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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 970469, 14 pages
Invariant Sets of Impulsive Differential Equations with Particularities in ω-Limit Set
Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska Street 64, 01033 Kyiv, Ukraine
Received 19 January 2011; Accepted 14 February 2011
Academic Editor: Elena Braverman
Copyright © 2011 Mykola Perestyuk and Petro Feketa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 .
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 where the function is defined in some subset of the -dimensional Euclidean space , continuous and satisfies a Lipschitz condition. Let be an invariant toroidal manifold of the system. While investigating the trajectories that begin in the neighborhood of the manifold , it is convenient to make the change of variables from Euclidean coordinates to so-called local coordinates , , where is a point on the surface of an -dimensional torus and is a point in an -dimensional Euclidean space . The change of variables is performed in a way such that the equation, which defines the invariant manifold , transforms into , in the new coordinates. In essence, the manifold , is the -dimensional torus in the space . The character of stability of the invariant torus is closely linked with stability of the set , : from stability, asymptotic stability, and instability of the manifold , there follow the stability, asymptotic stability, and instability of the torus , 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 . Theory of the existence and perturbation, properties of smoothness, and stability of invariant sets of systems defined in are considered in .
The main object of investigation of this paper is the system of differential equations, defined in the direct product of an -dimensional torus and an -dimensional Euclidean space that undergo impulsive perturbations at the moments when the phase point meets a given set in the phase space. Consider the system where , , is a continuous -periodic with respect to each of the components , vector function that satisfies a Lipschitz condition for every . are continuous -periodic with respect to each of the components , square matrices; , are continuous (piecewise continuous with first kind discontinuities in the set ) -periodic with respect to each of the components , vector functions.
We regard the point as a point of the -dimensional torus so that the domain of the functions , and is the torus . We assume that the set is a subset of the torus , which is a manifold of dimension defined by the equation for some continuous scalar -periodic with respect to each of the components , function.
The system of differential equations defines a dynamical system on the torus . Denote by the solution of (2.3) that satisfies the initial condition . The Lipschitz condition (2.2) guarantees the existence and uniqueness of such solution. Moreover, the solutions satisfies a group property  for all and .
Denote by , the solutions of the equation that are the moments of impulsive action in system (2.1). Let the function be such that the solutions exist, since otherwise, system (2.1) would not be an impulsive system. Assume that uniformly with respect to , where is the number of the points in the interval . Hence, the moments of impulsive perturbations satisfy the equality [10, 11]
Together with system (2.1), we consider the linear system that depends on as a parameter. We obtain system (2.7) by substituting for 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 , which has a period with respect to each of the components , , such that the function is a solution of system (2.7) for every .
We call a point an -limit point of the trajectory if there exists a sequence in so that The set of all -limit points for a given trajectory is called -limit set of the trajectory and denoted by .
Referring to system (2.7), the matrices and , that influence the behavior of the solution of the system (2.7), depend not only on the functions and but also on the behavior of the trajectories . Moreover, in , 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 that satisfies some conditions in the domain , where , Since the Lyapunov function has to satisfy some conditions not on the whole surface of the torus but only in the -limit set , it is interesting to consider system (2.1) with specific properties in the domain .
3. Main Result
Consider system (2.1) assuming that the matrices and are constant in the domain : Therefore, for every 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 and . Denote by Similar systems without impulsive perturbations have been considered in .
Theorem 3.1. Let the moments of impulsive perturbations be such that uniformly with respect to there exists a finite limit If the following inequality holds then system (2.1) has an asymptotically stable invariant set.
Proof. Consider a homogeneous system of differential equations
that depends on as a parameter. By , we denote the fundamental matrix of system (3.6), which turns into an identity matrix at the point , that is, . It can be readily verified  that satisfies the equalities
for all and . Rewrite system (3.6) in the form
The fundamental matrix of the system (3.6) may be represented in the following way :
where is the fundamental matrix of the homogeneous impulsive system with constant coefficients
that depends on as a parameter. Taking into account that the matrix satisfies the estimate 
for every and some , where , we obtain
It follows from (3.2) that for arbitrary small and , there exists a moment such that for all . Hence, multiplying (3.12) by , utilizing (3.13), and weakening the inequality, we obtain Using the Gronwall-Bellman inequality for piecewise continuous functions , we obtain the estimate for the fundamental matrix of the system (3.6) where Choosing and so that , the following estimate holds for all and some , .
