Continuous Piecewise Linear Systems Research Articles

Limit cycles in a rotated family of generalized Liénard systems allowing for finitely many switching lines

Analytic rotated vector fields have four significant properties: as the rotated parameter $\alpha$ changes, the amplitude of each stable (or unstable) limit cycle varies monotonically, each semi-stable limit cycle bifurcates at most two limit cycles, the isolated homoclinic loop (if exists) disappears while a unique limit cycle with the same stability arises or no closed orbits arise oppositely, and a unique limit cycle arises near the weak focus (if exists). In this paper, we prove that the four properties remain true for a rotated family of generalized Liénard systems having finitely many switching lines. Furthermore, we discuss variational exponent and use it to formulate multiplicity of limit cycles. Then we apply our results to give exact number of limit cycles to a continuous piecewise linear system with three zones and answer to a question on the maximum number of limit cycles in an SD oscillator.

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

This article concerns the global dynamics of a continuous planar piecewise linear differential system with three zones. We give global phase portraits in the Poincare disc and classify bifurcation diagrams under certain parametric conditions, when the dynamics of central linear zone is anti-saddle. Rich dynamical behaviors are demonstrated, from which we observe homoclinic loops appearing in three linear zones and limit cycles occurring in three linear zones which surround a node or node-focus.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/83/abstr.html

Model reduction of a periodically forced slow–fast continuous piecewise linear system

AbstractIn this paper, singular perturbation theory is exploited to obtain a reduced-order model of a slow–fast piecewise linear 2-DOF oscillator subjected to harmonic excitation. The nonsmooth nonlinearity of piecewise linear nature is studied in the case of bilinear damping as well as with bilinear stiffness characteristics. We propose a continuous matching of the locally invariant slow manifolds obtained in each subregion of the state space, which yields a reduced-order model of the same nature as the full dynamics. The frequency-response curves obtained from the full system and the reduced-order models suggest that the proposed reduction method can capture nonlinear behaviors such as super- and subharmonic resonances.

Fractal dimensions of continuous piecewise linear iterated function systems

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.

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

In this paper, we show that any switching hypersurface of [Formula: see text]-dimensional continuous piecewise linear systems is an [Formula: see text]-dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines.

Global dynamics of a degenerate planar piecewise linear differential system with three zones

We study the dynamics of a continuous planar piecewise linear differential system with three zones separated by two parallel lines. We provide a complete study on the maximum number of limit cycles by geometric techniques when the system is degenerate in the left and central zones. The bifurcation diagrams and global phase portraits in the Poincaré disc are given. It is shown that a degenerate continuous planar piecewise linear differential system can have scabbard bifurcation, grazing bifurcation and double limit cycle bifurcation. Moreover, we detect three new scabbard bifurcations, such as both the limit cycle with two zones and the limit cycle with three zones can be bifurcated from a singular continuum with infinite length.

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

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 boundary equilibrium bifurcation curves, degenerate boundary equilibrium bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curves. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results.

Limit cycles of planar piecewise linear Hamiltonian differential systems with two or three zones

In this paper, we study the existence of limit cycles in continuous and discontinuous planar piecewise linear Hamiltonian differential system with two or three zones separated by straight lines and such that the linear systems that define the piecewise one have isolated singular points, i.e. centers or saddles. In this case, we show that if the planar piecewise linear Hamiltonian differential system is either continuous or discontinuous with two zones, then it has no limit cycles. Now, if the planar piecewise linear Hamiltonian differential system is discontinuous with three zones, then it has at most one limit cycle, and there are examples with one limit cycle. More precisely, without taking into account the position of the singular points in the zones, we present examples with the unique limit cycle for all possible combinations of saddles and centers.

On rigorous integration of continuous piecewise linear systems

In this work, rigorous methods to compute solutions of continuous piecewise linear systems are presented. A general technique, based on integrating a perturbed linear system, is described to compute enclosures of trajectories crossing planes where the vector field in not smooth. Algorithms to compute enclosures of trajectories passing arbitrarily close to unstable equilibria are proposed. The usefulness of the methods is shown by proving the existence of trapping regions enclosing three different types of chaotic attractors existing in the Chua’s circuit.

Global Analysis of an Asymmetric Continuous Piecewise Linear Differential System with Three Linear Zones

In [Chen et al., 2020], the third author and other coauthors studied global dynamics of the following system: [Formula: see text] in the parameter region [Formula: see text]. To study completely the piecewise linear system, we consider the parameter region [Formula: see text] in this paper. Firstly, we study the local dynamics, such as the bifurcations of equilibria (including the equilibrium at infinity). Secondly, the number and stability of limit cycles are studied completely. Then, we analyze the existence of upper and lower saddle connections and homoclinic loops. Moreover, we show that there are no heteroclinic loops in this parameter region. Finally, we give the bifurcation diagram and all global phase portraits on the Poincaré disc are given as well as some numerical examples.

A new simple proof for Lum–Chua’s conjecture

The already proved Lum–Chua’s conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel characterization for Poincaré half-maps in planar linear systems. This proof is very short and straightforward, because this characterization avoids the inherent flaws of the usual methods to study piecewise linear systems (the appearance of large case-by-case analysis due to the unnecessary discrimination between the different spectra of the involved matrices, to deal with transcendental equations forced by the implicit occurrence of flight time, …). In addition, the application of the characterization allow us to prove that if a limit cycle exists, then it is hyperbolic and its stability is determined by a simple relationship between the parameters. To the best of our knowledge, the hyperbolicity of the limit cycle and this simple expression for its stability have not been pointed out before.

