Small-amplitude Limit Cycles Research Articles

Maximal canards in a slow–fast Rosenzweig–MacArthur model with intraspecific competition among predators

Using geometric singular perturbation theory, this paper investigates the canard phenomenon of a slow–fast Rosenzweig–MacArthur model. The model incorporates intraspecific competition among predators, assuming predator reproduction occurs much slower than prey. We demonstrate the occurrence of maximal canards between attracting and repelling slow manifolds as a bifurcation parameter varies. Additionally, we derive an analytic expression to approximate the bifurcation parameter value at which a maximal canard occurs. The method employed for this analysis relies on the blowup method. This involves finding a quasihomogeneous blowup map to desingularize the nonnormally hyperbolic point. Subsequently, charts are utilized to express the blowup in local coordinates, calculate local data, investigate the dynamics of the blown-up vector fields, and establish connections across charts. Furthermore, we provide numerical simulations to illustrate the canard explosion phenomenon in the model. Through parameter variation, we observe that the model transitions from a small amplitude limit cycle to small amplitude canard cycles (canards without a head), then to large amplitude canard cycles (canards with a head), and finally to a large amplitude relaxation cycle.

A Polynomial System of Degree Four with an Invariant Square Containing At Least Five Limit Cycles

We consider a class of polynomial systems of degree four with four real invariant straight lines that form a square, called this an invariant square, and also that contains in its interior at least five small amplitude limit cycles for a certain choice of the parameters. Moreover, we will obtain the necessary and sufficient conditions for the critical point inside the square to be a center.

Limit cycle bifurcation from a zero-Hopf equilibrium for a class of 3-dimensional Kolmogorov systems

A zero-Hopf equilibrium point p of a 3-dimensional autonomous differential system in R3 is an equilibrium point such that the eigenvalues of the linear part of the system at p are 0 and ±ωi with ω≠0. A zero-Hopf bifurcation takes place when from a zero-Hopf equilibrium bifurcate some small-amplitude limit cycles moving the parameters of the system.Polynomial Kolmogorov systems are the natural extension of the polynomial Lotka–Volterra systems of degree 2 to higher degree. The Kolmogorov systems have been studied intensively due to their applications for modeling many natural phenomena.In this paper we characterize, up to first order in the averaging theory, the eight distinct zero-Hopf bifurcations which can exhibit the class of 3-dimensional Kolmogorov systems of degree 3, providing an explicit approximation of the bifurcated small-amplitude limit cycles, together with information about their kind of stability.From each one of this eight different zero-Hopf bifurcations emerges one or two limit cycle using the averaging theory of first order. Moreover we provide an explicit example of each one of these eight zero-Hopf bifurcations.

An upper bound for the number of small-amplitude limit cycles in non-smooth Liénard system

The non-smooth Liénard system, a well-known nonlinear model, appears in a natural way in physics, chemistry, biology, and so on, in which periodic phenomena play a relevant role. In this paper, we investigate the small-amplitude limit cycles generated by the Hopf bifurcation of the non-smooth Liénard system. By utilizing the Picard–Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles for a generic non-smooth Liénard system. Finally, an example is given to illustrate the efficiency of the theoretical results.

Bifurcation of Limit Cycles from a Focus-Parabolic-Type Critical Point in Piecewise Smooth Cubic Systems

In this paper, we investigate the maximum number of small-amplitude limit cycles bifurcated from a planar piecewise smooth focus-parabolic type cubic system that has one switching line given by the x-axis. By applying the generalized polar coordinates to the parabolic subsystem and computing the Lyapunov constants, we obtain 11 weak center conditions and 9 weak focus conditions at (0,0). Under these conditions, we prove that a planar piecewise smooth cubic system with a focus-parabolic-type critical point can bifurcate at least nine limit cycles. So far, our result is a new lower bound of the cyclicity of the piecewise smooth focus-parabolic type cubic system.

A polynomial system of degree four with an invariant triangle containing at least four small amplitude limit cycles

In this work, the existence of a polynomial system of degree four with an invariant triangle containing at least four small-amplitude limit cycles is proved. This result improves the result obtained in [2].

