Homoclinic Loop Research Articles

General study on limit cycle bifurcation near a double eight figure loop

In this article, we explore the bifurcation problem of limit cycles near the double eight figure loop (compound cycle with a 2-polycycle connecting two homoclinic loops). A general theory is established to find the lower bound of the maximal number of limit cycles (isolated periodic orbits) near the double eight figure loop. The Liénard system, a well-known nonlinear dynamical model, appears in a natural way in physics, chemistry, engineering, and so on, where periodic phenomena play a relevant role. As an application, we investigate an $(n+1)$ th-order generalized Liénard system and prove the system has at least $7[\frac{n}{6}]+2[\frac{r}{2}]-[\frac{r}{4}]$ limit cycles near the double eight figure loop for any $n\geq5$ and $r=\rm mod(n,6)$ , and their distribution is also gained.

Dynamics of a slow-fast Leslie-Gower predator-prey model with prey harvesting.

For the Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, the known results are on the saddle-node bifurcation and the Hopf bifurcation of codimensions 1, the Bogdanov-Takens bifurcations of codimensions 2 and 3, and on the cyclicity of singular slow-fast cycles. Here, we focus on the global dynamics of the model in the slow-fast setting and obtain much richer dynamical phenomena than the existing ones, such as global stability of an equilibrium; an unstable canard cycle exploding to a homoclinic loop; coexistence of a stable canard cycle and an inner unstable homoclinic loop; and, consequently, coexistence of two canard cycles: a canard explosion via canard cycles without a head, canard cycles with a short head and a beard and a relaxation oscillation with a short beard. This last one should be a new dynamical phenomenon. Numerical simulations are provided to illustrate these theoretical results.

Bounding the number of limit cycles for perturbed piecewise linear Hamiltonian system

In this paper, we investigate the number of limit cycles for piecewise linear Hamiltonian system with a homoclinic loop under perturbations of piecewise smooth polynomials. By studying the number of zeros of the first order Melnikov function, we obtain the exact number of limit cycles bifurcating from the periodic annulus, which improves the results given by Liu and Xiong [13], Yang and Zhang [20].

Proof of two conjectures for perturbed piecewise linear Hamiltonian systems

In this paper, we study the number of limit cycles bifurcating from the centers of piecewise linear Hamiltonian systems having either a homoclinic loop or a heteroclinic loop under the perturbations of piecewise smooth polynomials. By investigating the Chebyshev properties of generating functions of the first order Melnikov functions, we obtain the sharp bounds of the number of limit cycles bifurcating from the periodic annuluses, which confirm the conjectures proposed by Liang, Han and Romanovski (2012) and Liang and Han (2016).

Stability and codimension 2 bifurcations in an [formula omitted] model with incubation delay

The aim of this paper is to investigate the complex nonlinear dynamics, such as the existence and effect of various types of bifurcations, in an infectious disease model system. An SIR delay model that accounts for saturated incidence, saturated treatment, and self-protection in the susceptible population is considered. The model system has a disease free equilibrium, and its local stability is determined by the basic reproduction number for any delay value. The existence of multiple endemic equilibria is obtained, and these equilibria may switch stability due to variation of bifurcation parameters. When incubation delay is zero, Bogdanov–Takens bifurcation occurs, which implies the existence of Hopf bifurcation, saddle–node, and homoclinic bifurcations. Thus, the disease will persist within or on the homoclinic loop and otherwise dies out. We also find the existence of backward bifurcation and forward (transcritical) bifurcation. The delay-induced stability switches are observed due to Hopf bifurcations, leading to endemic bubbles. Incubation delay affects the stability of equilibria when the model system has multiple endemic equilibria, so multi-stability is observed. Due to incubation delay, the two frequency Hopf–Hopf bifurcation and Bogdanov–Takens bifurcation are found. Thus, both codimension 1 and codimension 2 bifurcations in the effect of incubation delay are exhibited in this study. Numerically, we notice that with increasing incubation delay, the endemic equilibrium either changes stability (from stable to unstable or unstable to stable) or remains unstable with different frequencies of oscillations.

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.

