Uniqueness Of Limit Cycles Research Articles

Global Dynamics of a Planar Filippov System with Symmetry

Chen [2016a, 2016b] studied global dynamics of the Filippov systems [Formula: see text], respectively. To study the global dynamics of [Formula: see text] completely, since the dynamics of [Formula: see text] is very simple, we are only interested in the global dynamics of [Formula: see text] in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system [Formula: see text] is totally different from systems [Formula: see text].

A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport

We present a new discontinuous ordinary differential equation (ODE) model of the glacial cycles. Model trajectories flip from a glacial to an interglacial state, and vice versa, via a switching mechanism motivated by ice sheet mass balance principles. Filippov’s theory of differential inclusions is used to analyze the system, which can be viewed as a nonsmooth geometric singular perturbation problem. We prove the existence of a unique limit cycle, corresponding to the Earth’s glacial cycles. The diffusive heat transport component of the model is ideally suited for investigating the competing temperature gradient and transport efficiency feedbacks, each associated with ice-albedo feedback. It is the interplay of these feedbacks that determines the maximal extent of the ice sheet. In the nonautonomous setting, model glacial cycles persist when subjected to external forcing brought on by changes in Earth’s orbital parameters over geologic time. The system also exhibits various bifurcation scenarios as key parameters vary.

The dynamics and event-collision bifurcations in switched control systems with delayed switching

We analyse a switched control system with negative feedback and delayed switching. In particular, we consider the effects of small and arbitrary delays in the switching decision function on the asymptotic dynamics of the system. In the absence of time delay, the phase space contains a set of points which under the action of the system flow are bounded, and trajectories rooted at these points converge to neutrally stable pseudo-equilibria in finite time. This structure is destroyed under the introduction of time delay. For a sufficiently small time delay, the bounded trajectories converge to a unique small scale limit cycle attractor. This is shown by means of the so-called delayed switching lines. For larger delay times, we observe event-collision bifurcations, symmetry breaking bifurcations, homoclinic bifurcations and multistability. For larger time delays, the delayed switching lines play an important role as they may be used to determine the stability properties of limit cycle attractors. By means of the discontinuity mapping, we show why following an event-collision bifurcation the stability of a limit cycle attractor may be radically altered. Our numerical test-bed model we consider here may be used on the macroscopic scale as a model for human neuromuscular control during quite standing or target tracking. It is interesting that much of the complex dynamics we uncover here occurs in the parameter range of delay time of around 150 ms, which is a typical processing time of neurocontrol systems of healthy human subjects during the control of, for example, quite standing.

Global dynamics of Chua Corsage Memristor circuit family: fixed-point loci, Hopf bifurcation, and coexisting dynamic attractors

This paper presents an in-depth and rigorous mathematical analysis of a family of nonlinear dynamical circuits whose only nonlinear component is a Chua Corsage Memristor (CCM) characterized by an explicit seven-segment piecewise-linear equation. When connected across an external circuit powered by a DC battery, or a sinusoidal voltage source, the resulting circuits are shown to exhibit four asymptotically stable equilibrium points, a unique stable limit cycle spawn from a supercritical Hopf bifurcation along with three static attractors, four coexisting dynamic attractors of an associated non-autonomous nonlinear differential equation, and four corresponding coexistingpinched hysteresis loops. Thebasin of attractions of the above static and dynamic attractors is derived numerically via global nonlinear analysis. When driven by a battery, the resulting CCM circuit exhibits a contiguous fixed-point loci, along with its DC V–I curve described analytically by two explicit parametric equations. We also proved the fundamental feature of theedge of chaos property; namely, it is possible to destabilize a stable circuit (i.e., without oscillation) and make it oscillate, by merely adding a passive circuit element, namely $$L >0$$. The CCM circuit family is one of the few known example of a strongly nonlinear dynamical system that is endowed with numerous coexisting static and dynamic attractors that can be studied both experimentally, and mathematically, via exact formulas.

Elemental access to limit cycle existence in Biomath education.

