Existence of Center for Planar Differential Systems with Impulsive Perturbations

We present amethod that uses successor functions in ordinary differential systems to address the “center-focus” problemof a class of planar systems that have an impulsive perturbation. By deriving solution formulae for impulsive systems, several interesting criteria for distinguishing between the center and the focus of linear and nonlinear planar systems with state-dependent impulsions are established. The conditions describing the stability of the focus of the considered models are also given. The computing methods presented here aremore convenient for determining the center of impulsive systems than those in the literature.Numerical examples are given to show the effectiveness of the theoretical results.


Introduction
Many differential evolutionary processes that arise in physical, chemical, and biological phenomena are characterized by the fact that they experience an abrupt change of state at certain moments in time.These processes can be modeled using impulsive differential equations.It is known, for example, that many phenomena involve thresholds and exhibit impulsive attributes, including bursting rhythm models (medicine and biology), optimal control models (economics), pharmacokinetics, and frequency modulated systems [1][2][3].The theory of impulsive differential systems is much richer than the corresponding theory of systems without impulsive attributes.In general, there are three kinds of impulsive differential systems: (i) systems with impulses at fixed intervals, (ii) systems containing impulses with variable timing, and (iii) autonomously impulsive systems.Over the past few decades, the majority of research has been concentrated on systems with impulses at fixed intervals [4][5][6], while the other two kinds of impulsive differential systems were relatively less studied due to the difficulties caused by state-dependent impulsion.However, there have been significant developments in the theory of determining the center-focus for differential dynamical systems without impulsive attributes; the reader can refer to the monographs [7,8].The center problem consists of distinguishing between a center and a focus at the origin of the system.In the case of a focus, the related problem is to analyze its stability and determine its highest possible order.Since the time of Poincaré [9], who initially defined the notion of the center for a real system of differential equations in the plane, the center-focus problem has been further extended by Liapunov [10], Bendixson [11], and Frommer [12].Over the past couple decades, various kinds of techniques and algorithms have been developed, and extensive analyses have been consequently produced.Some results about center-focus problem for polynomial differential systems are presented in [13][14][15][16].About the topic of discontinuous systems, there are also a lot of literatures; see, for example, [17][18][19].
There are many results regarding the bifurcation of impulsive differential systems, for example, [20][21][22].However, to the best of our knowledge, the topic of the center-focus for systems with state-dependent impulsion has received less attention.In [23], the center-focus problem and the Hopf bifurcation of dynamical systems with impulses at variable intervals were studied by constructing a Poincaré map.An extension of these results to three-dimensional impulsive differential systems was presented in [24].However, according to existing techniques, it is not easy to distinguish between the center and the focus in dynamical systems with 2 Advances in Mathematical Physics a state-dependent impulsion.Motivated by this apparent lack of insight, in this paper we devoted our attention to a class of planar impulsive differential systems using successor functions.Our aim is to derive conditions for the existence of the center and focus and to analyze the stability of the focus in dynamical systems with a state-dependent impulsion.At the same time, a computational process will be presented.Due to the complexity caused by an impulse-jump in the systems, it is a challenge to obtain the solution of nonlinear impulsive systems using impulsive jumps of series solutions.In the case of linear impulsive systems, we extend the results in [23] to a more generalized scenario, and a convenient center-focus criterion is given.In the case of nonlinear impulsive systems, we propose a relatively simple method for distinguishing between the center and the focus of the systems, which is completely different from those in the existing literature.
The rest of this paper is organized as follows.In Section 2, the impulsive differential systems are formulated, and new definitions for the center and the focus for systems with a state-dependent impulsion are presented.The criteria for distinguishing between the center and the focus in linear and nonlinear impulsive systems are established by extension via the use of successor functions in Sections 3 and 4, respectively.Two numerical examples are given to demonstrate the effectiveness of the derived results in Section 5. Finally, our conclusions are drawn in Section 6.
The following conventions will be used throughout this paper.The transpose of matrix  will be denoted by   and the set of positive integer numbers by  + .The symbols sqrt() and exp() denote √ and   , respectively.

