Centers of discontinuous piecewise smooth quasi-homogeneous polynomial differential systems

In this paper we investigate the center problem for the discontinuous piecewise smooth quasi-homogeneous but non-homogeneous polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi-homogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi-homogeneous cubic and quartic polynomial differential systems

The number of limit cycles of Josephson equation

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $\beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi=\alpha$ are studied, where $\phi\in \mathbb S^{1}$ and $(\alpha,\beta,\gamma)\in \mathbb R^{3}$. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear.Comment: 25 pages, 15 figure

Classification on Boundary-Equilibria and Singular Continuums of Continuous Piecewise Linear Systems

In this paper, we show that any switching hypersurface of n -dimensional continuous piecewise linear systems is an (n−1) -dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines

Global dynamics of a SD oscillator

In this paper we derive the global bifurcation diagrams of a SD oscillator which exhibits both smooth and discontinuous dynamics depending on the value of a parameter a. We research all possible bifurcations of this system, including Pitchfork bifurcation, degenerate Hopf bifurcation, Homoclinic bifurcation, Double limit cycle bifurcation, Bautin bifurcation and Bogdanov-Takens bifurcation. Besides we prove that the system has at most five limit cycles. At last, we give all numerical phase portraits to illustrate our results

Centers of discontinuous piecewise smooth quasi-homogeneous polynomial differential systems

In this paper we investigate the center problem for the discontinuous piecewise smooth quasi-homogeneous but non-homogeneous polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi-homogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi-homogeneous cubic and quartic polynomial differential systems

Global dynamics of a SD oscillator

In this paper we derive the global bifurcation diagrams of a SD oscillator which exhibits both smooth and discontinuous dynamics depending on the value of a parameter a. We research all possible bifurcations of this system, including Pitchfork bifurcation, degenerate Hopf bifurcation, Homoclinic bifurcation, Double limit cycle bifurcation, Bautin bifurcation and Bogdanov-Takens bifurcation. Besides we prove that the system has at most five limit cycles. At last, we give all numerical phase portraits to illustrate our results

The limit cycles of the Higgins-Selkov systems

In this paper, we investigate the problem of limit cycles for general Higgins-Selkov systems with degree n+ 1. In particular, we first prove the uniqueness of limit cycles for a general Liénard system, which allows for discontinuity. Then, by changing the Higgins-Selkov systems into Liénard systems, theorems and some techniques for Liénard systems can be applied. After, we prove the nonexistence of limit cycles if the bifurcation parameter is outside an open interval. Finally, we complete the analysis of limit cycles for the Higgins-Selkov systems showing its uniqueness