Piecewise linear differential systems without equilibria produce limit cycles?

Abstract

Abstract. In this article we study the planar piecewise differential systems formed by two linear differential systems separated by a straight line, such that both linear differential have no equilibria, neither real nor virtual.When the piecewise differential system is continuous, we show that the system has no limit cycles. But when the piecewise differential system is discontinuous, we show that it can have at most one limit cycle.

The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line

“…However when the planar vector field is discontinuous, the adaptation of the 16th Hilbert's problem on the maximum number of existing limit cycles, is an open problem. In the last years many authors have worked in this problem, trying to determine how many limit cycles can appear in planar systems separated by a straight line, see for instance [1][2][3][4]8,[10][11][12][13][14][16][17][18][21][22][23][24][25][26][27][28][29] .…”

“…Moreover for seven of these fifteen classes of discontinuous piecewise differential systems the upper bound on the maximum number obtained is reached. More precisely, it was known that discontinuous piecewise differential systems formed by two linear isochronous centers separated by a straight line cannot have limit cycles, see 27 . If one of the systems is a linear isochronous center and the other is a quadratic isochronous center then their maximum number of limit cycles is studied in Theorem 1.…”

Second order adaptive time varying sliding mode control for synchronization of hidden chaotic orbits in a new uncertain 4-D conservative chaotic system

“…Recently many chaotic/hyperchaotic systems with hidden attractors have been reported (Jafari et al, 2017; Kingni et al, 2017; Korneev et al, 2017; Pham et al, 2017; Singh and Roy, 2017a,b,d; Sprott et al, 2017; Tlelo-Cuautle et al, 2017). Some hidden attractor chaotic systems with no equilibrium point have also been reported recently (Escalante-Gonzalez et al, 2017; Feng and Pan, 2017; Hoang et al, 2017; Kiseleva et al, 2017; Llibre and Teixeira, 2016; Ojoniyi and Njah, 2016). Most of the reported hidden attractor chaotic systems with no equilibrium point are of a dissipative nature.…”

Piecewise linear differential systems with only centers can create limit cycles?

“…It remains open to know if three is the maximum number of limit cycles that such systems can exhibit. In [18] we deal with discontinuous piecewise linear differential systems separated by a straight line such that each linear differential system has neither real nor virtual equilibria, and we show that such systems can have at most one limit cycle.…”