Piecewise linear differential systems with two real saddles

The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems

“…Figure 2 shows the effect of the orthogonalization of the change of coordinates (2). From the unfolding (3), for b = 0, we find nine different scenarios in which the two-fold singularity can be unfolded in a such way that is it possible to observe a change of stability in a sliding segment.…”

“…There are analogous results for piecewise smooth systems, for the case of continuous systems see for example [6,7,26,27], and for the case of discontinuous systems see [1,8,11,12,14,18]. In the discontinuous ones we can have more than one limit cycle, either all crossing cycles or including one sliding cycle, and in fact, the determination of the number of limit cycle has been the subject of several recent papers, see [2,3,4,10,15,16,17,20,22,23,24].…”

Piecewise linear differential systems without equilibria produce limit cycles?

“…Up to now we know that there are discontinuous systems with at least three limit cycles, see for instance [2,4,3,6,8,9,10,11,12,13,14,15,22,17,18,19,20,22,24].…”

“…Again we consider 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, defined by (1) and (2). But now we assume that this piecewise differential systems are discontinuous, i.e.…”

On the Number of Limit Cycles of a Piecewise Quadratic Near-Hamiltonian System

“…As we have seen, studying the existence and number of limit cycles is one of the main problems for piecewise smooth systems; see [8][9][10][11][12][13][14][15][16]. Limit cycles of piecewise smooth linear differential systems defined on two half-planes separated by a straight line = 0 or = 0 have been studied recently in [8][9][10][11], from which one can find that 3 limit cycles can appear for piecewise smooth linear systems.…”