Homoclinic Orbits Research Articles

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Semiclassical perturbations of single-degree–of–freedom Hamiltonian systems I: Separatrix splitting

We study semiclassical perturbations of single-degree-of-freedom Hamiltonian systems possessing hyperbolic saddles with homoclinic orbits, and provide a sufficient condition for the separatrices to split, using a Melnikov-type approach. The semiclassical systems give approximations of the expectation values of the positions and momenta to the semiclassical Schrödinger equations with Gaussian wave packets as the initial conditions. The occurrence of separatrix splitting explains a mechanism for the existence of trajectories to cross the separatrices on the classical phase plane in the expectation value dynamics. Such separatrix splitting does not occur in standard systems of Hagedorn and Heller for the semiclassical Gaussian wave packet dynamics as well as in the classical systems. We illustrate our theory for the potential of a simple pendulum and give numerical computations for the stable and unstable manifolds in the semiclassical system as well as solutions crossing the separatrices.

Bifurcations and Homoclinic Orbits of a Model Consisting of Vegetation-Prey-Predator Populations.

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov-Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions.

Dynamical behaviors in perturbative longitudinal vibration of microresonators under the parallel-plate electrostatic force

The dynamic model for perturbative longitudinal vibration of microresonators subjected to the parallel-plate electrostatic force, which can be converted into a cubic oscillator with nonlinear polynomials, is established in this manuscript. The orbits and global dynamical behaviors of the cubic oscillator at full state are studied both analytically and numerically. The expressions of homoclinic orbits and subharmonic orbits are obtained analytically by solving the Hamilton system. The scenarios of phase portraits and equilibria are given. With the Melnikov method, the critical value of chaos arising from homoclinic intersections is derived analytically. The investigation yields intriguing dynamical phenomena, including the controllable frequencies that regulate the system without inducing chaos. The conditions for the occurrence of subharmonic bifurcations of integer order are presented with the subharmonic Melnikov method. Besides, the results indicate that the system does not undergo fractional order subharmonic bifurcation and it can reach a chaotic state through a finite number of integer order subharmonic bifurcations. On the basis of theoretical analysis, some numerical simulations including time histories, phase portraits, bifurcation diagrams, Poincaré cross-sections, Lyapunov exponential spectrums and basins of attractor are given, which are consistent with theoretical results.

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.

Snapback Repellers, Computational Chaos and Extreme Multistability in Discrete-Time Memristor Murali–Lakshmanan–Chua Circuit

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves.

New features of circular geodesics in Kalb–Ramond gravity: an autonomous dynamical system approach

Kalb–Ramond gravity is expected to show the signatures of the low energy limit of the Lorentz violating extension of the standard model of particle physics. To understand the theory further, we study the circular geodesics belonging to a class of exact solutions [found recently by Lessa, Silva, Maluf, Almeida (LSMA)] by posing them as a Hamiltonian dynamical autonomous system. The analysis on the phase space for relevant values of the Lorentz violating parameters reveals for the first time a wealth of novel information that include prediction of homoclinic orbits, saddle points, separatrices that are not available from the conventional analysis. Their image on the actual physical space is discussed.

Sliding homoclinic orbits and chaotic dynamics in a class of 3D piecewise-linear Filippov systems

Sliding homoclinic orbits and chaotic dynamics in a class of 3D piecewise-linear Filippov systems

Homoclinic behavior around a degenerate heteroclinic cycle in a Lorenz-like system

In this work, we analyze a degenerate heteroclinic cycle that appears in a Lorenz-like system when one of the involved equilibria changes from real saddle to saddle-focus. First, from a theoretical model based on the construction of a Poincaré return map, we demonstrate that an infinite number of homoclinic connections arise from the point of the parameter plane where the degenerate heteroclinic cycle appears. The subsequent numerical study not only illustrates the presence of the first homoclinic orbits in the infinite succession but also allows to find other important local and global organizing centers of codimension two (Bogdanov–Takens bifurcations, degenerate homoclinic and heteroclinic connections, T-points) and three (triple-zero bifurcation, doubly-degenerate heteroclinic cycles, degenerate T-points).

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.

