Planar Piecewise Linear Systems Research Articles

Markus–Yamabe conjecture for planar piecewise linear refracting system

Markus–Yamabe conjecture for planar piecewise linear refracting system

Crossing Limit Cycles in Planar Piecewise Linear Systems Separated by a Nonregular Line with Node–Node Type Critical Points

In this paper, we investigate the existence and number of crossing limit cycles in a class of planar piecewise linear systems with node–node type critical points defined in two zones separated by a nonregular line formed by two rays emanated from the origin [Formula: see text], which are the positive [Formula: see text]- and [Formula: see text]-axes. We focus our attention on the existence of two-point crossing limit cycles, which intersect the switching line at two points. We obtain sufficient conditions under which the system has two two-point crossing limit cycles which intersect only one of the two rays. Moreover, we construct examples to show that this class of systems can have two, three or four two-point crossing limit cycles.

Two-parametric unfoldings for planar invisible double-tangency singularities.

We consider the family of planar discontinuous piecewise linear systems in two regions with a straight line as boundary and one tangency point in each region. First, we recall a seven-parametric canonical form for such a family. Next, we select two parameters as bifurcation parameters. One of them represents the distance between the tangency points. It is known that this bifurcation parameter unfolds the called pseudo-Hopf bifurcation. The coefficient that determines the stability of the crossing limit cycle that emerges from this bifurcation mechanism, called the first Lyapunov coefficient, is considered our second bifurcation parameter. Finally, with these two bifurcation parameters, we give a two-parametric unfolding for the invisible fold-fold and focus-fold singularities. The bifurcation diagram for each unfolding consists of two curves of bifurcation points: a curve of pseudo-Hopf bifurcation points and a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin (pB) bifurcation because of the dynamical behavior that is same as the Bautin bifurcation for smooth dynamical systems.

Limit Cycles of a Planar Piecewise Linear System with an Improper Node

This paper is concerned with the number of limit cycles for a planar piecewise linear (PWL) system with two zones separated by a straight line. Assume that one of the subsystems of the PWL system has an improper node. The number of limit cycles for saddle-improper node type, focus-improper node type and center-improper node type (the focus or the center is a virtual or boundary equilibrium) are studied. First, we introduce displacement functions and study the number of zeros of displacement functions for different types. Then, we give the parameter regions where the exact number of limit cycles is one or two (at least two) for different types.

Periodic trajectories in planar discontinuous piecewise linear systems with only centers and with a nonregular switching line

In this paper periodic trajectories of dynamical systems presenting discontinuities are studied. The considered model consists of two distinct linear differential systems, each one containing a single equilibrium point of centre type. Each system is defined on disjoint regions of the plane, the separation line is a union of two half-straight lines contained on the coordinate axes. The obtained differential system is non-smooth, so we apply Filippov’s theory to study the transitions from one dynamical system to another. The combination of the two linear plus the Filippov system acting on the separation line generates a nonlinear regime observed by the presence of limit cycles, sliding and tangential periodic trajectories as well as the coexistence of such objects. In theorem we establish the location, stability and hyperbolicity of limit cycles for certain classes of the considered model. In theorem we perform the global analysis of a representative model through bifurcation theory to analyse the birth of limit cycles, sliding periodic trajectories and tangential ones. We also provide some results addressing the coexistence of periodic trajectories and two potential physical interpretations of the model considered in the paper, one addressing nonlinear oscillations and the other considering slow-fast systems of neuron models. The main techniques employed to obtain the results are first integrals, Poincaré half return maps, and elements of bifurcation theory.

Bifurcation Analysis of a Planar Piecewise Linear System of Focus–Focus Type

This paper is devoted to discussing the sliding bifurcation phenomena of a planar piecewise linear system with two zones separated by a straight line. We consider those systems for which in each zone there is precisely one real focus. By introducing Poincaré map and defining proper displacement function, some results on the existence of crossing periodic orbits, sliding periodic orbits, sliding-crossing periodic orbits and sliding homoclinic connections are obtained. Moreover, we establish some conditions for the simultaneous occurrence of crossing and sliding periodic orbits and show a concrete example.

Global Dynamics of Planar Piecewise Linear Inelastic systems having straight lines as switching manifolds

In this paper we classify the phase portrait and the limit sets of a special class of piecewise smooth vector fields in the plane with different configurations of straight lines as switching manifolds. We study the behavior at infinity through a Poincaré compactification as well as the relations between canonical regions and vector fields which are defined over the switching regions. Results addressing the global behavior of trajectories, tangency points, vector fields over the switching manifolds, and equilibrium and pseudo-equilibrium points at the finite and the infinite part are stated. In particular, we prove the existence of 123 distinct phase portraits for the class of piecewise smooth vector fields that we consider.

Existence of Four-Intersection-Point Limit Cycles with Only Saddles Separated by Two Parallel Straight Lines in Planar Piecewise Linear Systems

In this paper, we study a family of planar piecewise linear systems with saddles separated by two parallel lines, and mainly investigate the existence of four-intersection-point limit cycles. We provide complete conclusions on the existence of a special four-intersection-point limit cycle and a heteroclinic loop. And, based on these results, we give some sufficient conditions for the existence of general four-intersection-point limit cycles. Some examples are given to illustrate the main results.

