Uniqueness Of Limit Cycles Research Articles

Hopf bifurcation and limit cycle of the two‐variable Oregonator model for Belousov–Zhabotinsky reaction

This paper is concerned with a two‐variable Oregonator model for Belousov–Zhabotinsky Reaction. Llibre and Oliveira (2022) proved that Oregonator model with an unstable node or focus has at least one limit cycle in the positive quadrant, and the system has a unique stable limit cycle for some values of the parameters. It is shown in the present paper that the positive equilibrium is not a center, and that if the system has a positive equilibrium which is a weak focus, then its order is at most 2. There exist some parameter values such that system has small limit cycle around the positive equilibrium.

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.

Continuous dependence of stationary distributions on parameters for stochastic predator–prey models

AbstractThis research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0.

Bifurcations of traveling wave solutions of a generalized Burgers–Fisher equation

We deal with the traveling wave solutions (including periodic wave solutions, solitary wave solutions and kink or antikink solutions) of a generalized Burgers–Fisher equation via studying the dynamics of a related Liénard system. We obtain necessary and sufficient conditions for the Liénard system to have a unique limit cycle, a homoclinic loop and a heteroclinic loop, respectively.

Canard phenomena for a slow-fast predator-prey system with group defense of the prey

In this study, we consider that the predator reproduces much slower than the prey and propose a slow-fast predator-prey system with the group defense of the prey, which is described by the Ivlev-type functional response. We analyze the canard phenomena (relaxation oscillation and canard cycles) of our proposed model by applying the singular perturbation theory. Firstly, we apply the blow-up techniques and visualize the behavior near the special fold point, which allows us to prove the existence of the unique attractive limit cycle and imply the appearance of the canard phenomenon when crossing the fold point, and the numerical simulations verify the theoretical results. Then we further use the singular perturbation theory to investigate the existence and cyclicity of the canard cycle without head and the existence of the canard cycle with head. Our main results of the complex canard phenomenon reveal the fact that some certain species in the ecosystem suddenly burst back many years after being about extinct.

Nonexistence and Uniqueness of Limit Cycles in a Class of Three-Dimensional Piecewise Linear Differential Systems

During the last twenty years there has been increasing interest in studying the piecewise differential systems, mainly due to their many applications in natural science and technology. Up to now the most studied differential systems are in dimension two, here we study them in dimension three. One of the main difficulties for studying these differential systems consists in controlling the existence and nonexistence of limit cycles, and the numbers when they exist. In this paper, we study the nonsymmetric limit cycles for a family of three-dimensional piecewise linear differential systems with three zones separated by two parallel planes. For this class of differential systems we study the nonexistence, existence and uniqueness of their limit cycles.

On the cubic Kukles systems with an algebraic limit cycle of degree two

This paper is concerned with cubic Kukles system of the form x˙=−y, y˙=x+λy+a1x2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7y3, where λ,ai∈R, i=1,2,…,7, under the assumption that the above system has an algebraic periodic orbit of degree 2. It is shown that such an algebraic periodic orbit surrounds a center if and only if λ=0. If λ≠0, then it is the unique limit cycle. Moreover, we provide all the possibilities for the global phase portrait in the Poincaré disk of cubic Kukles system with an algebraic periodic orbit of degree 2.

Hybrid Control of Self-Oscillating Resonant Converters

We describe parallel and series resonant converters (PRC and SRC) via a unified set of input-dependent coordinates whose dynamics are intrinsically hybrid. We then propose hybrid feedback showing a self-oscillating behavior whose amplitude and frequency can be adjusted by a reference input ranging from zero to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\pi $ </tex-math></inline-formula> . For any reference value in that range, we give a Lyapunov function certifying the existence of a unique nontrivial hybrid limit cycle whose basin of attraction is global except for the origin. Our results are confirmed by experimental results on an SRC prototype.

Dynamic analysis of a predator–prey model of Gause type with Allee effect and non-Lipschitzian hyperbolic-type functional response

In this work, we study a predator–prey model of Gause type, in which the prey growth rate is subject to an Allee effect and the action of the predator over the prey is determined by a generalized hyperbolic-type functional response, which is neither differentiable nor locally Lipschitz at the predator axis. This kind of functional response is an extension of the so-called square root functional response, used to model systems in which the prey have a strong herd structure. We study the behavior of the solutions in the first quadrant and the existence of limit cycles. We prove that, for a wide choice of parameters, the solutions arrive at the predator axis in finite time. We also characterize the existence of an equilibrium point and, when it exists, we provide necessary and sufficient conditions for it to be a center-type equilibrium. In fact, we show that the set of parameters that yield a center-type equilibrium, is the graph of a function with an open domain. We also prove that any center-type equilibrium is stable and it always possesses a supercritical Hopf bifurcation. In particular, we guarantee the existence of a unique limit cycle, for small perturbations of the system.

