holds for all t ∈ [ 0 , τ ] , where ρ q is a time σ -reparametrization.

From the aforementioned definition, it is obvious that the quasi-continuous dependence property is a natural generalization of the standard continuous dependence on the initial conditions. For simplicity, we denote the quasi-continuous dependence by QCD property in the sequel. In [16], a crucial proposition was established by the author. It is shown that for impulsive dynamical systems, the QCD property is equivalent to the continuity of ψ . Hence, in this article the QCD property holds for our impulsive system ( Ω , φ ˜ ) by previous assumption that ψ is continuous on Ω .

3 Main results

The concepts of asymptotically orbital stability and asymptotically Zhukovskiǐ stability are different in their dynamical properties. However, for an impulsive periodic orbit, the following result holds.

Theorem 3.1

Let Γ be a periodic orbit in an impulsive dynamical system ( Ω , φ ˜ ) . Then, Γ is asymptotically orbitally stable if and only if Γ is asymptotically Zhukovskiǐ quasi-stable.

Proof

Assume that Γ = p 1 ∗ R + with period τ and order k , where p 1 ∈ N . Then, Γ can be written as

Γ = p 1 ∗ [ 0 , t 1 ) ∪ p 2 ∗ [ 0 , t 2 ) ∪ ⋯ ∪ p k ∗ [ 0 , t k ) ,

where p i ∈ N , ψ ( p i ) = t i ( i = 1 , 2 , … , k ) , and ∑ i = 1 k t i = τ . Clearly, we have I ( p i ∗ t i ) = p i + 1 for i = 1 , 2 , … , k − 1 and I ( p k ∗ t k ) = p 1 ; the solution segments { p i ∗ [ 0 , t i ) : i = 1 , 2 , … , k } are pairwise disjoint.

For a given ε > 0 , by the quasi-continuous dependence, there exists a θ ∈ ( 0 , ε ) such that if q ∈ B θ ( p 1 ) ; then d ( p 1 ∗ t , q ∗ ρ q ( t ) ) < ε holds for all t ∈ [ 0 , τ ] , where ρ q is a time reparametrization. Now, assume that Γ = p 1 ∗ R + is asymptotically orbitally stable; then for the aforementioned θ , there is a δ ∈ ( 0 , θ ) such that if q ∈ B δ ( p 1 ) , we have q ∗ R + ⊂ U ( Γ , θ ) and d ( q ∗ t , Γ ) → 0 as t → + ∞ . Without loss of generality, let ε be sufficiently small so that U ( Γ , ε ) is composed of k pairwise disjoint components, i.e., k disjoint tubes.

Let q ∈ B δ ( p 1 ) , and we define a time reparametrization τ q as follows. Write L = N ∩ B θ ( p 1 ) , then we have q 1 = q ∗ ρ q ( τ ) ∈ L . Clearly, it follows from q 1 ∗ R + ⊂ U ( Γ , θ ) that there exists a time reparametrization ρ q 1 so that d ( p 1 ∗ t , q 1 ∗ ρ q 1 ( t ) ) < θ holds for all t ∈ [ 0 , τ ] . Thus, inductively, there exist two sequences { q n } and { ρ q n } satisfying d ( p 1 ∗ t , q n ∗ ρ q n ( t ) ) < θ and q n ∗ ρ q n ( τ ) = q n + 1 for n = 1 , 2 , … , where { ρ q n } are time reparametrizations. Let q 0 = q and ρ q 0 = ρ q . Then, we define τ q ( t ) = ρ q n ( t − n τ ) for t ∈ [ n τ , ( n + 1 ) τ ] ( n = 0 , 1 , 2 , … ) , and it is easy to see that τ q is a time reparametrization. Now, we obtain d ( p 1 ∗ t , q ∗ ρ q ( t ) ) < θ < ε , and certainly d ( p 1 ∗ t , q ∗ ρ q ( t ) ) → 0 as t → + ∞ . So, Γ = p 1 ∗ R + is asymptotically Zhukovskiǐ quasi-stable.

Conversely, from definitions, it is easy to see that if Γ is asymptotically Zhukovskiǐ quasi-stable, then Γ is asymptotically orbitally stable. Thus, the proof is completed.□

From the proof of Theorem 3.1, it is easy to see that the following statement is true.

Corollary 3.2

Let Γ = p ∗ R + be a eventually periodic orbit in an impulsive dynamical system ( Ω , φ ˜ ) , Then, Γ is asymptotically orbitally stable if and only if Γ is asymptotically Zhukovskiǐ quasi-stable.

Proof

We just need to show the necessity of the statement since the sufficiency is obvious by the definitions of asymptotically orbital stability and asymptotically Zhukovskiǐ quasi-stability.