This paper originated from the desire to develop elementary calculus based tools to empower students, not necessarily with a strong mathematical background, to test predator-prey related models for boundedness of solutions and for the existence of limit cycles. There are several well-known methods available to prove, or disprove, the existence of bounded solutions to systems of differential equations. These methods rely on LiГ©nard's theorem or using Dulac or Lyaponov functions. The level of mathematics required in the study of differential equations is not addressed in the courses presented on the first year level, and students in biology, ecology, economics and other fields are often not suitably equipped to perform these advanced techniques.The conditions under which a unique limit cycle exists in predator-prey systems is considered a primary problem in mathematical ecology. A great deal of mathematical effort has gone into trying to establish simple, yet general, theorems which will allow one to decide whether a given set of nonlinear equations has a limit cycle or not. We introduce a method to first determine the boundedness of solution trajectories in such a way that the transformation to a LiГ©nard system or the use of a Dulac function can be avoided. Once boundedness of trajectories has been established, the nature of the equilibrium points reduces to simple eigenvalue analysis. The Elemental Limit Cycle method (ELC) provides elementary criteria to evaluate the nature of the pivotal functions of a system which will indicate boundedness and may be applicable to more general models.

On the existence of a stable limit cycle to a piecewise linear system in R2

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type “focus‐focus” and “focus‐center.” Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case “focus‐center,” we use a pure geometric approach. In the case “focus‐focus,” we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two‐dimensional system describing the qualitative behavior of a two‐dimensional excitable membrane model.

On the existence and uniqueness of limit cycles in planar refracted systems

On the existence and uniqueness of limit cycles in planar refracted systems

Monetary policy in the unique growth cycle of post Keynesian systems

This paper deals with the following two issues in a post Keynesian system: the existence and uniqueness of a growth cycle and the effectiveness of a counter-cyclical monetary policy for stabilization. It is found that the unique growth cycle, represented by the unique limit cycle, can be observed if the rate of interest is set to a constant level while that the long-run equilibrium can be globally asymptotically stable if an appropriate counter-cyclical monetary policy is conducted.

Limit cycles in periodically driven open quantum systems

We investigate the long-time behavior of quantum N-level systems that are coupled to a Markovian environment and subject to periodic driving. As our main result, we obtain a general algebraic condition ensuring that all solutions of a periodic quantum master equation with Lindblad form approach a unique limit cycle. Quite intuitively, this criterion requires that the dissipative terms of the master equation connect all subspaces of the system Hilbert space during an arbitrarily small fraction of the cycle time. Our results provide a natural extension of Spohn’s algebraic condition for the approach to equilibrium to systems with external driving. Moreover, our theory leads to a rigorous condition for the emergence of dissipative discrete time crystals and covers also classical, periodically modulated Markov jump processes.

Dissipative periodic and chaotic patterns to the KdV–Burgers and Gardner equations

We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems and Shil'nikov's analysis. The stability of the equilibrium point is considered, and Hopf bifurcations are investigated after a certain scaling that reduces the parameter space of a three-mode dynamical system which now depends only on two parameters. The Hopf curve divides the two-dimensional space into two regions. On the left region the equilibrium point is stable leading to dissapative periodic orbits. While changing the bifurcation parameter given by the velocity of the traveling waves, the equilibrium point becomes unstable and a unique stable limit cycle bifurcates from the origin. This limit cycle is the result of a supercritical Hopf bifurcation which is proved using the Lyapunov coefficient together with the Routh-Hurwitz criterion. On the right side of the Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using Shil'nikov's theorem which requires the construction of a homoclinic orbit, while for the Gardner equation the supercritical Hopf bifurcation leads only to a stable periodic orbit.

Hopf-like boundary equilibrium bifurcations involving two foci in Filippov systems

This paper concerns two-dimensional Filippov systems — ordinary differential equations that are discontinuous on one-dimensional switching manifolds. In the situation that a stable focus transitions to an unstable focus by colliding with a switching manifold as parameters are varied, a simple sufficient condition for a unique local limit cycle to be created is established. If this condition is violated, three nested limit cycles may be created simultaneously. The result is achieved by constructing a Poincaré map and generalising analytical arguments that have been employed for continuous systems. Necessary and sufficient conditions for the existence of pseudo-equilibria (equilibria of sliding motion on the switching manifold) are also determined. For simplicity only piecewise-linear systems are considered.

