Heteroclinic Orbits Research Articles

Periodic orbits around 216-Kleopatra asteroid modelled by a dipole-segment

Asteroid 216-Kleopatra presents a ham-bone shape, that is to say, it is quite elongated body with two protuberances on its end-points, which makes of it the perfect candidate to model its shape and mass distribution by a Dipole-Segment, consisting on a massive segment with two spherical masses at the end-points of the rod and that is rotating about its center of mass with uniform velocity. Once obtained an approximation of the actual values of the involved parameters from recent results of Burov and coworkers, we obtain the complete map of families of symmetric periodic orbits, and describe the orbits of the different families, its parametric evolution, stability and also we compute the heteroclinic orbits connecting unstable isosceles equilibria.

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

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

Global bifurcation analysis and derivation of an optimal harvesting policy of a prey–predator model with Holling type-IV functional scheme

ABSTRACT This paper establishes a prey–predator model with a Holling type IV functional response and proportional harvesting in prey species and nonlinear harvesting in predator species. We provide a completely local and global bifurcation analysis of the model to characterize some meaningful results that may emerge from the interaction of biological resources. The proposed model exhibits various complex behaviors involving heteroclinic and homoclinic orbits, and we prove analytically that it also undergoes transcritical, Hopf, and Bogdanov–Takens bifurcations in the prey–predator plane. Moreover, the system is unfolded to compute the bifurcation curves, highlighting the critical dynamics by perturbing two bifurcation parameters near the cusp. Furthermore, from the economic point of view, the bionomic equilibrium of the system is studied, and optimal harvesting policy is derived using the Pontryagin’s maximum principle. Numerical simulations are presented to verify our analytical results.

A complete analysis of a slow–fast modified Leslie–Gower predator–prey system

In this paper, we transform a modified Leslie–Gower predator–prey system into the corresponding fast–slow version by assuming that prey reproduces much faster than predator, and then perform a complete analysis about its dynamics. More specifically, we find the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its (or their) location, and then we further fully determine its (or their) dynamics under explicit conditions. Besides, by converting the slow–fast system into its slow–fast normal form, we are able to characterize its rich dynamics completely, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits and global attraction of equilibrium. Moreover, the sufficient conditions to guarantee these various rich dynamics are explicitly given, including the explicit conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which generally cannot be explicitly derived in the existing literatures. Additionally, the cyclicity of diverse canard cycles is found under explicit conditions. Of particular interest is that we show the existence and uniqueness of one canard cycle without head whose cyclicity is at most two under explicit parameters conditions. Our results complement and enrich the previous work (relaxation oscillation, and global attraction of a boundary equilibrium) about this fast–slow system. We also employ numerical simulations to illustrate our theoretical analysis.

Dynamics of the epidemiological Predator-Prey system in advective environments.

This paper aims to establish the existence of traveling wave solutions connecting different equilibria for a spatial eco-epidemiological predator-prey system in advective environments. After applying the traveling wave coordinates, these solutions correspond to heteroclinic orbits in phase space. We investigate the existence of the traveling wave solution connecting from a boundary equilibrium to a co-existence equilibrium by using a shooting method. Different from the techniques introduced by Huang, we directly prove the convergence of the solution to a co-existence equilibrium by constructing a special bounded set. Furthermore, the Lyapunov-type function we constructed does not need the condition of bounded below. Our approach provides a different way to study the existence of traveling wave solutions about the co-existence equilibrium. The existence of traveling wave solutions between co-existence equilibria are proved by utilizing the qualitative theory and the geometric singular perturbation theory. Some other open questions of interest are also discussed in the paper.

Dynamics of a restricted (6 + 1)-vortex problem

This paper investigates the dynamics of a restricted (6 + 1)-vortex problem. The existence of nested equilateral triangle configurations depends on the position of two triangles, which can be divided into two situations: staggered arrangement and symmetrical arrangement. Firstly, we provide the existence conditions for the configuration and further discuss the stability of the configuration numerically. Secondly, we have established the global dynamics of the restricted point-vortex problem with configurations under different arrangements, including the dynamic behavior of equilibrium points, periodic orbits, homoclinic orbits, and heteroclinic orbits.

Bistable click mechanism for dipteran flight robot

The flight mechanism of the dipteran insects is commonly used to design the flapping robot. However, the nonlinear dynamics of the buckling click mechanism of the flapping robot with bistability still unclear. In this paper, a novel flapping robot model with the click mechanism proposed based on the bistability of snap-through buckling. The nonlinear governing equations of the flight robot are obtained by using the Euler–Lagrange equation. The nonlinear potential energy, moment of the elastic force, equilibrium bifurcation, as well as equilibrium stability are investigated to show the bistable characteristics. The transitional and persistent sets of bifurcation regions in the parameter plane and the corresponding phase portraits are obtained with single and double well behaviors. Then, the periodic response of the free flight are defined by the analytical solution of three kinds of elliptical functions, as well as the amplitude frequency responses are investigated by numerical integration. Based on the topological equivalent, the chaotic thresholds of the homo-clinic and heteroclinic orbits for the chaotic vibration of the perturbed robot system are derived. Finally, the prototype of nonlinear flapping robot is manufactured and the experimental system is setup. The nonlinear static moment of force curves, periodic response and dynamic flight vibration of dipteran robot system are carried out. It is shown that the test results are agree well with the theoretical analysis and numerical simulation. Those results have the potential application for the structure design of the efficient flight robot.