Let Γ be eventually periodic. That means there exists a t 0 > 0 such that Γ 1 = p 1 ∗ R + is periodic, where p 1 = p ∗ t 0 . Assume that Γ is asymptotically orbitally stable, so is Γ 1 . Thus, by Theorem 3.1, we have Γ 1 which is asymptotically Zhukovskiǐ quasi-stable. That is, for every ε > 0 , there is a θ ∈ ( 0 , ε ) such that if q 1 ∈ B θ ( p 1 ) , then d ( p 1 ∗ t , q 1 ∗ ρ 1 ( t ) ) < ε for each t ∈ R + and d ( p 1 ∗ t , q 1 ∗ ρ 1 ( t ) ) → 0 as t → + ∞ , where ρ 1 is a time reparametrization. For the aforementioned t 0 > 0 and θ > 0 , by quasi continuous dependence of Γ , there exists a δ > 0 such that if q ∈ B δ ( p ) , then d ( p ∗ t , q ∗ ρ 0 ( t ) ) < θ holds for t ∈ [ 0 , t 0 ] , where ρ 0 is a time reparametrization. Set q 1 = q ∗ τ 0 ( t 0 ) , and it is clear that q 1 ∈ B θ ( p 1 ) . Given a q ∈ B δ ( p ) , we define a time reparametrization as follows. Let ρ q ( t ) = τ 0 ( t ) if t ∈ [ 0 , t 0 ] and ρ q ( t ) = ρ 1 ( t − t 0 ) if t ∈ [ t 0 , + ∞ ) . It is easy to verified that ρ q is a time reparametrization. Certainly, we have d ( p ∗ t , q ∗ ρ q ( t ) ) < ε for all t ∈ R + and d ( p ∗ t , q ∗ ρ q ( t ) ) → 0 as t → + ∞ . Thus Γ is asymptotically Zhukovskiǐ quasi-stable.

Now, we turn to consider orbitally stability, recurrence, and Zhukovskiǐ quasi-stability in a planar impulsive dynamical system as follows.□

Theorem 3.3

Let p 0 ∈ Ω , Γ = p 0 ∗ R + be the positive orbit of p 0 . If Γ is φ ˜ -recurrent, then Γ is orbitally stable if and only if Γ is Zhukovskiǐ quasi-stable.

Proof

Suppose p 0 is φ ˜ -recurrent, then for any ε > 0 , there exists a T > 0 such that γ + ( p 0 ) ⊂ U ( p 0 ∗ [ 0 , T ] , ε ) . By the quasi-continuous dependence, for the ε > 0 and T > 0 earlier, there exists a θ ∈ ( 0 , ε ) such that for every q ∈ B θ ( p 0 ) , we have d ( p 0 ∗ t , q ∗ ρ q ( t ) ) < ε for each t ∈ [ 0 , T ] , where ρ q is a time reparametrization. Furthermore, we can find a δ 1 > 0 , such that if q ∈ B δ 1 ( p 0 ) , then d ( p 0 ∗ t , q ∗ ρ q ( t ) ) < θ holds for each t ∈ [ 0 , T ] . By the orbitally stability of Γ , for the θ above, there is a δ 2 > 0 such that if q ∈ B δ 2 ( p 0 ) , then q ∗ R + ⊂ U ( Γ , θ ) .

Setting δ = min { δ 1 , δ 2 } . For any q ∈ B δ ( p 0 ) , we define a time reparametrization τ q as follows. Let p 1 = p 0 ∗ T and q 1 = q ∗ ρ q ( T ) ; thus, we have d ( p 1 , a 1 ) < θ . Clearly, q 1 ∗ R + ⊂ U ( Γ , θ ) implies there exists a time reparametrization ρ 1 such that d ( p 1 ∗ t , q 1 ∗ ρ 1 ( t ) ) < θ holds for all t ∈ [ 0 , T ] . Thus, inductively there exist three sequences { p n } , { q n } , and { ρ n } such that d ( p n ∗ t , q n ∗ ρ n ( t ) ) < θ for each t ∈ [ 0 , T ] and p n + 1 = p n ∗ T , q n + 1 = q n ∗ ρ n ( T ) for n = 1 , 2 , … , where { ρ n } are time reparametrizations. Let q 0 = q and ρ 0 = ρ q . Then, we define τ q ( t ) = ρ n ( t − n T ) for t ∈ [ n T , ( n + 1 ) T ] ( n = 0 , 1 , 2 , … ) , and it is easy to see that τ q is a time reparametrization and d ( p 0 ∗ t , q ∗ τ q ( t ) ) < ε for every t ∈ R + ; then, Γ = p 0 ∗ R + is Zhukovskiǐ quasi-stable.

Conversely, assume that Γ is a Zhukovskiǐ quasi-stable orbit. That is, for any ε > 0 , there exists a δ > 0 such that, if q ∈ B δ ( p 0 ) , d ( p 0 ∗ t , q ∗ τ q ( t ) ) < ε for all t ∈ R + , where τ q is a time reparametrization. That means q ∗ R + ⊂ U ( Γ , ε ) , so Γ is orbitally stable and the proof is completed.□

Acknowledgements

The authors thank the anonymous referees for their suggestions which led to improvements in the exposition.