BIFURCATION ANALYSIS OF AN SIR MODEL WITH SATURATED INCIDENCE RATE AND STRONG ALLEE EFFECT

In this paper, rich dynamics and complex bifurcations of an SIR epidemic model with saturated incidence rate and strong Allee effect are investigated. First, the existence of disease-free and endemic equilibria is explored, and we prove that the system has at most three positive equilibria, which exhibit different types such as hyperbolic saddle and node, degenerate unstable saddle (node) of codimension 2, degenerate saddle-node of codimension 3 at disease-free equilibria, and cusp, focus, and elliptic types Bogdanov–Takens singularities of codimension 3 at endemic equilibria. Second, bifurcation analysis at these equilibria are investigated, and it is found that the system undergoes a series of bifurcations, including transcritical, saddle node, Hopf, degenerate Hopf, homoclinic, cusp type Bogdanov–Takens of codimensional 2, and focus and elliptic type Bogdanov–Takens bifurcation of codimension 3 which are composed of some bifurcations with lower codimension. The system shows very rich dynamics such as the coexistence of multiple periodic orbits and homoclinic loops. Finally, numerical simulations are conducted on the theoretical results.

Bifurcation Analysis of a Predator–Prey Model with Alternative Prey and Prey Refuges

In this paper, we study the codimensions of Hopf bifurcation and Bogdanov–Takens bifurcation of a predator–prey model with alternative prey and prey refuges, which was proposed by Chen et al. [ 2023 ]. The results show that the predator–prey model can undergo a supercritical Hopf bifurcation or a Bogdanov–Takens bifurcation of codimension two under certain parameter conditions. It means that there are some predator–prey models with an alternative prey and prey refuges which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Hopf bifurcation is at most one and codimension of Bogdanov–Takens bifurcation is at most two.

Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

Existence and Number of Figure-Eight Loops in Planar Sector-Wise Linear Systems with Saddle–Saddle Dynamics

In this paper, we investigate the existence of one type of homoclinic double loops (i.e. figure-eight loops) in a family of planar sector-wise linear systems with saddle–saddle dynamics. We obtain necessary and sufficient conditions for the existence of a figure-eight loop. Moreover, we prove that such systems can have simultaneously three types of invariant sets: a figure-eight loop, a homoclinic loop and three different types of periodic orbits. We also provide an example to show that a crossing limit cycle can bifurcate from this figure-eight loop.

Predator–prey model with sigmoid functional response

AbstractThe sigmoid functional response in the predator–prey model was posed in 1977. But its dynamics has not been completely characterized. This paper completes the classification of the global dynamics for the classical predator–prey model with the sigmoid functional response, whose denominator has two different zeros. The dynamical phenomena we obtain here include global stability, the existence of the heteroclinic and homoclinic loops, the consecutive canard explosions via relaxation oscillation, and the canard explosion to a homoclinic loop among others. As we know, the last one is a new dynamical phenomenon, which has never been reported previously. In addition, with the help of geometric singular perturbation theory, we solve the problem of connection between stable and unstable manifolds from different singularities, which has not been well settled in the published literature.

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.

Double homoclinic bifurcations by perturbing a class of cubic Z2-equivariant polynomial systems with nilpotent singular points

This paper focuses on the study of the limit cycle bifurcation of a general cubic Z2-equivariant Hamiltonian differential system with two nilpotent singular points. Initially, the necessary and sufficient conditions for the occurrence of two nilpotent singularities are provided in the unperturbed system, along with a complete description of all possible phase portraits on the plane. Subsequently, using a combination of the Melnikov function method and various analytical techniques, we derive the approximate expansion of the first order Melnikov function at the interval endpoint, as well as formula for its coefficients. These results can be utilized to analyze double homoclinic loop bifurcation of the perturbed system, finding the lower bound of limit cycles.

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

Saddle-node bifurcation and Bogdanov-Takens bifurcation of a SIRS epidemic model with nonlinear incidence rate

The Bogdanov-Takens bifurcation of the SIRS epidemic model with nonlinear incidence rate was studied by Ruan and Wang (2003) [11], Tang et al. (2008) [13] and Lu et al. (2019) [9] in recent years. The results in the mentioned papers showed that the SIRS epidemic model with nonlinear incidence rate kI2/(1+ωI2) can undergo a Bogdanov-Takens bifurcation of codimension two. In this paper we study the SIRS epidemic model with nonlinear incidence rate kIp/(1+ωIq) for general p and q. The bifurcation analysis indicates that there is a saddle-node or a cusp of codimension two for various parameter values and the model can undergo a saddle-node bifurcation or a Bogdanov-Takens bifurcation of codimension two if suitable bifurcation parameters are selected. It means that there are some SIRS epidemic models which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Bogdanov-Takens bifurcation is at most two.