Globally Exponentially Attracting Sets and Heteroclinic Orbits Revealed

Motivated by the open problems on the global dynamics of the generalized four-dimensional Lorenz-like system, this paper deals with the existence of globally exponentially attracting sets and heteroclinic orbits by constructing a series of Lyapunov functions. Specifically, not only is a family of mathematical expressions of globally exponentially attracting sets derived, but the existence of a pair of orbits heteroclinic to S0 and S± is also proven with the aid of a Lyapunov function and the definitions of both the α-limit set and ω-limit set. Moreover, numerical examples are used to justify the theoretical analysis. Since the obtained results improve and complement the existing ones, they may provide support in chaos control, chaos synchronization, the Hausdorff and Lyapunov dimensions of strange attractors, etc.

Chaotic dynamics of granules-beam coupled vibration: Route and threshold

Although granular materials are the second most processed in industry after water, the theoretical study of granules-structure interactions is not as advanced as that of fluid-structure interactions due to the lack of a unified view of the constitutive relation of granular materials. In the previous work [1], the theoretical model of granules-beam coupled vibration was developed and verified by experiments. However, it was also found that the system exhibits significant stiffness-softening Duffing characteristics even under micro-vibration, which implies that chaotic responses may be reached under certain conditions. To reveal the route and critical conditions for the system to enter chaos, in the present work, the chaotic dynamics of the system are studied. In qualitative analysis, Melnikov method is applied to analyze the instability behavior of the perturbed heteroclinic orbit of the system, thus the critical condition for the system to enter Smale horseshoe chaos, i.e., the Melnikov criterion, is obtained. The validity of the criterion is verified numerically. In experimental studies, the existence of chaotic responses and the route to chaos for granules-beam coupled vibrations is revealed. The experimental results suggest that the system response first experiences symmetry-breaking then undergoes a complete period-doubling cascade, and finally enters single-scroll chaos. In addition, although the Additional Dynamic Load (ADL) generated by granular media is highly complicated, a general and simple evolution pattern of the chaos threshold is found by parametric experiments, which is also supported by the Melnikov criterion. In short, the chaotic dynamics of granules-beam coupled vibration is revealed, which is a contribution to the engineering vibration aspect. On the theoretical side, an impressive result lies in that for the first time, the Melnikov criterion of such fractional-order systems is obtained in a global and closed form, which provides, respectively, improvement suggestions and reference results for the existing and future research on chaotic dynamics of fractional-order systems, especially of the Duffing-type systems.

Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales–Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homoclinic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.

Revealing More Hidden Attractors from a New Sub-Quadratic Lorenz-Like System of Degree 6 5

In the sense that the descending powers of some certain variables may widen the range of parameters of self-excited and hidden attractors, this technical note proposes a new three-dimensional Lorenz-like system of degree [Formula: see text]. In contrast to the previously studied one of degree [Formula: see text], the newly reported one creates more hidden Lorenz-like attractors coexisting with the unstable origin and a pair of stable node-foci in a broader range of parameters, which confirms the generalization of the second part of the celebrated Hilbert’s 16th problem once more. In addition, some other dynamics, i.e. Hopf bifurcation, the generic and degenerate pitchfork bifurcation, invariant algebraic surface, first integral, singularly degenerate heteroclinic cycle with nearby chaotic attractor, ultimate bounded set and existence of a pair of heteroclinic orbits, are discussed.

On a Family of Hamilton–Poisson Jerk Systems

In this paper, we construct a family of Hamilton–Poisson jerk systems. We show that such a system has infinitely many Hamilton–Poisson realizations. In addition, we discuss the stability and we prove the existence of periodic orbits around nonlinearly stable equilibrium points. Particularly, we deduce conditions for the existence of homoclinic and heteroclinic orbits. We apply the obtained results to a family of anharmonic oscillators.

Applying Lin's method to constructing heteroclinic orbits near the heteroclinic chain

In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under ‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits and connected by three hyperbolic saddle points . Assume that the degeneracy of the unperturbed heteroclinic orbit is , the splitting index is . By applying Lin's method, we construct heteroclinic orbits connected and near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from to . Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices , and then obtain the existence of the heteroclinic orbits which backward asymptotic to and forward asymptotic to .

