Homoclinic Orbits Research Articles

Minimal Chaotic Networks of Linear Neuron-Like Elements with Single Rectification: Three Prototypes

Chaotic oscillations induced by single rectification in networks of linear neuron-like elements are examined on three prototype models: one nonautonomous system and two autonomous systems. The first is a system of coupled neurons with periodic input; the second is a system of three coupled neurons with six couplings; the third is a ring of four unidirectionally coupled neurons with one reverse coupling. In each system, the output function of one neuron is ramp and that of the others is linear. Each system is piecewise linear and the phase space is separated into two domains by a single border. Steady states, periodic solutions and homoclinic orbits are derived rigorously and their stability is evaluated with the eigenvalues of the Jacobian matrices. The bifurcation analysis of the three systems shows that chaotic attractors could be generated through cascades of period-doubling bifurcations of periodic solutions.

Melnikov-type method for chaos in a class of hybrid piecewise-smooth systems with impact and noise excitation under unilateral rigid constraint

The Melnikov method for chaos is developed in a class of hybrid piecewise-smooth systems with impact and noise excitation under unilateral rigid constraint, the corresponding unperturbed Hamiltonian system has the three-piecewise homoclinic orbit. This homoclinic orbit crosses the first and second switching manifolds continuously, then jumps symmetrically with x-axis on the third switching manifold through the rigid impact. It will not cross the third switching manifold and enter the next zone under unilateral rigid constraint. Especially, the perturbed trajectory crosses the first switching manifold through the elastic impact rule expressed by the reset map. Afterwards, the random non-smooth Melnikov function is obtained, the chaotic criteria with and without noise excitation are derived. Furthermore, the theoretical results obtained are applied to analyse the chaos in an actual non-smooth mechanical system with impacts, dry friction and noise excitation.

Bifurcations for counterintuitive post-inhibitory rebound spike related to absence epilepsy and Parkinson disease

Seizures are caused by increased neuronal firing activity resulting from reduced inhibitory effect and enhancement of inhibitory modulation to suppress this activity is used as a therapeutic tool. However, recent experiments have shown a counterintuitive phenomenon that inhibitory modulation does not suppress but elicit post-inhibitory rebound (PIR) spike along with seizure to challenge the therapeutic tool. The nonlinear mechanism to avoid the PIR spike can present theoretical guidance to seizure treatment. This paper focuses on identifying credible bifurcations that underlie PIR spike by modulating multiple parameters in multiple theoretical models. The study identifies a codimension-2 bifurcation called saddle--node homoclinic orbit (SNHOB), which is an intersection between saddle node bifurcation on invariant cycle (SNIC) and other two bifurcations. PIR spike cannot be evoked for the SNIC far from the SNHOB but induced for the SNIC close to the SNHOB, which extends the bifurcation condition for PIR spike from the well-known Hopf to SNIC. Especially, in a thalamic neuron model, increases of conductance of T-type Ca2+ (T Ca) channel induce SNIC bifurcation approaching to the SNHOB to elicit PIR spikes, closely matching experimental results of the absence seizure or Parkinson diseases. Such results imply that, when inhibition is employed to relieve absence seizure and Parkinson diseases related to PIR spike, modulating SNIC to get far from the SNHOB to avoid PIR spike is the principle. The study also addresses the complex roles of T Ca current and comprehensive relationships between PIR spike and nonlinear conceptions such as bifurcation types and shapes of threshold curve.

Complex dynamics of the simplest neuron model: Singular chaotic Shilnikov attractor as specific oscillatory neuron activity

We study complex dynamics of the Chialvo model that is the simplest neuron-type model in form of a four-parameter family of two-dimensional noninvertible maps (endomorphisms). Main elements of bifurcation diagram in the plane of two parameters have been constructed in which regions corresponding to both quasi-periodic and chaotic oscillations are selected. We also indicate special regions corresponding to singular discrete chaotic Shilnikov attractors that we consider as a new type of the so-called snap-back repellers (over an unstable focus). The study of time series was carried out in which there were classified patterns of specific oscillatory activities in the cases when homoclinic orbits to the unstable focus exists and, when such orbits were not yet formed but a strange attractor already exists. New dynamical characteristics are proposed, with the help of which it is possible to assess the level of distinctness of atypical oscillatory activity.

Homoclinic orbit and the violation of the chaos bound around a black hole with anisotropic matter fields

We study the homoclinic orbit and the violation of chaos bound, which are obtained by particle motions around a black hole that coexist with anisotropic matter fields. The homoclinic one is associated with an unstable local maximum of the effective potential. By perturbing a particle located slightly away from the homoclinic one, we numerically compute Lyapunov exponents indicating the sensitivity of the initial value. Our results demonstrate that the violation of the chaos bound increases with higher angular momentum, and the anisotropic matter gives rise to violating the chaos bound further, even in the case of the nonextremal black hole. We utilize the Hamiltonian-Jacobi formalism to explicitly illustrate how the geodesic motion of a particle can be integrable in the procedure of obtaining our findings.