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 of the system (2.1) may be represented as 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 (2.4), and (2.6), one can show that the function satisfies the equation for and has discontinuities at the points . It means that the function is a solution of the system (2.7). Hence, defines the invariant set of system (2.1).
Let us prove the asymptotic stability of the invariant set. Let be an arbitrary solutions of the system (2.7), and is the solution that belongs to the invariant set. The difference of these solutions admits the representation Taking into account estimate (3.17), the following limit exists It proves the asymptotic stability of the invariant set .
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 and be continuous -periodic with respect to each of the components , square matrices. Consider the perturbed system
Theorem 4.1. Let the moments of impulsive perturbations be such that uniformly with respect to , there exists a finite limit and the following inequality holds Then there exist sufficiently small constants and such that for any continuous -periodic with respect to each of the components , functions and such that system (4.1) has an asymptotically stable invariant set.
Proof. The constants and exist since the matrices and are continuous functions defined in the torus , which is a compact manifold.
Consider the impulsive system that corresponds to system(4.1) that depends on as a parameter. The fundamental matrix of the system (4.5) may be represented in the following way where is the fundamental matrix of the system (3.6). Then taking estimate (3.17) into account, Using the Gronwall-Bellman inequality for piecewise continuous functions, we obtain the estimate for the fundamental matrix of the system (4.5) Let the constants and be such that . Hence, the matrix satisfies the estimate for all and some , . As in Theorem 3.1, from estimate (4.9), we conclude that the system (4.1) has an asymptotically stable invariant set , which admits the representation
Consider the nonlinear system of differential equations with impulsive perturbations of the form where , , is a continuous -periodic with respect to each of the components , vector function and satisfies Lipschitz conditions (2.2); and are continuous -periodic with respect to each of the components , functions that have continuous partial derivatives with respect to up to the second order inclusively. Taking these assumptions into account, system (4.11) may be rewritten in the following form: where
, , , , , and . We assume that the matrices and are constant in the domain : and the inequality holds.
We will construct the invariant set of system (4.12) using an iteration method proposed in . As initial invariant set , we consider the set , as —the invariant set of the system where is the invariant set on -step.
Using Theorem 4.1, the invariant set , may be represented as where is the fundamental matrix of the homogeneous system that depends on as a parameter and satisfies the estimate for all and some , only if Let us prove that the invariant sets belong to the domain . Denote by Since the torus is a compact manifold, such constants and exist. Analogously to , using the representation (4.16) and estimate (4.18), we obtain that where is a minimum gap between moments of impulsive actions. Condition (3.4) guarantees that such constant exists. Assume that the constants and are such that .
Let us obtain the conditions for the convergence of the sequence . For this purpose, we estimate the difference and take into account that the functions satisfy the relations for all , . Hence, the difference is the invariant set of the linear impulsive system Then, taking (4.21) into account, Let the functions and satisfy the Lipschitz condition with constants and correspondingly. Then Assuming that the constants and are so small that we conclude that the sequence converges uniformly with respect to and Thus, the invariant set admits the representation where is the fundamental matrix of the homogeneous system that depends on as a parameter and satisfies the estimation for all and some , . The following assertion has been proved.
Theorem 4.2. Let the matrices and be constant in the domain : uniformly with respect to , there exists a finite limit and the following inequality holds where Then there exist sufficiently small constants and and sufficiently small Lipschitz constants and such that for any continuous -periodic with respect to each of the components , matrices and , which have continuous partial derivatives with respect to up to the second order inclusively, such that and for any system (4.11) has an asymptotically stable invariant set.
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 that has specific properties in the -limit set of the trajectories . We have proved that it is sufficient to impose some restrictions on system (2.1) only in the domain to guarantee the existence and asymptotic stability of the invariant set.
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