Stability and Hopf bifurcation of an epidemiological model with effect of delay the awareness programs and vaccination: analysis and simulation

The present research is concerned with studying the media coverage delay impact and vaccine impact to control outbreaks the infectious disease in the population, we suggested that our mathematical modeling divide the population into three classes: susceptible persons S(t), vaccinated person V(t) and infected persons I(t). The susceptible individuals are also divided due to media programs into awareness and unawareness of disease danger and its mode of transmission. We studied the influence of time delay in response to the media program on awareness of the epidemic. We first discussed the existence of the equilibrium points of the system and obtain sufficient conditions for the local asymptotic stability of all equilibrium points. Further, the existence of Hopf bifurcation is shown near endemic equilibrium. It is interesting to note that there exists at least one limit cycle around the unstable endemic equilibrium. In particular, sufficient conditions for a unique stable limit cycle have been presented. Finally, we verified the feasibility of the results by numerical simulation and give a brief conclusion to generalize that the delay effect of media and vaccination can cause unexpected dynamic predictions for interacting populations.

Global dynamics of a Leslie–Gower predator–prey model with square root response function

A Leslie–Gower predator–prey model with square root response function is considered. We prove that the origin is unstable using blow-up method, and show that if the positive equilibrium is locally stable, then it is globally asymptotically stable based on the divergency criterion. Also, if the positive equilibrium is unstable, the system admits a unique stable limit cycle. A conclusion is finally given to show the influence of the square root response function on the dynamic behaviors of the system.

On the existence of limit cycles and invariant surfaces for sewing piecewise linear differential systems on [formula omitted

We consider a class of three-dimensional piecewise linear differential systems with two zones separated by a plane. In this class we show that there exist differential systems having: (i) a unique limit cycle; (ii) a unique one–parameter family of periodic orbits; (iii) scrolls; (iv) invariant cylinders.

Non-existence and uniqueness of limit cycles in a class of generalized Liénard equations

Non-existence and uniqueness of limit cycles in a class of generalized Liénard equations

Dynamics of a Generalized Rayleigh System

Consider the first order differential system given by $$\begin{aligned} \begin{array}{l} {\dot{x}}= y, \qquad {\dot{y}}= -x+a(1-y^{2n})y, \end{array} \end{aligned}$$ where a is a real parameter and the dots denote derivatives with respect to the time t. Such system is known as the generalized Rayleigh system and it appears, for instance, in the modeling of diabetic chemical processes through a constant area duct, where the effect of adding or rejecting heat is considered. In this paper we characterize the global dynamics of this generalized Rayleigh system. In particular we prove the existence of a unique limit cycle when the parameter $$a\ne 0$$ .

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.

The unique limit cycle in post Keynesian theory

This paper establishes not only the existence and uniqueness of a limit cycle (representing persistent growth cycles) but also the convergence of every non-equilibrium solution path (describing the state of the economy) to the unique limit cycle in post Keynesian theory.

Relay Self-Oscillations for Second Order, Stable, Nonminimum Phase Plants

We study a relay feedback system (RFS) having an ideal relay element and a linear, time-invariant, second-order plant. The relay element is modeled as an ideal on-off switch. And the plant is modeled using a transfer function that as follows: first, is Hurwitz stable, second, is proper, third, has a positive real zero, andfourth, has a positive dc gain. We analyze this RFS using a state-space description, with closed-form expressions for the state trajectory from one switching time to the next. We prove that the state transformation from one switching time to the next, first, has a Schur stable linearization, and first, is a contraction mapping. Then using the Banach contraction mapping theorem, we prove that all trajectories of this RFS converge asymptotically to a unique limit cycle. This limit cycle is symmetric, and is unimodal as it has exactly two relay switches per period. This result helps understand the behavior of the relay autotuning method, when applied to second-order plants with no time delay. We also treat the case where the plant either has no finite zero, or has exactly one zero that is negative.

Demand cycles and heterogeneous conformity preferences

We explore the dynamics of demand for n designs of a good when agents have preferences for (anti-)conformity. Agents differ in their social status and each agent seeks to imitate those of higher status and to distinguish herself from those of lower status, relative to her own status. In each period, every agent chooses a design given each agent's demand in the previous period. We show that demand dynamics resemble fashion cycles: The demand for designs is repetitively bell-shaped over time, and, when positively demanded, a design trickles from high- to low-status individuals. At least for n=3, the demand dynamics converge to a unique limit cycle. We obtain a similar (though weaker) convergence result for n=4, and simulations suggest that the result holds for n=4 and 5.

On the uniqueness and expression of limit cycles in planar polynomial differential system via monotone iterative technique

ABSTRACT This paper investigates the limit cycles for planar differential system where are homogeneous polynomials of degree n. Using the method of upper and lower solutions we show that this system has a unique limit cycle under some suitable conditions. Moreover, by employing the monotone iterative technique, we provide a computational method to obtain the expression, or the asymptotical one, of this limit cycle in the polar form. The error estimation of the asymptotical solutions is also provided.

Existence and Uniqueness of Limit Cycles for a Class of Nonlinear Systems

Existence and Uniqueness of Limit Cycles for a Class of Nonlinear Systems