Limit cycles and bifurcations in a class of planar piecewise linear systems with a nonregular separation line

In this paper, we consider a class of planar piecewise linear differential systems with a nonregular separation line, which can be transformed to a normal form with only 5 parameters under some conditions. We study the coexistence of crossing limit cycles and sliding limit cycles by establishing Poincaré maps for two subsystems that have the same Jacobi matrix. Using expressions and properties of Poincaré maps, we can clarify the relationship between trajectories of two subsystems. Our main results reveal that (1) the numbers of crossing limit cycles and sliding limit cycles can be 2; (2) the coexistence number of limit cycles can be 3. Moreover, by a complete discussion of classification and analysis, we show saddle-node bifurcation and critical crossing cycle bifurcation existing in generic Filippov systems. Finally, four numerical examples are given to verify the correctness of the obtained theoretical results.

Properties of Poincaré half-maps for planar linear systems and some direct applications to periodic orbits of piecewise systems

This paper deals with fundamental properties of Poincaré half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincaré half-maps, their series expansions (Taylor and Newton–Puiseux) at the tangency point and at infinity, the relative position between the graph of Poincaré half-maps and the bisector of the fourth quadrant, and the sign of their second derivatives. All these properties are essential to understand the dynamic behavior of planar piecewise linear systems. Accordingly, we also provide some of their most immediate, but non-trivial, consequences regarding periodic orbits.

The global dynamics of linear refracting systems of focus–node or center–node type

We investigate the global dynamics of discontinuous planar piecewise linear refracting systems with a straight line of separation for focus–node (FN) type, center–node (CN) type, focus–improper node (FN′) type and center–improper node (CN′) type. By complete studying singularities, the invariant manifolds, limit cycles and all the possible configuration of orbits, we give the sufficient and necessary conditions of these system existing limit cycle for all parameters regions and prove that this four types of systems have at most one limit cycle. In addition, we also obtain that these refracting systems have 26 kinds of global topological phase portraits in the Poincaré disk.

Impact limit cycles in the planar piecewise linear hybrid systems

This paper aims to study impact limit cycles in the planar piecewise linear hybrid systems formed by center type vector fields and reset maps on the impact surfaces. Motivated by Llibre & Teixeira, 2018, where an open problem was posed: Piecewise linear differential systems with only centers can create limit cycles? We answer this problem for piecewise linear hybrid systems separated by one or two parallel straight lines. By using Poincaré map and first integral, we present an estimate of the maximum number of two-zone and three-zone impact limit cycles. When the hybrid systems are separated by a unique straight line, they can have at most 1 two-zone impact limit cycle. When they are separated by two parallel straight lines, we show that such hybrid systems can have at most 2 three-zone impact limit cycles. Furthermore, we employ some numerical examples to illustrate our main results and show that this upper bound can indeed be reached.

Global Dynamics of a Planar Piecewise Linear Refracting System of Node-Node Types

In this paper, we study global dynamics of a planar piecewise linear refracting system with a straight line of separation of node-node type, improper node-improper node type and improper node-node type, respectively. We obtain the conditions of the uniqueness and nonexistence of limit cycles of the refracting system and provide classifications of the topological phase portraits in the Poincaré disk.

Sliding Bifurcation in Planar Piecewise Smooth Systems with Two Zones Separated by an Ellipse

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.

A degenerate planar piecewise linear differential system with three zones

In (Euzébio et al., 2016 [10] ; Chen and Tang, 2020 [8] ), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F ( x ) − y , y ˙ = g ( x ) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincaré disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve.

Limit Cycles in the Discontinuous Planar Piecewise Linear Systems with Three Zones

Limit Cycles in the Discontinuous Planar Piecewise Linear Systems with Three Zones

Global phase portraits of planar piecewise linear refracting systems of saddle–saddle type

This paper deal with the global dynamics of planar piecewise linear refracting systems of saddle–saddle type with a straight line of separation. We investigate the singularities, limit cycles, homoclinic orbits, heteroclinic orbits and make the classification of global phase portraits in the Poincaré disk for the refracting systems. We prove that these systems have 18 topologically different global phase portraits.

Limit Cycles in Discontinuous Planar Piecewise Linear Systems Separated by a Nonregular Line of Center–Center Type

Many natural phenomena can be modeled as discontinuous dynamical systems separated by a nonregular line. The number and distribution of limit cycles in discontinuous linear systems are important topics for research. In this paper, we focus on the limit cycles created by discontinuous planar piecewise linear systems separated by a nonregular line of center–center type, and prove that such systems have at most two limit cycles, which can be reached. Furthermore, the two limit cycles are nested and intersect the separation line at two points or four points, that is, either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points.

The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.

Impact Limit Cycles in the Planar Piecewise Linear Hybrid Systems

Impact Limit Cycles in the Planar Piecewise Linear Hybrid Systems