Sliding Homoclinic Bifurcations in a Class of Three-Dimensional Piecewise Affine Systems

This paper studies sliding homoclinic bifurcations in a class of symmetric three-zone three-dimensional piecewise affine systems. The systems have one parameter and the unperturbed systems have a pair of sliding homoclinic orbits to a saddle. Based on the analysis of the one-dimensional Poincaré maps, two types of sliding cycles are obtained from the sliding homoclinic bifurcations of the systems. In addition, two examples of sliding homoclinic orbits and sliding cycles are provided with simulations to illustrate the effectiveness of the theorems.

Dynamics and Chaos of Convective Fluid Flow

In this paper, a mathematical model of fluid flows in a convective thermal system is developed, and a five-dimensional dynamical system is developed for the investigation of the convective fluid dynamics. The analytical solutions of periodic motions to chaos of the convective fluid flows are developed for steady-state vortex flows, and the corresponding stability and bifurcations of periodic motions in the five-dimensional dynamical system are studied. The harmonic frequency-amplitude characteristics for periodic flows are obtained, which provide energy distribution in the parameter space. Analytical homoclinic orbits for the convective fluid flow systems are developed for the asymptotic convection through the infinite-many homoclinic orbits in the five-dimensional dynamical system. The dynamics of fluid flows in the convective thermal systems are revealed, and one can use such methodology to predict atmospheric and oceanic phenomena through thermal convections.

Modeling the Nonmonotonic Immune Response in a Tumor–Immune System Interaction

Tumor–immune system interactions are very complicated, being highly nonlinear and not well understood. A large number of tumors can potentially weaken the immune system through various mechanisms such as secreting cytokines that suppress the immune response. In this paper, we propose a tumor–immune system interaction model with a nonmonotonic immune response function and adoptive cellular immunotherapy (ACI). The model has a tumor-free equilibrium and at most three tumor-presence equilibria (low, moderate and high ones). The stability of all equilibria is studied by analyzing their characteristic equations. The consideration of nonmonotonic immune response results in a series of bifurcations such as the saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. In addition, numerical simulation results show the coexistence of periodic orbits and homoclinic orbits. Interestingly, along with various bifurcations, we also found two bistable scenarios: the coexistence of a stable tumor-free as well as a high-tumor-presence equilibrium and the coexistence of a stable-low as well as a high-tumor-presence equilibrium, which can show symmetric and antisymmetric properties in a range of model parameters and initial cell concentrations. The new findings indicate that under ACI, patients can possibly reach either a stable tumor-free state or a low-tumor-presence state in the presence of nonmonotonic immune response once the immune system is activated.

Canard Cycles and Homoclinic Orbit of a Leslie–Gower Predator–Prey Model with Allee Effect and Holling Type II Functional Response

Canard Cycles and Homoclinic Orbit of a Leslie–Gower Predator–Prey Model with Allee Effect and Holling Type II Functional Response

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.

Bifurcation of the travelling wave solutions in a perturbed (1 + 1)-dimensional dispersive long wave equation via a geometric approach

Choosing ${\kappa }$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$ -dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa }\in \left (0,1-\frac {\sqrt 3}{3}\right )\cup \left (1+\frac {\sqrt 3}{3},2\right )$ , while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt {15}/3$ for $\kappa =0$ or $\kappa =2$ . The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.

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.

The stability of smooth solitary waves for the [formula omitted]-family of Camassa–Holm equations

The b-family of Camassa–Holm (b-CH) equation is a one-parameter family of PDEs, which includes the completely integrable Camassa–Holm and Degasperis–Procesi equations but possesses different Hamiltonian structures. Motivated by this, we study the existence and the orbital stability of the smooth solitary wave solutions with nonzero constant background to the b-CH equation for the special case b=1, whose the Hamiltonian structure is different from that of b≠1. We establish a connection between the stability criterion for the solitary waves and the monotonicity of a singular integral along the corresponding homoclinic orbits of the spatial ODEs. We verify the latter analytically using the framework for the monotonicity of period function of planar Hamiltonian systems, which shows that the smooth solitary waves are orbitally stable. In addition, we find that the existence and orbital stability results for 0<b<1 are similar to that of b>1, particularly the stability criteria are the same. Finally, combining with the results for the case b>1, we conclude that the solitary waves to the b-CH equation is structurally stable under the variation of the parameter b>0.