A degenerate planar piecewise linear differential system with three zones

Global studies on a continuous planar piecewise linear differential system with three zones

“…Proposition 3.2. Consider a second-order memristor oscillator with a flux-controlled memristor, as modeled by system (5) with (6). Under the design condition…”

“…In recent years, there has been tremendous interest in developing piecewise linear differential systems, see [2,4,5,6,7,8,9,10,11,15,16,17,20,21,22,23,24,26,27,30,33] and references therein. In applied science and engineering, piecewise linear differential systems can model a large number of nonlinear problems.…”

“…Global phase portraits in the Poincaré disc with single equilibrium was showed in [20]. A lot of effort has been devoted to characterize the global dynamics of CPWL differential systems with three linear zones separated by two parallel straight lines, see [5,6,7,8,9,10,11,15,17,23,24,25,27,30] and references therein. However, there are still many cases that have not been studied.…”

“…When the determinant for the Jacobian matrix of its central zone vanishes (d c = 0), we call system (1) with three zones degenerate planar piecewise linear differential systems. The study of the global dynamics of such systems has been completely investigated when t r t l < 0, d r > 0 and d l > 0, see [6,10,15]. The conditions of d r > 0 and d l > 0 mean that the dynamics of left and right zones of system (1) are anti-saddles.…”

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

“…As we see, the planar continuous piecewise linear system with the switching lines is actually the simplest nonsmooth dynamical system. For some results of continuous piecewise linear systems we refer to [1,5,6,15,16,17,18,19] and for some boundary equilibria and singular continuums of continuous piecewise linear systems we refer to [3,4,9,14]. As far as our knowledge goes, there are no complete classifications on phase portraits of boundary-equilibria and singular continuums for continuous piecewise linear systems.…”