On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

doi:10.1016/j.nonrwa.2013.02.004

Nonlinear Analysis: Real World Applications

Volume 14, Issue 5, October 2013, Pages 2002-2012

https://doi.org/10.1016/j.nonrwa.2013.02.004 Get rights and content

Highlights

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Non-symmetric planar continuous piecewise-linear differential systems are studied.
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Some results about the existence and uniqueness of their limit cycles are given.
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For systems with three linear zones and no symmetries new results are obtained.
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For systems with two linear zones a shorter proof of known results is achieved.
•
The application to the McKean model of a single neuron activity is described.

Abstract

Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems.

New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.

A relevant application to the McKean piecewise linear model of a single neuron activity is included.

Section snippets

Introduction and statement of main results

For planar differential systems, the analysis of the possible existence of limit cycles and their uniqueness is a problem which has attracted the interest of many works in the past. For smooth systems, good classical references in the field are the books [1], [2]. The restriction of this problem to polynomial differential equations is the well-known 16th Hilbert’s problem [3]. Since Hilbert’s problem turns out to be a strongly difficult one, Smale [4] has particularized it to Liénard

Applications

We analyze here a celebrated continuous piecewise linear model in mathematical biology. Following [20], the equations for the two-dimensional McKean model of a single neuron activity take the form $C \dot{v} = f (v) - w + I,$ $\dot{w} = v - γ w,$ where $v$ stands for the voltage, $w$ is the gating variable and $f (v) = {\begin{matrix} - v & if v < a / 2, \\ v - a & if a / 2 \leq v \leq (1 + a) / 2, \\ 1 - v & if v > (1 + a) / 2 . \end{matrix}$ Here, $C > 0, γ > 0, I$ is a constant drive, and $f (v)$ is a CPWL caricature of the cubic FitzHugh–Nagumo nonlinearity $f (v) = v (1 - v) (v - a)$ , provided that $0 < a < 1$ , see [20] for more

The Massera’s method for uniqueness of limit cycles

We review in this section a geometrical argument which is usually known as Massera’s method; it will allow us, after adequate adaptations, to show the uniqueness of limit cycles in the CPWL differential systems considered in this paper, when they satisfy certain hypotheses. Uniqueness results for limit cycles are typically rather involved; see [1], [2], for a review on the subject. Here we reformulate in a specific way the simple and elegant idea proposed by J.L. Massera in his brief note

Preliminary results and uniqueness of limit cycles

In this section we give some preliminary results and also include uniqueness results for possible periodic orbits. Note that if we make in system (1) the change $X = - x, Y = - y$ we get the system $\dot{X} = \tilde{F} (X) - Y,$ $\dot{Y} = \tilde{g} (X) + δ,$ where the new functions $\tilde{F}$ and $\tilde{g}$ are obtained from that given in (2) by interchanging the subscripts L and R. Thus, there is no loss of generality in assuming $δ \geq 0$ hereafter.

Our first result is just a preparation lemma, reducing by one the number of parameters and looking for a more

Existence of limit cycles and proof of main results

In this section we will use as main tools the Poincaré return map to show the existence of periodic orbits. We start by considering systems with three linearity zones given in (12), (13), (14) with $x_{L} < 0 < x_{R}$ for which the conditions $a_{L}, a_{R} < 0, a_{C} > 0$ and $b_{L}, b_{R} \geq 0$ hold. We will use the positive and negative parts of the $y$ -axis as domain and range for defining two different half-return maps, namely a right half-return map $P_{R}$ and a left half-return map $P_{L}$ .

We start by studying the qualitative properties of

Acknowledgments

J. Llibre is partially supported by a MICINN/FEDER grant number MTM2008-03437, by an AGAUR grant number 2009SGR-410 and by ICREA Academia. M. Ordóñez is partially supported by a MICINN/FEDER grant number ECO2010-17766. E. Ponce is partially supported by a MICINN/FEDER grant number MTM2009-07849 and Junta de Andalucía grant number P08-FQM-03770; he also wants to acknowledge Prof. Marco Sabatini for his helpful suggestions and discussions on Massera’s method.