Memory Asymmetry Creates Heteroclinic Orbits to Nash Equilibrium in Learning in Zero-Sum Games

Learning in games considers how multiple agents maximize their own rewards through repeated games. Memory, an ability that an agent changes his/her action depending on the history of actions in previous games, is often introduced into learning to explore more clever strategies and discuss the decision-making of real agents like humans. However, such games with memory are hard to analyze because they exhibit complex phenomena like chaotic dynamics or divergence from Nash equilibrium. In particular, how asymmetry in memory capacities between agents affects learning in games is still unclear. In response, this study formulates a gradient ascent algorithm in games with asymmetry memory capacities. To obtain theoretical insights into learning dynamics, we first consider a simple case of zero-sum games. We observe complex behavior, where learning dynamics draw a heteroclinic connection from unstable fixed points to stable ones. Despite this complexity, we analyze learning dynamics and prove local convergence to these stable fixed points, i.e., the Nash equilibria. We identify the mechanism driving this convergence: an agent with a longer memory learns to exploit the other, which in turn endows the other's utility function with strict concavity. We further numerically observe such convergence in various initial strategies, action numbers, and memory lengths. This study reveals a novel phenomenon due to memory asymmetry, providing fundamental strides in learning in games and new insights into computing equilibria.

Heteroclinic Orbits for Singular Hamiltonian Systems

We are concerned with the existence of heteroclinic orbits for singular Hamiltonian systems of second order $\ddot{q}(t) + \nabla V(t, \,q)=0 $ where $V(t,\,q)$ is periodic in $t$ and has a singularity at a point ${q=e}$. Suppose $V$ possesses a global maximum $\overline V$ on $\mathbb R \times \mathbb R ^N\setminus\{e\}$ and $V(t,\,x)= \overline{V}$ if and only if $x\in \mathcal{M}$ where $\mathcal{M}$ contains at least two points and consists only of isolated points. Under these and suitable conditions on $V$ near $q=e$ and at infinity, we show for each $a_0^{}\in \mathcal M$, the existence of at least one heteroclinic orbit joining $a_0^{}$ to $\mathcal M \setminus\{a_0^{}\}$. Two different settings are studied. For the first, the usual strong force condition of Gordon near the singularity is assumed. For the second, the potential $V$ behaves near $q=e$ like $-\frac1{|q-e|^\alpha}$ with $0<\alpha<2$ (the weak force case). In both cases the existence of heteroclinic orbits $q\colon\mathbb R \to\mathbb R^N\setminus\{e\}$ is obtained via a minimization of the corresponding action functional.

Pseudo and true singularly degenerate heteroclinic cycles of a new 3D cubic Lorenz-like system

In contrast to the coexistence of infinitely many pseudo and true singularly degenerate heteroclinic cycles with nearby two-scroll hyperchaotic Lorenz-like attractors coined in four-dimensional systems, this note reports the ones with nearby bifurcated chaotic attractors in a new 3D cubic Lorenz-like system. Except for the rabbit head/boat shape two-scroll attractors, there exist globally exponentially asymptotically stable parabolic type equilibria and a pair of heteroclinic orbits. In addition, numerical simulations also validate the correctness of the theoretical analysis.

Analytic Solutions of 2D Cubic Quintic Complex Ginzburg-Landau Equation

The dynamical behaviour of traveling waves in a class of two-dimensional system whose amplitude obeys the two-dimensional complex cubic-quintic Ginzburg-Landau equation is deeply studied as a function of parameters near a subcritical bifurcation. Then, the bifurcation method is used to predict the nature of solutions of the considered wave equation. It is applied to reduce the two-dimensional complex cubic-quintic Ginzburg-Landau equation to the quintic nonlinear ordinary differential equation, easily solvable. Under some constraints of parameters, equilibrium points are obtained and phase portraits have been plotted. The particularity of these phase portraits obtained for new ordinary differential equation is the existence of homoclinic or heteroclinic orbits depending on the nature of equilibrium points. For some parameters, one has the orbits starting to one fixed point and passing through another fixed point before returning to the same fixed point, predicting then the existence of the combination of a pair of pulse-dark soliton. One has also for other parameters, the orbits linking three equilibrium points predicting the existence of a dark soliton pair. These results are very important and can predict the same solutions in many domains, particularly in wave phenomena, mechanical systems, or laterally heated fluid layers. Moreover, depending on the values of parameter systems, the analytical expression of the solutions predicted is found. The three-dimensional graphs of these solutions are plotted as well as their 2D plots in the propagation direction.

Arnold diffusion in Hamiltonian systems on infinite lattices

AbstractWe consider a system of infinitely many penduli on an ‐dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next‐to‐nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasi‐periodic tori with transverse heteroclinic orbits. We implement this approach in an infinite dimensional setting, both in the space of bounded ‐sequences and in spaces of decaying ‐sequences. Key steps in the proof are an invariant manifold theory for hyperbolic tori and a Lambda Lemma for infinite dimensional coupled map lattices with decaying interaction.

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.