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

Abstract

This paper is concerned with the global dynamics of a continuous planar piecewise linear differential system with three zones, where the dynamic of the one of the exterior linear zones is saddle and the remaining one is anti-saddle. We give all global phase portraits in the Poincaré disc and the complete bifurcation diagram including scabbard bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curve. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results.

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

“…So all discussions for system (1.1) were restricted in the region G 1 . When d c ≤ 0, global phase portraits in the Poincaré disc and bifurcation diagrams of system (1.1) in the region G 1 were presented [21]. Therefore, in this study we continue to explore global dynamics of system (1.1) in the region G 1 when d c > 0.…”

“…Hence, the vector field (F (x) − y, g(x)) of system (5.29) is rotated about t c [22,Definition 1.6] or [38,Definition 3.3]. From [21,Lemma 5.3], it follows that the manifolds W + N and W − N of the saddle point N l of system (5.29) rotate clockwise when t c increases and t l , d c , d l , α are fixed, where W + N is the stable manifold of the right-hand side of N l and W − N is the unstable manifold of the right-hand side of N l . This indicates that system (5.29) exhibits at most one homoclinic loop.…”

“…However, the unstable homoclinic loop breaks when t c = φ(α) ± ε, even ε > 0 is sufficiently small. According to [21,Lemma 5.3], the manifolds W + E l and W − E l rotate clockwise when t c increases and t r , t l , d r , d c , d l , α are fixed. When t c = φ(α) − ε and t c = φ(α) + ε, the relative locations of the manifolds W + E l and W − E l are depicted in Figure 17(a) and (c), respectively.…”

“…Considerable attention has been devoted to characterizing global dynamics of system (1.1) [6,7,9,10,17,20,24,25,26,27,29]. Jia-Su-Chen [21] once investigated global dynamics of system (1.1) in the region:…”

Global dynamics and bifurcations of Filippov SIR epidemic model with general nonlinear transmission and discontinuous control

“…[9,12,21,25,28,34,41,45,46], H.C. [14,23,24] and I.C. [22,25,26,44] and the other control [1,3,10,18,40,42]. Nevertheless, in this manuscript, we design a SIR epidemic system to address the D.T.C.…”