Hydra effect and global dynamics of predation with strong Allee effect in prey and intraspecific competition in predator

We consider a predator-prey system with strong Allee effect in prey and intraspecific competition in predator. We divide the parameter space into several mutually exclusive regions, and for each region we give a rather complete description of the global dynamics including equilibrium stability, Hopf bifurcation and its directions, existence of a heteroclinic orbit loop, a homoclinic orbit loop and limit cycles. In comparison with systems without the Allee effect in prey and systems without the intraspecific competition in predator, we find that the Allee effect in prey and the over-exploitation of the prey by the predator may drive the extinction of two species, but the intraspecific competition in the predator can generate “hydra effect” that regulates the population size of both predator and prey by suppressing the over-exploitation of prey, improving the probability of survivals of two species.

Limit cycle bifurcations near a double 2-polycycle on the cylinder

In this paper, we consider the bifurcation problem of limit cycles near a double 2-polycycle for a cylinder near-Hamiltonian system. We first study the stability of a general homoclinic loop of type II on the cylinder, and then obtain certain results on the number and distribution of limit cycles for the cylinder near-Hamiltonian system near the double 2-polycycle by using the Melnikov function method and the method of stability-changing of a homoclinic loop of type I or II. Furthermore, we provide a way to find alien limit cycles for cylinder near-Hamiltonian system. As applications, we obtain the number of limit cycles of a class of cylinder near-Hamiltonian systems.

On the number of limit cycles near a homoclinic loop with a nilpotent cusp of order m

This paper investigates the expansions of Melnikov functions near a homoclinic loop with a nilpotent cusp of order m. It presents a methodology for calculating all coefficients in these expansions, which can be employed to study the problem of limit cycle bifurcation. As an application, by utilizing the obtained results, the paper rigorously establishes that a polynomial Liénard system of degree n+1 has at least n+[n4] limit cycles near the homoclinic loop with a nilpotent cusp of order one. This work not only updates and generalizes existing results, but also provides a rigorous application of the obtained findings in the context of limit cycle bifurcation.

Semi-analytical estimates for the chaotic diffusion in the Second Fundamental Model of Resonance. Application to Earth’s navigation satellites

We discuss the applicability of the Melnikov and Landau–Teller theories in obtaining semi-analytical estimates of the speed of chaotic diffusion in systems driven by the separatrix-like stochastic layers of a resonance belonging to the ‘second fundamental model’ (SFM) (Henrard and Lemaitre, 1983). Stemming from the analytic solution for the SFM in terms of Weierstrass elliptic functions, we introduce stochastic Melnikov and Landau–Teller models allowing to locally approximate chaotic diffusion as a sequence of uncorrelated ‘jumps’ observed in the time series yielding the slow evolution of an ensemble of trajectories in the space of the adiabatic actions of the system. Such jumps occur in steps of one per homoclinic loop. We show how a semi-analytical determination of the probability distribution of the size of the jumps can be arrived at by the Melnikov and Landau–Teller approximate theories. Computing also the mean time required per homoclinic loop, we arrive at estimates of the chaotic diffusion coefficient in such systems. As a concrete example, we refer to the long-term diffusion of a small object (e.g. Earth navigation satellite or space debris) within the chaotic layers of the so-called 2g+h lunisolar resonance, which is of the SFM type. After a suitable normal form reduction of the Hamiltonian, we compute estimates of the speed of diffusion of these objects, which compare well with the results of numerical experiments.

Limit cycle bifurcations near double homoclinic and double heteroclinic loops in piecewise smooth systems

In this paper, the number and distributions of limit cycles bifurcating from a double homoclinic loop and a double heteroclinic loop of piecewise smooth systems with three zones are considered. By introducing a suitable Poincaré map near the double homoclinic loop, three criteria are derived to judge its inner and outer stability. Then through stability-changing method, bifurcation theorems of limit cycles near the double homoclinic and double heteroclinic loops for non-symmetric and symmetric piecewise near-Hamiltonian systems are established. A piecewise linear Z2-equivariant system is presented as an application and five limit cycles are obtained, three of which are alien limit cycles.