Preliminaries
Consider the following planar differential systems with an impulsive perturbation: where  = [ 1 ,  2 ]  ∈  2 ;  is a known 2 × 2 constant matrix; () :  2 →  2 is a real analytical vector function in the neighborhood of the origin such that (0) = 0; and  0 is a subset of  2 .The phase point of (1) moves between two consecutive intersections with the set  0 along the trajectories of (1).When the solution to (1) meets the set  0 at the moment , the point () has a jump Δ = , and ( + ) = () + .
The following assumptions will be imposed throughout the paper.
( 1 ) Set  0 = ⋃  =1   ,  ∈  + , where   are half-lines starting at the origin in the planar coordinate.( 2 ) The Jacobian matrix of () has a pair of conjugate complex roots at the origin.The roots are denoted as  ± ,  > 0, where  is complex unit.
Next, one must make the transformation  1 =  cos ,  2 =  sin .Then (1) is transformed, in polar coordinates, to the following differential system that possesses impulsive attributes: where Similar to the concepts of the center and focus in the ordinary differential systems [7,25,26], the definition of the center and focus for impulsive differential systems (1) is given as follows.
Theorem 2. Let  +1 = 2 +  0 .For  ∈ ( −1 ,   ],  = 1, 2, . . .,  + 1, the solution to system (5a) and ( 5b) with an initial value of ( 0 ,  0 ) can be formulated as where  0 <  1 and Proof.When  = 1,  ∈ ( 0 ,  1 ], the solution to system (5a) and (5b) is It follows that This solution intersects with  =  1 at point ( 1 ,  1 ), where From (5b), we have So, when  = 2,  ∈ ( 1 ,  2 ].The solution to system (5a) and (5b) is given by from which we obtain which implies that (6) holds when  = 1 and  = 2. Now, suppose that ( 6) is true when  = .Namely, as This solution intersects with  =   at point (  ,   ), where it gives that from (5b) We can conclude that when  ∈ (  ,  +1 ], the solution to system (5a) and (5b) is it follows that where This implies that (6) holds when  =  + 1.Using the reasoning of mathematical induction, we can immediately conclude that ( 6) is true for any  ∈ ( −1 ,   ],  = 1, 2, . . ., + 1.The proof is complete. According to Definition 1 and Theorem 2, the following theorem holds, and it can distinguish between the center and the focus of a linear impulsive differential system.Theorem 3. Denote (,  0 ,  0 ) as the solution to system (5a) and (5b) with an initial value ( 0 ,  0 ), and let  +1 = ( +1 ,  0 ,  0 ).Then the origin of system ( 4) is Remark 4. When  ̸ = 0, the origin is only a focus of the differential part in system (4).However, according to Theorem 3, the origin may become a center of the impulsive differential system (4) when  ̸ = 0. Whether the focus is stable or not has no dependency on the sign of .Therefore, the conclusion in Theorem 3 is actually an extension of the theory of centers and foci in ordinary differential systems.
Remark 5. When  0 = 0 and  ∈ (  ,  +1 ], the solution (6) yields which is the solution of the nonperturbed system in [23].Therefore, Theorem 2 of this paper extends the existing results to a more generalized case.

Nonlinear Impulsive Differential Systems
In this section, successor functions are used to study the center-focus problem of a class of nonlinear impulsive differential systems.We will consider the following nonlinear impulsive differential systems: where  = [ ] is a real analytical vector function in the neighborhood of the origin such that () = (‖‖) and   (0, 0) = 0,  = 1, 2.  0 and  are the same as in (4).
Remark 8.In order to obtain the solution to nonlinear impulsive differential systems, we use the impulsive jumps of the series solutions in differential systems.This method is fundamentally different from that in [23], and our method plays an important role in distinguishing between the center and the focus of impulsive differential systems.

Numerical Examples
In this section, two numerical examples are given to illustrate the theoretical results presented in the previous sections.
Proposition 12. Assume that  0 is sufficiently small.If  0 > 1, the origin is an unstable focus of system (57a) and (57b).If  0 < 1, the origin is a stable focus of system (57a) and (57b).Otherwise, the coefficient of  0 is zero and the coefficient of  2 0 is positive in (65) when  0 = 1; thus the origin is an unstable focus of system (57a) and (57b).Numerical simulations are displayed in Figure 2. The initial states of system (57a) and (57b) are  1 (0) = 0.2,  2 (0) = 0. Here, we find a stable focus for  0 = 0.5 and an unstable focus for  0 = 2.1; see Figures 2(a) and 2(b).We find that the origin of the system (57a) and (57b) is still an unstable focus when  0 = 1; see Figure 2(c).

Conclusion
In this paper, we studied the center-focus problem of planar differential systems that have a state-dependent impulsion.A criterion for distinguishing between the center and the focus in linear and nonlinear impulsive differential systems was established using successor functions.Our results are potentially useful if applied to the theory of impulsive differential systems.An extension of these results to threedimensional impulsive differential systems is an interesting topic that will be considered in future research.