On Non Liénard Type with One Limit Cycle

In the present paper, we have investigated the non Lienard system. We have shown that limit cycles may bifurcate at the origin. Bendixons theorem has been used in our study to prove non-existence of limit cycles. We have also proved that the system has unique limit cycle through change of the parameters.

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.

The number and stability of limit cycles for planar piecewise linear systems of node–saddle type

The objective of this paper is to study the number and stability of limit cycles for planar piecewise linear (PWL) systems of node–saddle type with two linear regions. Firstly, we give a thorough analysis of limit cycles for Liénard PWL systems of this type, proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle. These conditions can be easily verified directly according to the parameters in the systems, and play an important role in giving birth to two limit cycles for general PWL systems. In this step, the tool of a Bendixon-like theorem is successfully employed to derive the existence of a limit cycle. Secondly, making use of the results gained in the first step, we obtain parameter regions where the general PWL systems have at least one, at least two and no limit cycles respectively. In addition for the general PWL systems, some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively. Finally, some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles.

Limit cycles of a Liénard system with symmetry allowing for discontinuity

This paper presents new results on the limit cycles of a Liénard system with symmetry allowing for discontinuity. Our results generalize and improve the results in [32, Theorem 1 and 2] or the monograph [33, Chapter 4, Theorem 5.2]. The results in [33] are only valid for the smooth system. We emphasize that our main results are valid for discontinuous systems. Moreover, we show the presence and an explicit upper bound for the amplitude of the two limit cycles, and we estimate the position of the double-limit-cycle bifurcation surface in the parameter space. Until now, there is no result to determine the amplitude of the two limit cycles. The existing results on the amplitude of limit cycles guarantee that the Liénard system has a unique limit cycle. Finally, some applications and examples are provided to show the effectiveness of our results. We revisit a co-dimension-3 Liénard oscillator (see [22,34]) in Application 1. Li and Rousseau [22] studied the limit cycles of such a system when the parameters are small. However, for the general case of the parameters (in particular, the parameters are large), the upper bound of the limit cycles remains open. We completely provide the bifurcation diagram for the one-equilibrium case. Moreover, we determine the amplitude of the two limit cycles and estimate the position of the double-limit-cycle bifurcation surface for the one-equilibrium case. Application 2 is presented to study the limit cycles of a class of the Filippov system.

Existence and uniqueness of growth cycles in post Keynesian systems

In this paper, we explore the existence and uniqueness of a limit cycle (or of a growth cycle) in a post Keynesian system in which investment is determined by the expected rate of profit. We find that a periodic orbit appears if the marginal effect of the expected rate of profit on investment is strong enough and that it is a unique limit cycle if the speed of revisions of expectation is fast enough.

Estimates of Size of Cycle in a Predator-Prey System

We consider a Rosenzweig–MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are of independent interest.

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.

Existence and uniqueness of limit cycles in planar system of Liénard type

This paper is a study of one of the most beautiful phenomena in dynamical systems limit cycle. In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory having the property that at least one other trajectory spirals into it as time approaches either positive or negative infinity. In this paper, existence and uniqueness of limit cycles for a generalized Lienard system will be studied. Moreover, some examples will be presented to illustrate our results.

Quadratic systems with a symmetrical solution

In this paper we study the existence and uniqueness of limit cycles for so-called quadratic systems with a symmetrical solution: dx(t)/dt = P2(x, y) ≡ a00+ a10x + a01y + a20x2+ a11xy + a02y2dy(t)/dt = Q2(x, y) ≡ b00+ b10x + b01y + b20x2+ b11xy + b02y2where (x, y) ∈ ℝ2t ∈ ℝ, aijbij∈ ℝ, i.e. a real planar system of autonomous ordinary differential equations with linear and quadratic terms in the two independent variables. We prove that a quadratic system with a solution symmetrical with respect to a line can be of two types only. Either the solution is an algebraic curve of degree at most 3 or all solutions of the quadratic system are symmetrical with respect to this line. For completeness we give a new proof of the uniqueness of limit cycles for quadratic systems with a cubic algebraic invariant, a result previously only available in Chinese literature. Together with known results about quadratic systems with algebraic invariants of degree 2 and lower, this implies the main result of this paper, i.e. that quadratic systems with a symmetrical solution have at most one limit cycle which if it exists is hyperbolic.