Perturbation of a piecewise regular-singular Liénard system

In this paper we discuss the perturbation of a generalized Liénard system, defined in two regions. For x≤0 the system is regular, while for x > 0 the system is singularly perturbed. An essential difference with other papers is that a degenerate canard point appears which requires a blow-up analysis.We prove that under generic conditions the perturbed system has at most one small-amplitude limit cycle and at most N canard limit cycles, where N is the number of local extrema of the function F(x) in the singular part of the Liénard system. For critical values of the control parameters the mechanism of a canard explosion is discussed.As an application we prove that the upper bound for the number of limit cycles in a Gause predator-prey system with a general monotonic functional response function with cut-off is two times the number of maxima of the natural prey growth function after the cut-off.

Bifurcation of Limit Cycles and Center in 3D Cubic Systems with Z3-Equivariant Symmetry

This paper focuses on investigating the bifurcation of limit cycles and centers within a specific class of three-dimensional cubic systems possessing Z3-equivariant symmetry. By calculating the singular point values of the systems, we obtain a necessary condition for a singular point to be a center. Subsequently, the Darboux integral method is employed to demonstrate that this condition is also sufficient. Additionally, we demonstrate that the system can bifurcate 15 small amplitude limit cycles with a distribution pattern of 5−5−5 originating from the singular points after proper perturbation. This finding represents a novel contribution to the understanding of the number of limit cycles present in three-dimensional cubic systems with Z3-equivariant symmetry.

Bifurcations driven by generalist and specialist predation: mathematical interpretation of Fennoscandia phenomenon.

In this paper, we revisit a predator-prey model with specialist and generalist predators proposed by Hanski et al. (J Anim Ecol 60:353-367, 1991) , where the density of generalist predators is assumed to be a constant. It is shown that the model admits a nilpotent cusp of codimension 4 or a nilpotent focus of codimension 3 for different parameter values. As the parameters vary, the model can undergo cusp type (or focus type) degenerate Bogdanov-Takens bifurcations of codimension 4 (or 3). Our results indicate that generalist predation can induce more complex dynamical behaviors and bifurcation phenomena, such as three small-amplitude limit cycles enclosing one equilibrium, one or two large-amplitude limit cycles enclosing one or three equilibria, three limit cycles appearing in a Hopf bifurcation of codimension 3 and dying in a homoclinic bifurcation of codimension 3. In addition, we show that generalist predation stabilizes the limit cycle driven by specialist predators to a stable equilibrium, which clearly explains the famous Fennoscandia phenomenon.

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.

Limit cycles in a switching Liénard system

<p style='text-indent:20px;'>In this paper, we consider a class of quadratic switching Liénard systems with three switching lines. We give an algorithm for computing the Lyapunov constants of this system. Based on this method, we obtain a center condition and three limit cycles bifurcating from the focus <inline-formula><tex-math id="M1">\begin{document}$ (0,0) $\end{document}</tex-math></inline-formula>. Further, an example of quadratic switching systems is constructed to show the existence of six limit cycles bifurcating from the center. This is a new low bound on the maximal number of small-amplitude limit cycles obtained in such quadratic switching systems.</p>

Twenty-Six Crossing Limit Cycles Around One Singular Point in a Cubic Switching System

In this paper, we obtain a lower bound for the maximum number of crossing limit cycles for cubic planar switching systems with two regions separated by a straight line. By pseudo Hopf bifurcation and [Formula: see text]-order Lyapunov constants, we prove that there exist cubic near-integrable switching systems with at least 26 small-amplitude limit cycles bifurcating from an elementary center after suitable perturbations. To the best of our knowledge, this is the best lower bound so far for the cubic class by using a first order analysis.

Configuration of Ten Limit Cycles in a Class of Cubic Switching Systems

The bifurcation of limit cycles is an important part in the study of switching systems. The investigation of limit cycles includes the number and configuration, which are related to Hilbert’s 16th problem. Most researchers studied the number of limit cycles, and only few works focused on the configuration of limit cycles. In this paper, we develop a general method to determine the configuration of limit cycles based on the Lyapunov constants. To show our method by an example, we study a class of cubic switching systems, which has three equilibria: (0,0) and (±1,0), and compute the Lyapunov constants based on Poincaré return map, then find at least 10 small-amplitude limit cycles that bifurcate around (1,0) or (−1,0). Using our method, we determine the location distribution of these ten limit cycles.

