Jump bifurcations in some degenerate planar piecewise linear differential systems with three zones

Phase portraits of continuous piecewise linear Liénard differential systems with three zones

“…Systems (1) satisfying (2) and (3) have been studied extensively, see for instance [7,9,10,14,15,17,21]. For the symmetrical cases, that is a = 0, k 1 = k 3 and l 1 = l 3 , see [9] for the bifurcation sets and existence of the limit cycles, see [10] for the amplitude and period of the limit cycles, see [17] for the global phase portraits and bifurcation diagrams.…”

“…In [15] the authors studied the uniqueness and non-uniqueness of limit cycles for the more general cases, they showed that systems (1) satisfying (2) and (3) can have at leat two limit cycles for the cases either k 1 > 0, k 2 < 0, k 3 < 0, or k 1 < 0, k 2 < 0, k 3 > 0. In [7] the authors found that systems (1) satisfying (2) and 3can exist jump bifurcations when l 2 = 0. In [21] it is investigated the boundary equilibrium bifurcations of systems (1) satisfying (2) and (3).…”

Modeling Regulation of Economic Sustainability in Energy Systems with Diversified Resources

“…It should be noted that the generalizations declared by them of the applicability of the model raise doubts about their application to technical and especially power systems. The proposal studied the nonlinear processes of the periodic development of such systems are more pronounced, more specifically, reflecting the specifics of this study of linear differential systems in proposals for estimating jump-like bifurcations of Euzébio [7]. However, in the three indicated works, the tasks of interaction of objects whose processes have a high degree of diversification of resources are not solved.…”

“…It has been established that such domestic waste begins to emit a mixture of gases containing up to 50% of methane and hydrogen sulfide, and when burning waste, much more harmful gases occur. It is 20 times more harmful than carbon dioxide in creating the greenhouse effect [5][6][7][8][9][10][11][12]. In this case, it is possible to use functional mathematical models with dependences of a continuous type on time.…”

Poincaré recurrence theorem for non-smooth vector fields

“…However, that is not the case. Indeed, lots of applications in engineering leads to NSVF's having no sliding or escaping motions, that is, considering only sewing and tangency points (or even only sewing points), see [10] and references therein. Other known fact from NSVF's which is distinct to smooth dynamical system is that, surprisingly, even linear NSVF without sliding or escaping can present limit cycles (see, for instance, [17]).…”