Nine Limit Cycles in a 5-Degree Polynomials Liénard System

In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five.

Habitat loss causes long transients in small trophic chains

Transients in ecology are extremely important since they determine how equilibria are approached. The debate on the dynamic stability of ecosystems has been largely focused on equilibrium states. However, since ecosystems are constantly changing due to climate conditions or to perturbations such as the climate crisis or anthropogenic actions (habitat destruction, deforestation, or defaunation), it is important to study how dynamics can proceed till equilibria. In this contribution we investigate dynamics and transient phenomena in small food chains using mathematical models. We are interested in the impact of habitat loss in ecosystems with vegetation undergoing facilitation. We provide a thorough dynamical study of a small food chain system given by three trophic levels: vegetation, herbivores, and predators. The dynamics of the vegetation alone suffers a saddle-node bifurcation, causing extremely long transients. The addition of a herbivore introduces a remarkable number of new phenomena. Specifically, we show that, apart from the saddle node involving the extinction of the full system, a transcritical and a supercritical Hopf-Andronov bifurcation allow for the coexistence of vegetation and herbivores via non-oscillatory and oscillatory dynamics, respectively. Furthermore, a global transition given by a heteroclinic bifurcation is also shown to cause a full extinction. The addition of a predator species to the previous systems introduces further complexity and dynamics, also allowing for the coupling of different transient phenomena such as ghost transients and transient oscillations after the heteroclinic bifurcation. Our study shows how the increase of ecological complexity via addition of new trophic levels and their associated nonlinear interactions may modify dynamics, bifurcations, and transient phenomena.

Period-Increasing Indicates Loss of Stability

We first present a mathematical theory on period-increasing as an indicator of stability loss in a dynamical system of any dimension not less than two, and show that in two-dimensional autonomous ODEs, the period-increasing is closely associated with homoclinic or heteroclinic bifurcation. Then we demonstrate that the proposed period-increasing regime shift can occur in some ecological models.

Approximations of the Heteroclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two. Application to a Liénard Equation

A dynamical system possessing an equilibrium point with two zero eigenvalues is considered. We assume that a degenerate Bogdanov–Takens bifurcation with symmetry of order two is present and, in the parameter space, a curve of heteroclinic bifurcation values emerges from the codimension two bifurcation point. Using a blow-up transformation and a perturbation method, we obtain second order approximations both for the heteroclinic orbits and for the curve of heteroclinic bifurcation values. Applications of our results for the truncated normal form and for a Liénard equation are presented. Some numerical simulations illustrating the accuracy of our results are performed.

Trans-heteroclinic bifurcation: a novel type of catastrophic shift

Global and local bifurcations are extremely important since they govern the transitions between different qualitative regimes in dynamical systems. These transitions or tipping points, which are ubiquitous in nature, can be smooth or catastrophic. Smooth transitions involve a continuous change in the steady state of the system until the bifurcation value is crossed, giving place to a second-order phase transition. Catastrophic transitions involve a discontinuity of the steady state at the bifurcation value, giving place to first-order phase transitions. Examples of catastrophic shifts can be found in ecosystems, climate, economic or social systems. Here we report a new type of global bifurcation responsible for a catastrophic shift. This bifurcation, identified in a family of quasi-species equations and named as trans-heteroclinic bifurcation , involves an exchange of stability between two distant and heteroclinically connected fixed points. Since the two fixed points interchange the stability without colliding, a catastrophic shift takes place. We provide an exhaustive description of this new bifurcation, also detailing the structure of the replication–mutation matrix of the quasi-species equation giving place to this bifurcation. A perturbation analysis is provided around the bifurcation value. At this value the heteroclinic connection is replaced by a line of fixed points in the quasi-species model. But it is shown that, if the replication–mutation matrix satisfies suitable conditions, then, under a small perturbation, the exchange of heteroclinic connections is preserved, except on a tiny range around the bifurcation value whose size is of the order of magnitude of the perturbation. The results presented here can help to understand better novel mechanisms behind catastrophic shifts and contribute to a finer identification of such transitions in theoretical models in evolutionary biology and other dynamical systems.

Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback

The heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback are studied by Melnikov method. The Melnikov function is analytically established to detect the necessary conditions for generating chaos. Through the analysis of the analytical necessary conditions, we find that the influences of the delayed displacement feedback and delayed velocity feedback are separable. Then the influences of the displacement and velocity feedback parameters on heteroclinic bifurcation and threshold value of chaotic motion are investigated individually. In order to verify the correctness of the analytical conditions, the Duffing oscillator is also investigated by numerical iterative method. The bifurcation curves and the largest Lyapunov exponents are provided and compared. From the analysis of the numerical simulation results, it could be found that two types of period-doubling bifurcations occur in the Duffing oscillator, so that there are two paths leading to the chaos in this oscillator. The typical dynamical responses, including time histories, phase portraits, and Poincare maps, are all carried out to verify the conclusions. The results reveal some new phenomena, which is useful to design or control this kind of system.