The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems

This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function H=12y2+12x2−23x3+a4x4(a≠0) under two types of polynomial perturbations of degree m, respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least [3m−14] small limit cycles near the center, where m≤101, and that the related near-Hamiltonian system with general polynomial perturbations can have at least m+[m+12]−2 small-amplitude limit cycles, where m≤16. Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, [·] represents the integer function.

Bifurcation Analysis for a Class of Cubic Switching Systems

In this paper, we consider a class of switching systems perturbed by cubic homogeneous polynomials. This class of systems is separated by a straight line: [Formula: see text], and has three equilibria: [Formula: see text] and [Formula: see text] which are in the separation line. A new version of the Gasull–Torregrosa method based on Poincaré return maps is presented, and used to compute the Lyapunov constants. Based on this method, a complete classification on the center conditions is obtained for the studied class of systems. Furthermore, by perturbing the cubic switching integral system with cubic homogeneous polynomials, we show that at least ten small-amplitude limit cycles are obtained around one of the centres. This is a new lower bound for the number of limit cycles bifurcating from a center in such switching systems with cubic homogeneous nonlinearities.

Bifurcation of limit cycles from a parabolic–parabolic type critical point in a class of planar piecewise smooth quadratic systems

In this paper we study small amplitude limit cycles bifurcated from a parabolic–parabolic (PP) type critical point of planar piecewise smooth quadratic systems having exactly one switching line given by the x-axis. Besides the non-smoothness, more difficulties arise when the critical point has a parabolic contact. For such kind of critical point, to make sure that the return map is analytic, one has to use the generalized polar coordinates instead of the classic polar coordinates to compute the corresponding Lyapunov constants. Consequently, much more complicated computations are involved. By the recent results of Novaes and Silva, the index of the first nonzero Lyapunov constant for a PP type critical point is always an even number 2ℓ+2 with ℓ≥1. We call the corresponding weak focus (0,0) is of order ℓ. We obtain eight center conditions and six conditions under which (0,0) is a weak focus of order 6. Furthermore, we prove that at least seven limit cycles can bifurcate from (0,0).

Dynamics of a cubic differential advertising model

In this paper a cubic differential advertising model with a singular point O(0,0) is studied. The dynamics of the system is investigated and the global phase portraits are obtained. A small amplitude limit cycle is emitted by the Hopf super critical bifurcation.

Bifurcation for a class of piecewise cubic systems with two centers

In this paper, a class of symmetric cubic planar piecewise polynomial systems are presented, which have two symmetric centers corresponding to two period annuli. By perturbation and considering piecewise first order Melnikov function, we show the existence of 18 limit cycles (not small-amplitude limit cycles) with the configuration ( 9 , 9 ) bifurcating from the two period annuli and 22 small-amplitude limit cycles with the configuration ( 11 , 11 ) , respectively.

Limit Cycles from Hopf Bifurcation in Nongeneric Quadratic Reversible Systems with Piecewise Perturbations

In this paper, we consider a nongeneric quadratic reversible system with piecewise polynomial perturbations. We use the expansion of the first order Melnikov function to obtain the maximal number of small-amplitude limit cycles produced by Hopf bifurcation in the perturbed systems.

Hopf bifurcation in 3-dimensional polynomial vector fields

In this work we study the local cyclicity of some polynomial vector fields in R3. In particular, we give a quadratic system with 11 limit cycles, a cubic system with 31 limit cycles, a quartic system with 54 limit cycles, and a quintic system with 92 limit cycles. All limit cycles are small amplitude limit cycles and bifurcate from a Hopf type equilibrium. We introduce how to find Lyapunov constants in R3 for considering the usual degenerate Hopf bifurcation with a parallelization approach, which enables to prove our results for 4th and 5th degrees.