Funding information: This study was funded by The National Natural Science Foundation of China (Grant Nos. 12061037, and 41801219) and the High-level Personnel of Special Support Program of Xiamen University of Technology (No. 4010520009).
Author contributions: The authors read and approve the final manuscript.
Conflict of interest: The authors state no conflict of interest.

References

[1] M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer, New York, 2010. 10.1007/978-1-4419-6581-3Search in Google Scholar

[2] W. M. Haddad, V. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, Princeton, NJ, 2006. 10.1515/9781400865246Search in Google Scholar

[3] A. N. Michel, L. Hou, and D. Liu, Stability of Dynamical Systems: Continuous Discontinuous and Discrete Systems, Birkhäuser, Boston, 2008. Search in Google Scholar

[4] S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl. 150 (1990), 120–128. 10.1016/0022-247X(90)90199-PSearch in Google Scholar

[5] S. K. Kaul, On impulsive semidynamical systems-II. Recursive properties, Nonlinear. Anal. 16 (1991), 635–645. 10.1016/0362-546X(91)90171-VSearch in Google Scholar

[6] S. K. Kaul, Stability and asymptotic stability in impulsive semidynamical systems, J. Appl. Math. Stochastic Anal. 7 (1994), 509–523. 10.1155/S1048953394000390Search in Google Scholar

[7] V. F. Rozhko, A class of almost periodic motions in pulsed systems, Differ. Uravn. 8 (1972), 2012–2022. Search in Google Scholar

[8] V. F. Rozhko, Lyapunov stability in discontinuous dynamical systems, Differ. Uravn. 11 (1975), 1005–1012. Search in Google Scholar

[9] K. Ciesielski, On semicontinuity in impulsive systems, Bull. Pol. Acad. Sci. Math. 52 (2004), 71–80. 10.4064/ba52-1-8Search in Google Scholar

[10] K. Ciesielski, On stability in impulsive dynamical systems, Bull. Pol. Acad. Sci. Math. 52 (2004), 81–91. 10.4064/ba52-1-9Search in Google Scholar

[11] K. Ciesielski, On time reparametrizations and isomorphisms of impulsive dynamical systems, Ann. Polon. Math. 84 (2004), 1–25. 10.4064/ap84-1-1Search in Google Scholar

[12] E. M. Bonotto and M. Federson, Topological conjugation and asymptotic stability in impulsive semi dynamical systems, J. Math. Anal. Appl. 326 (2007), 869–881. 10.1016/j.jmaa.2006.03.042Search in Google Scholar

[13] E. M. Bonotto, Flows of characteristic 0+ in impulsive semidynamical systems, J. Math. Anal. Appl. 332 (2007), 81–96. 10.1016/j.jmaa.2006.09.076Search in Google Scholar

[14] E. M. Bonotto and M. Federson, Limit sets and Poincaré-Bendixson theorem in impulsive semidynamical systems, J. Differential Equations 8 (2007), 23–41. 10.1016/j.jde.2008.02.007Search in Google Scholar

[15] E. M. Bonotto and N. G. Grulha, Lyapunov stability of closed set in impulsive semidynamical systems, Electron. J. Differential Equations 78 (2010), 2–18. Search in Google Scholar

[16] C. Ding, Limit sets in impulsive semidynamical systems, Topol. Methods Nonlinear Anal. 43 (2014), 97–115. 10.12775/TMNA.2014.007Search in Google Scholar

[17] K. Li, C. Ding, F. Wang, and J. Hu, Limit set maps in impulsive semidynamical systems, J. Dyn. Control Syst. 20 (2014), 47–58. 10.1007/s10883-013-9204-5Search in Google Scholar

[18] C. Ding, Lyapunov quasi-stable trajectories, Fund. Math. 220 (2013), 139–154. 10.4064/fm220-2-4Search in Google Scholar

[19] J. Hu, K. Li, Z. Luo, and C. Ding, Zhukovskiǐ quasi-stable orbits of impulsive dynamical systems, Qual. Theory Dyn. Syst. 16 (2017), 635–643. 10.1007/s12346-016-0208-xSearch in Google Scholar

[20] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Springer, Berlin, 2002. 10.1007/978-3-642-62006-5_2Search in Google Scholar

[21] E. M. Bonotto and M. Z. Jimenez, On impulsive semidynamical systems: minimal, recurrent and almost periodic motions, Topol. Methods Nonlinear Anal. 44 (2014), 121–141. 10.12775/TMNA.2014.039Search in Google Scholar

Received: 2022-02-25

Revised: 2022-06-21

Accepted: 2022-06-23

Published Online: 2022-09-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2022-0478

Keywords for this article

periodicity; recurrence; orbital stability; Zhukovskiǐ quasi-stability

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