Period-1 Motions to Twin Spiral Homoclinic Orbits in the Rössler System

Abstract In this paper, period-1 motions to twin spiral homoclinic orbits in the Rössler system are presented. The period-1 motions varying with a system parameter are predicted semi-analytically through an implicit mapping method, and the corresponding stability and bifurcations of the period-1 motions are determined through eigenvalue analysis. The approximate homoclinic orbits are obtained, which can be detected through the periodic motions with the positive and negative infinite large eigenvalues. The two limit ends of the bifurcation diagram of the period-1 motion are at twin spiral homoclinic orbits. For comparison, numerical and analytical results of stable period-1 motion are presented. The approximate spiral homoclinic orbits are demonstrated for a better understanding of complex dynamics of homoclinic orbits. Herein, only initial results on periodic motions to homoclinic orbits are presented for the Rössler system. In fact, the Rössler system has rich complex dynamics existing in other high-dimensional nonlinear systems. Thus, the further studies of bifurcation trees of periodic motions to infinite homoclinic orbits will be completed in sequel.

A new vibro-impact bistable oscillator with an adjustable rigid wall

In this paper, a new vibro-impact bistable oscillator under vicious damping and external harmonic excitation is proposed by adding an adjustable rigid wall to one side of the archetypal smooth and discontinuous (SD) oscillator and an impact mapping is given to describe energy loss during collisions. In the unperturbed ideal case, there exist rich and perfect geometrical structure including grazing phenomena such as periodic orbits and homoclinic orbits tangent to the wall due to its adjustability as a constraint. Perturbed dynamics is studied to show the influence of the constraint on the original SD oscillator through the Lyapunov exponent diagrams, bifurcation diagrams, and the corresponding phase diagrams and attractors. Some new non-smooth chaotic attractors and periodic orbits are found to have grazing or transversal interaction with the unilateral rigid wall. Due to the effect of the introduced impact mapping and different locations of the rigid wall, the finger-like chaotic attractors of the SD oscillator will be suppressed and changed into different non-smooth periodic motions by varying the non-dimensional parameter which reflects the degree of the springs compressed, while the periodic motions of the SD oscillator will become new non-smooth chaotic attractors or different non-smooth periodic motions.

Out-of-plane chaotic motion and suppression for tethered tug-debris systems with thrust perturbation

This paper investigates the characterization and suppression of chaotic motion in an out-of-plane tethered tug-debris system that is towed by a constant thrust with time-varying micro-perturbations at a fixed non-zero in-plane libration angle. These perturbations arise due to imperfections in the thruster’s mechanical system on the tug. Static equilibrium solutions and homoclinic orbits of the corresponding Hamiltonian system are derived, and a subcritical fork bifurcation is identified and presented. By the Melnikov method, it is found that the micro-fluctuation in the thrust could induce chaos in the system near unstable saddle points. Then, a chaotic parameter region is produced to identify chaos efficiently. The pure out-of-plane chaotic motion would appear, provided system parameters are in the region. A sliding mode controller is designed to suppress the chaotic oscillation by deploying and retrieving the tether. Finally, the impact of the system parameters on the chaotic region is estimated numerically. Simulation results demonstrate that chaotic motions exist and can be identified efficiently by the chaotic parameter region and that the control method is effective in suppressing the chaotic motion.

An Application of the Melnikov Method to a Piecewise Oscillator

In this paper we present a new application of the Melnikov method to a class of periodically perturbed Duffing equations where the nonlinearity is non-smooth as otherwise required in the classical applications. Extensions of the Melnikov method to these situations is a topic with growing interests from the researchers in the past decade. Our model, motivated by the study of mechanical vibrations for systems with ''stops'', considers a case of a nonlinear equation with piecewise linear components. This allows us to provide a precise analytical representation of the homoclinic orbit for the associated autonomous planar system and thus obtain simply computable conditions for the zeros of the associated Melnikov function.

Existence of Running Solutions in a Relativistic Tricomi’s Equation Using Perturbation Theory

We use perturbation methods to establish the existence of a second kind periodic solution (running solution) of a nonlinear Tricomi’s equation type under relativistic effects. First, we estimate conditions for the existence of either an equilibrium point or a second-kind periodic solution through the average method, where we assumed the nonlinear part as a positive perturbation. Then, we use the Melnikov function to estimate conditions for the existence of running solutions, considering the persistence of the homoclinic orbits associated with the conservative equation.

