Homoclinic Cycles Research Articles

Global dynamics of a special class of planar sector-wise linear systems

In this article, we study the global dynamics of a special class of planar sector-wise linear differential systems with two subsystems being the same except for their equilibriums. Taking the position of one equilibrium as the bifurcation parameter, we provide a complete analysis about sliding cycle bifurcation and sliding homoclinic bifurcation. Moreover, we obtain the existence of all important separatrix orbits and their dependence on the bifurcation parameter, including sliding heteroclinic orbit, heteroclinic cycle, limit cycle, sliding homoclinic cycle and sliding cycle. For more information see https://ejde.math.txstate.edu/Volumes/2024/57/abstr.html

Identifying numerical bifurcation structures of codimensions 1 and 2 in interacting species system

The present study has focused on the examination of a Holling type III nonlinear mathematical model that incorporates the influence of fear and Michaelis–Menten‐type predator harvesting. The incorporation of fear with respect to the prey species has been observed to result in a reduction in the survival probability of the prey population and a concurrent reduction in the reproduction rate of the prey species. The existence and stability of ecologically significant equilibria have been ascertained through mathematical analysis. Emphasis within the proposed model primarily centers on numerical bifurcations of codimensions 1 and 2. Numerical validation has been performed on all simulated outcomes within the feasible range of parametric values. Dynamical characteristics of the model have subsequently undergone investigation through a series of numerical simulations, successfully revealing various forms of local and global bifurcations. In addition to the identification of saddle‐node, Hopf, Bogdanov–Takens, transcritical, cusp, homoclinic, and limit point cycle (LPC) bifurcations, the model has also demonstrated bistability and global asymptotic stability. These bifurcation phenomena serve as illustrative examples of the intricate dynamical behavior inherent to the model. Numerical validation through graphical representations has been utilized to elucidate the effects of factors such as fear, nonlinear predator harvesting, and predation rate on the dynamics of the interacting species under different parametric conditions.

Bifurcations and global dynamics of a predator–prey mite model of Leslie type

AbstractIn this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus‐type and cusp‐type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle‐node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature.

Chaotic behaviors and coexisting homoclinic cycles in a class of 3D piecewise systems

Investigating homoclinic trajectories and chaotic behaviors, is conducive to better understanding the fourteenth problem listed by Smale in his response to Arnold. This paper analytically detects coexisting homoclinic cycles without symmetry required in a class of three-dimensional (3D) piecewise systems, which is significantly different from the smooth case where a pair of homoclinic cycles with respect to one equilibrium point are generally symmetric. Moreover, it is rigorously shown that such coexisting cycles gives rise to chaos by means of analyzing a Poincaré map. An example finally is presented to illustrate the proposed results.

Bifurcation Analysis of a Modified Leslie–Gower Predator–Prey System

The Leslie–Gower model, a kind of predator–prey model with weak Allee effect, is studied in this paper. The existence and stability of non-negative equilibria are first discussed. Then, we investigate several bifurcation phenomena undergoing positive equilibria, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, etc. Some possible dynamical behaviors of this model are illustrated by numerical simulation. The bifurcation diagrams for the cases of codimensions 2 and 3 are given respectively. The coexistence of a periodic cycle and a homoclinic cycle, and two limit cycles enclosing an unstable equilibrium are also proved. This appears to be the first study of the Leslie–Gower model including the influence of weak Allee effect on prey.

Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response.

In this paper, we consider the dynamics of a slow-fast Bazykin's model with piecewise-smooth Holling type Ⅰ functional response. We show that the model has Saddle-node bifurcation and Boundary equilibrium bifurcation. Furthermore, it is also proven that the model has a homoclinic cycle, a heteroclinic cycle or two relaxation oscillation cycles for different parameters conditions. These results imply the dynamical behavior of the model is sensitive to the predator competition rate and the initial densities of prey and predators. In order to support the theoretical analysis, we present some phase portraits corresponding to different values of parameters by numerical simulation. These phase portraits include two relaxation oscillation cycles, an unstable relaxation oscillation cycle surrounded by a stable homoclinic cycle; the coexistence of a heteroclinic cycle and an unstable relaxation oscillation cycle. These results reveal far richer and much more complex dynamics compared to the model without different time scale or with smooth Holling type Ⅰ functional response.

Modeling the Dynamics of a Ratio-Dependent Leslie–Gower Predator–Prey System with Strong Allee Effect

In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results.

Dynamical analysis in a piecewise smooth predator–prey model with predator harvesting

The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator–prey model with predator harvesting. We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). It is shown that boundary equilibrium bifurcation and sliding–grazing bifurcation can happen as the threshold value varies. We provide analytical analysis to prove the existence of sliding limit cycles and sliding homoclinic cycles, the coexistence of them with standard limit cycles. Some numerical simulations are given to demonstrate our results.