Global dynamics of an asymmetry piecewise linear differential system: Theory and applications

In this paper, we have completely studied a four-parameter family of continuous piecewise linear differential systems with three linear regions and no symmetry in the compactification space of R2. The bifurcation diagram and the corresponding global phase portraits of the continuous piecewise linear system are derived. It is shown that the continuous piecewise linear system has rich dynamics such as three isolated equilibria, three limit cycles, a figure-eight loop and a homoclinic loop which likes a cuspidal loop in R2 for some parameters values, respectively. As applications of our results, we can obtain the global dynamics of continuous piecewise linear FitzHugh-Nagumo equation and non-symmetric memristor-based electronic oscillators, respectively.

Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

In this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.

Bifurcations from a center at infinity in 3D piecewise linear systems with two zones

We consider continuous piecewise linear systems in R3 with two zones under the assumption of having a linear center in the invariant manifold of the point at infinity. A specific conic projection is introduced, so that it is possible to analyze in a convenient way the dynamics near such a center at infinity in two qualitative different situations.In the semi-homogeneous case, such a center is associated to the existence of a continuum of invariant semi-cones sharing the vertex at the origin; perturbing the configuration it is possible to detect the bifurcation of a limit cycle at infinity leading to the bifurcation of isolated invariant semi-cones.For the non-homogeneous case, the cycles at infinity does not imply invariant semi-cones. However, the non-generic case when the center at infinity is associated to the existence of invariant cylinders for one of the involved vector fields, becomes rather interesting. It is possible then, by perturbing the other vector field, to get the bifurcation of a big limit cycle from infinity without destroying the center.

Saddle-node of limit cycles in planar piecewise linear systems and applications

In this article, we prove the existence of a saddle-node bifurcation of limit cycles in continuous piecewise linear systems with three zones. The bifurcation arises from the perturbation of a non-generic situation, where there exists a linear center in the middle zone. We obtain an approximation of the relation between the parameters of the system, such that the saddle-node bifurcation takes place, as well as of the period and amplitude of the non-hyperbolic limit cycle that bifurcates. We consider two applications, first a piecewise linear version of the FitzHugh-Nagumo neuron model of spike generation and second an electronic circuit, the memristor oscillator.

A multiple focus-center-cycle bifurcation in 4D discontinuous piecewise linear memristor oscillators

The dynamical richness of 4D memristor oscillators has been recently studied in several works, showing different regimes, from stable oscillations to chaos. Typically, only numerical simulations have been reported and so there is a lack of mathematical results. We focus our analysis in the existence of multiple stable oscillations in the 4D piecewise linear version of the canonical circuit proposed by Itoh and Chua (Int J Bifurc Chaos 18(11):3183–3206, 2008). This oscillator is modeled by a discontinuous piecewise linear dynamical system. By adding one parameter that stratifies the 4D dynamics, it is shown that the dynamics in each stratum is topologically equivalent to a 3D continuous piecewise linear dynamical system. Some previous results on bifurcations in such reduced system allow to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits.

The boundary focus–saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators

Among the boundary equilibrium bifurcations in planar continuous piecewise linear systems with two zones separated by a straight line, the focus–saddle bifurcation corresponds with a one-parameter transition from a situation without equilibria to a configuration with two equilibria, namely a focus and a saddle point. Depending on the dynamics of the two linear systems involved, the focus can appear surrounded by a limit cycle, by a saddle-loop (homoclinic connection) or by nothing else.After introducing a criticality coefficient whose sign discriminates the different possible situations, the focus–saddle bifurcation is quantitatively characterized for the first time. The analysis requires to work in a more general framework, as is the family of planar refracting linear systems with two zones, for which a new result about existence and uniqueness of limit cycles and saddle-loops is also shown.The achieved results are applied to the study of oscillations in an electronic circuit involving a single memristor cell, showing rigorously the appearance of limit cycles via the focus–saddle bifurcation analyzed in the paper.

On Discontinuous Piecewise Linear Models for Memristor Oscillators

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.

On the periodic solutions of the Michelson continuous and discontinuous piecewise linear differential system

Applying new results from the averaging theory for continuous and discontinuous differential systems, we study the periodic solutions of two distinct versions of the Michelson differential system: a Michelson continuous piecewise linear differential system and a Michelson discontinuous piecewise linear differential system. The tools here used can be applied to general nonsmooth differential systems.

Limit cycles in planar continuous piecewise linear systems

In this paper an asymmetric planar continuous piecewise linear differential system with three zones x˙=y−F(x),y˙=−g(x) is considered. The aim of this paper gives a completely study of limit cycles when this system satisfies such conditions and the uniqueness equilibrium does not lie in the central region. When (x−x0)g(x)>0 for ∀x ≠ x0 and y=F(x) is a Z-shaped curve, it owns at most two limit cycles, which exist between a linear Hopf bifurcation surface and a double limit cycle bifurcation surface. Moreover, we prove the conjectures proposed by Ponce et al. [27]. When the uniqueness equilibrium lies in the central region, this system has exactly one limit cycles by others. Finally, some numerical examples are demonstrated.