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Topological classifications of a piecewise linear Liénard system with three zones
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Llibre et al. in [J. Nonlinear Sci., 2019] studied the number of limit cycles of piecewise linear Liénard system with one anti-saddle at the central zone in some parameter regions. However, the exact number and hyperbolicity of limit cycles were still unclear as in these cases of parameter. The goal of this paper is to give the exact number of limit cycles and the hyperbolicity of limit cycles in these parameter cases, where the maximum number of limit cycles is two. Moreover, all corresponding global phase portraits in the Poincaré disc are obtained.
On the periodic orbits of the continuous–discontinuous piecewise differential systems with three pieces separated by two parallel straight lines
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During these last twenty years many papers have been published about the piecewise differential systems in the plane. The increasing interest on these kind of differential systems mainly is due to their big number of applications for modeling many natural phenomena. As usual one of the main difficulties for understanding the dynamics of the differential systems in the plane consists in controlling their periodic orbits and mainly their limit cycles. Therefore there are a big number of papers studying the existence or non-existence of periodic orbits of the continuous and discontinuous piecewise differential systems, but as far as we know this paper will be one of the first papers studying the periodic orbits of a class of piecewise differential systems such that in a part of the line of separation between the two differential systems forming the piecewise differential system is continuous and in the other part it is discontinuous.
Thus we shall study the piecewise differential systems separated by two parallel straight lines, in one of these straight lines the piecewise differential system is continuous and in the other discontinuous, moreover in the three pieces of these piecewise differential systems we put arbitrary Hamiltonian systems of degree one. We obtain that such kind of continuous–discontinuous piecewise differential systems cannot have limit cycles, but they can have a continuum of periodic orbits.
Global dynamics and bifurcation for a discontinuous oscillator with irrational nonlinearity
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The objective is to carry out thorough analysis of global dynamics and bifurcation for a discontinuous oscillator, showing the transition of dynamics between the piecewise smooth system (with irrational nonlinearity) and the piecewise linear system. The existence of one or three equilibrium sets is proven by the approach of Filippov. Based on non-smooth Lyapunov functions and their set-valued derivatives, asymptotical stability especially finite-time asymptotical stability is shown for the equilibrium sets. In particular, some conditions are established guaranteeing the uniqueness of an equilibrium set and its global finite-time asymptotical stability. Making use of the parametric representation, codimension-one bifurcation curves and a codimension-two bifurcation point are obtained. We give numerical simulations and bifurcation diagram to illustrate the transition of dynamics and verify the results. We further clarify some previous misconceptions in Li et al. (2016).
Limit cycles of generic piecewise center-type vector fields in R<sup>3</sup> separated by either one plane or by two parallel planes
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While the limit cycles of the piecewise differential systems in the plane $R^{2}$ have been studied intensively during these last twenty years, this is not the case for the limit cycles of the piecewise differential systems in the space $R^{3}$ .
The goal of this article is to study the continuous and discontinuous piecewise differential systems in $R^{3}$ , formed by linear vector fields similar to planar centers separated by one or two parallel planes. We call those “center-type” differential systems, which have two pure imaginary numbers and zero as eigenvalues. When these kinds of piecewise differential systems are continuous or discontinuous separated by one plane, then they have no limit cycles. Also, if they are continuous separated by two planes, then generically they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel planes, we show that generically they can have at most four limit cycles, and that there exist such systems with four limit cycles. The genericity here means that the statements hold in a residual set of the space of parameters associated to the differential system.
We recall that the same problem but for discontinuous piecewise differential systems in $R^{2}$ formed by linear differential centers separated by two parallel straight lines has at most one limit cycle.
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The HLBs detailed below are local bifurcations. For piecewise-smooth systems limit cycles can instead be created in global bifurcations such as ‘canard super-explosions’ where an equilibrium transitions instantaneously to a large amplitude limit cycle [79–82], For piecewise-linear systems a limit cycle may be created at infinity [83] and if there are two parallel switching manifolds a limit cycle intersecting both manifolds can arise [84–87]. Other bifurcations include those at which two equilibria and a local limit cycle emanate from a single point on a switching manifold.
For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this transition can be achieved in fundamentally different ways. This paper reviews 20 such ‘Hopf-like’ bifurcations for two-dimensional ODE systems with state-dependent switching rules. The bifurcations include boundary equilibrium bifurcations, the collision or change of stability of equilibria or folds on switching manifolds, and limit cycle creation via hysteresis or time delay. In each case a stationary solution changes stability and possibly form, and emits one limit cycle. Each bifurcation is analysed quantitatively in a general setting: we identify quantities that govern the onset, criticality, and genericity of the bifurcation, and determine scaling laws for the period and amplitude of the resulting limit cycle. Complete derivations based on asymptotic expansions of Poincaré maps are provided. Many of these are new, done previously only for piecewise-linear systems. The bifurcations are collated and compared so that dynamical observations can be matched to geometric mechanisms responsible for the creation of a limit cycle. The results are illustrated with impact oscillators, relay control, automated balancing control, predator–prey systems, ocean circulation, and the McKean and Wilson–Cowan neuron models.
On the limit cycles of the piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center
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Citation Excerpt :
Since the 1930s many books and papers study the piecewise differential systems, mainly due to their applications to mechanics and electrical circuits, see for instance [6,7,25,28]. The more studied piecewise differential systems are the continuous and discontinuous piecewise differential systems separated by a straight-line, see for instance [4,9–11,17–23]. A limit cycle is an isolated periodic orbit in the set of all periodic orbits of a differential system.
While the limit cycles of the discontinuous piecewise differential systems formed by two linear differential systems separated by one straight line have been studied intensively, and up to now there are examples of these systems with at most 3 limit cycles. There are almost no works studying the limit cycles of the discontinuous piecewise differential systems formed by one linear differential system and a quadratic polynomial differential system separated by one straight line.
In this paper using the averaging theory up to seven order we prove that the discontinuous piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center separated by one straight line can have 8 limit cycles. More precisely, at every order of the averaging theory from order one to order seven we provide the maximum number of limit cycles that can be obtained using the averaging theory.
Primary 34C05, 34A34.

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View full text

On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

Highlights

Abstract

Section snippets

Introduction and statement of main results

Applications

The Massera’s method for uniqueness of limit cycles

Preliminary results and uniqueness of limit cycles

Existence of limit cycles and proof of main results

Acknowledgments

J. Math. Anal. Appl.

J. Sound Vib.

J. Sound Vib.

J. Math. Anal. Appl.

J. Appl. Math. Mech.

Mathematische Probleme, Lecture, Second Internat. Congr. Math. (Paris, 1900)

Nachr. Ges. Wiss. Göttingen Math. Phys. Kl.

Bull. Amer. Math. Soc.

Mathematical problems for the next century

Math. Intelligencer

On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations

Nonlinearity

Cubic Liénard equations with linear damping

Nonlinearity

Global study of a family of cubic Liénard equations

Nonlinearity

On the uniqueness and nonexistence of limit cycles for predator–prey systems

Nonlinearity

Phase portraits of planar control systems

Nonlinear Anal.

Bifurcation sets of continuous piecewise linear systems with two zones

Internat. J. Bifur. Chaos