Chaos emerges from coexisting homoclinic cycles for a class of 3D piecewise systems

Accurately detecting a homoclinic cycle and chaos in a concrete system takes a huge challenge. This paper gives a sufficient condition of coexisting homoclinic cycles (degenerate) with respect to a saddle-focus in a class of three-dimensional piecewise systems with two switching manifolds. Furthermore, by constructing the Poincaré map, it is rigorously shown that chaotic behavior is induced by the coexisting cycles without necessary symmetry in the considered system, which is evidently different from the usual dynamics in smooth system theory that a pair of symmetrical homoclinic cycles can generate chaos but the asymmetric ones cannot. It finally provides an example to illustrate the effectiveness of the results established.

Persistent strange attractors in 3D polymatrix replicators

We introduce a one-parameter family of polymatrix replicators defined in a three-dimensional cube and study its bifurcations. For a given interval of parameters, each element of the family can be C2-approximated by a vector field whose flow exhibits suspended horseshoes and persistent strange attractors. The proof relies on the numerically observed Shilnikov homoclinic cycle to the interior equilibrium. We also describe the phenomenological steps responsible for the transition from regular to chaotic dynamics in the family (route to chaos).

Linking bifurcation analysis of Holling–Tanner model with generalist predator to a changing environment

AbstractBifurcation theory has been highly popular in the analysis of mathematical models. However, stability and bifurcation analyses are only for asymptotic dynamics while applied scientists care more about transient dynamics. In this paper, we first rigorously analyze Holling–Tanner model with generalist predators who have alternative food sources, and then discuss transient dynamics via a changing environment. For a constant environment, we provide a complete bifurcation analysis with high codimension. It is shown that the highest codimension of a nilpotent cusp is 3, and the model can undergo degenerate Bogdanov–Takens bifurcation of codimension 3. Moreover, by using resultant elimination to solve the semialgebraic varieties of Lyapunov coefficients, we show that a center‐type equilibrium is a weak focus with order at most 2, and the model can exhibit Hopf bifurcation of codimension 2. Our results indicate that generalist predators can cause not only richer dynamics and bifurcations, but also the extinction of prey for some positive initial densities. Numerical simulations, including the coexistence of a limit cycle and a homoclinic cycle, tristability, two limit cycles, are presented to illustrate the theoretical results. In a changing environment, the populations start along one stable state but can track unstable states or oscillations when the system crosses a bifurcation point, and then tend to another stable state or oscillations. This tracking on transient dynamics predicts regime shifts under environmental changes. When environmental conditions vary, the populations can track unstable states in the constant environment. The rate of environmental change determines how long the system tracks an unstable state although finally the solution under environmental change is attracted to a stable steady state or limit cycle. Finally, we focus on a periodic environment and find that the populations converge to a periodic solution or an invariant torus depending on both the initial environmental capacity and the amplitude of periodic fluctuation.

Degenerate Bogdanov-Takens bifurcation of codimension 4 in Holling-Tanner model with harvesting

In this paper, we revisit the Holling-Tanner model with constant-yield prey harvesting. It is shown that the highest codimension of a nilpotent cusp is 4, and the model can undergo degenerate Bogdanov-Takens bifurcation of codimension 4. Moreover, when the model has a center-type equilibrium, we show that it is a weak focus with order at least 3 and at most 4, and the model can exhibit Hopf bifurcation of codimension 3. Some algebraic methods including resultant elimination and pseudo-division are used to solve the semi-algebraic varieties of normal form coefficients or focal values. Our results indicate that constant-yield prey harvesting can cause not only richer dynamics and bifurcations, but also the coextinction of both populations with some positive initial densities. Finally, numerical simulations, including the coexistence of limit cycle and homoclinic cycle, and three limit cycles, are presented to illustrate the theoretical results.

Coexisting singular cycles in a class of three-dimensional three-zone piecewise affine systems

<p style='text-indent:20px;'>Detecting an isolated homoclinic or heteroclinic cycle is a great challenge in a concrete system, letting alone the case of coexisting scenarios and more complicated chaotic behaviors. This paper systematically investigates the dynamics for a class of three-dimensional (3D) three-zone piecewise affine systems (PWASs) consisting of three sub-systems. Interestingly, under different conditions the considered system can display three types of coexisting singular cycles including: homoclinic and homoclinic cycles, heteroclinic and heteroclinic cycles, homoclinic and heteroclinic cycles. Furthermore, it establishes sufficient conditions for the presence of chaotic invariant sets emerged from such coexisting cycles. Finally, three numerical examples are provided to verify the proposed theoretical results.</p>

A predator-prey model with cooperative hunting in the predator and group defense in the prey

<p style='text-indent:20px;'>In this paper we propose a predator-prey model with a non-differentiable functional response in which the prey exhibits group defense and the predator exhibits cooperative hunting. There is a separatrix curve dividing the phase portrait. The species with initial population above the separatrix result in extinction of prey in finite time, and the species with initial population below it can coexist, oscillate sustainably or leave the prey surviving only. Detailed bifurcation analysis is carried out to explore the effect of cooperative hunting in the predator and aggregation in the prey on the existence and stability of the coexistence state as well as the dynamics of system. The model undergoes transcritical bifurcation, Hopf bifurcation, homoclinic (heteroclinic) bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation, and through numerical simulations it is found that it possesses rich dynamics including bubble loop of limit cycles, and open ended branch of periodic orbits disappearing through a homoclinic cycle or a loop of heteroclinic orbits. Also, a continuous transition of different types of Hopf branches are investigated which forms a global picture of Hopf bifurcation in the model.</p>

Homoclinic Orbits in Several Classes of Three-Dimensional Piecewise Affine Systems with Two Switching Planes

The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of the symmetry. An analytic proof is provided using the concrete expression forms of the analytic solution, stable manifold, and unstable manifold. Meanwhile, a sufficient condition for the existence of two homoclinic orbits is also obtained. Furthermore, two concrete piecewise affine asymmetric systems with two homoclinic orbits have been constructed successfully, demonstrating the method’s effectiveness.

Dynamics in a Predator–Prey Model with Cooperative Hunting and Allee Effect

This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate p=p#, by investigating stable and unstable manifolds of saddles. When p>p#, both species go extinct, and when p<p#, there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays.

Triple point of synchronization, phase singularity, and excitability along the transition between unbounded and bounded phase oscillations in a forced nonlinear oscillator.

We report the discovery of a codimension-two phenomenon in the phase diagram of a second-order self-sustained nonlinear oscillator subject to a constant external periodic forcing, around which three regimes associated with the synchronization phenomenon exist, namely phase-locking, frequency-locking without phase-locking, and frequency-unlocking states. The triple point of synchronization arises when a saddle-node homoclinic cycle collides with the zero-amplitude state of the forced oscillator. A line on the phase diagram where limit-cycle solutions contain a phase singularity departs from the triple point, giving rise to a codimension-one transition between the regimes of frequency unlocking and frequency locking without phase locking. At the parameter values where the critical transition occurs, the forced oscillator exhibits a separatrix with a π phase jump, i.e., a particular trajectory in phase space that separates two distinct behaviors of the phase dynamics. Close to the triple point, noise induces excitable pulses where the two variants of type-I excitability, i.e., pulses with and without 2π phase slips, appear stochastically. The impacts of weak noise and some other dynamical aspects associated with the transition induced by the singular phenomenon are also discussed.

Coexistence of singular cycles in a new kind of 3D non-smooth systems with two discontinuous boundaries

The aim here is to study complex dynamical behavior in a new kind of non-smooth systems with two discontinuous boundaries in space $$\mathbb {R}^3$$ . This paper provides an applicable method for accurately detecting coexistence of singular cycles, including (a) homoclinic and homoclinic cycles, (b) homoclinic and heteroclinic cycles, (c) heteroclinic and heteroclinic cycles. It is further proposed some criteria for guaranteeing three types of coexistence in the considered system. Moreover, existence conditions of chaos induced by such coexistence are established. Finally, three numerical examples are provided to verify the validity of theoretical results.

Complex dynamics of a SIRS epidemic model with the influence of hospital bed number

<p style='text-indent:20px;'>In this paper, the nonlinear dynamics of a SIRS epidemic model with vertical transmission rate of neonates, nonlinear incidence rate and nonlinear recovery rate are investigated. We focus on the influence of public available resources (especially the number of hospital beds) on disease control and transmission. The existence and stability of equilibria are analyzed with the basic reproduction number as the threshold value. The conditions for the existence of transcritical bifurcation, Hopf bifurcation, saddle-node bifurcation, backward bifurcation and the normal form of Bogdanov-Takens bifurcation are obtained. In particular, the coexistence of limit cycle and homoclinic cycle, and the coexistence of stable limit cycle and unstable limit cycle are also obtained. This study indicates that maintaining enough number of hospital beds is very crucial to the control of the infectious diseases no matter whether the immunity loss population are involved or not. Finally, numerical simulations are also given to illustrate the theoretical results.</p>

Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.