Stability and chaotic dynamic analysis of Li-doped fullerene-IRMOF composite materials for hydrogen storage

Isoreticular metal organic framework (IRMOF) material is a type of ideal hydrogen storage material for its porosity and large surface area. Lithium doping and fullerene impregnation can enhance the hydrogen storage capacity of IRMOF efficiently. In this paper, the nonlinear dynamic characteristics of Li-doped fullerene-IRMOF composite material are qualitatively discussed in theory, which determine its thermal stability. A type of Li-doped fullerene-IRMOF composite material is proposed for hydrogen storage, and a new constitutive model is proposed to describe the hysteretic property of IRMOF. The system's nonlinear dynamic model is developed, and the fast-slow dynamic analysis method is introduced to interpret not only the system's jumping phenomena between the multi-pulse homoclinic and heteroclinic orbits but also its chaotic dynamics. The nonlinear dynamic responses of the system under deterministic and stochastic excitation are obtained respectively, and symmetrical and asymmetric orbital fusion phenomena have been discovered. The results of theoretical analysis and numerical simulation both show that the system has novel nonlinear dynamic characteristics such as double periodic motion, multi-pulse jumping orbits, and chaos; multi-pulse jumping phenomena are caused by the stiffness difference between fullerene and IRMOF, which form a fast-slow system; symmetrical fusion and asymmetrical fusion of system's orbits occur under different parameters. Safe, efficient and rapid hydrogen storage and release can be achieved through controlling the system's parameters to adjust the dynamic behavior of the system, which will be a development direction in the application of hydrogen energy.

The 0:1 resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcation

We consider the non-semisimple 0:1 resonance (i.e., the unperturbed equilibrium has two purely imaginary eigenvalues ( and ) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), this bifurcation essentially triggered by the reversible universal unfolding with respect to BCKV-restricted morphisms of the planar non-semisimple singularity (the is regarded as distinguished parameter with respect to the external parameter λ). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the -symmetry generated by the additional pair of imaginary eigenvalues to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g., 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by Broer et al. (Z Angew Math Phys 44: 389-432, 1993). An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation.

Bifurcation of backward and forward solitary waves in helicoidal Peyrard–Bishop–Dauxois model of DNA

In this work, we investigate the bifurcation of backward and forward solitary waves in helicoidal Peyrard–Bishop–Dauxois model of deoxyribonucleic acid (DNA). We start by showing that this model can adopt right-handed or left-handed behavior depending on the wave frequency that it propagates. Then, using the semi-discrete approximation, we show that the dynamics of modulated waves in the network is governed by a quintic nonlinear Schrödinger (QNLS). Applying the phase imprint technique, this QNLS equation is first reduced into a derivative QNLS, and then to a cubic–quintic Duffing oscillator equation. Based on the phase plane analysis, we present all phase portraits of the dynamical system. The obtained results show a number of new phase portraits which cannot exist without the phase imprint approach. The exact representations of the nonlinear localized waves corresponding to the homoclinic and heteroclinic orbits in the phase portrait of the dynamical system are given. These waves include bright soliton, kink and anti-kink solitons, dark soliton and grey solitons. In addition, the effects of phase imprint parameter on the wave-shape profile of these solutions are investigated. We find out that the phase imprint parameter considerably affects the amplitude and the width of each of the above enumerated solitary waves, while the phase imprint method leads to new type solutions.

Relative prevalence-based dispersal in an epidemic patch model.

In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: [Formula: see text] and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov-Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates [Formula: see text] (due to single contact) and [Formula: see text] (due to double exposures). In a connected environment, we establish a threshold [Formula: see text] between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when [Formula: see text] and patch 1 has a lower infection rate, our results indicate: (i) [Formula: see text] can be nonmonotonic in dispersal rates and [Formula: see text] ([Formula: see text] is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When [Formula: see text] and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.

The quantitative analysis of homoclinic orbits from quadratic isochronous systems

In this paper, we investigate a pair of symmetric homoclinic orbits given by a Perturbation-Incremental (PI) method for the quadratic isochronous systems, which can bifurcate from the periodic orbits of the quadratic isochronous centers. The existing researches on the isochronous system are mostly from a qualitative perspective, while we have studied the quadratic isochronous system through the quantitative analysis method. As a quantitative analysis method, the PI method can not only determine the position of the limit cycle and the homoclinic (heteroclinic) orbit, but also provide an analytical expression of the asymptotic solution. At the same time, this method combines the significant characteristics of the perturbation method and the parameter incremental method. By utilizing the perturbation method, we obtain the analytic expressions of the zero-order perturbation solutions of homoclinic orbits when the parameters of systems are small. Then, the approximate analytical expressions of homoclinic orbits can be obtained by using the parameter incremental method when they are iterated from small parameters to large parameters. The numerical results will be compared with those from the Runge–Kutta method. The theoretical results agree very well with numerical simulation.

Dynamics of the restricted vortex problem with a honeycomb configuration

In this paper, we have investigated the restricted 14-vortex problem with a honeycomb configuration similar to the football surface pattern. First, we give a sufficient condition for the existence of honeycomb configurations and obtain the instability of configurations. Then, we establish the equation of motion of the tracer particle for the restricted 14-vortex problem and analyze the stability and distribution of equilibrium points and singular points. As can be seen from the global phase diagram of system, there are only four types of orbits: equilibrium points, homoclinic orbits, heteroclinic orbits, and periodic orbits.

Sliding homoclinic orbits and bifurcations of three-dimensional piecewise affine systems

Sliding homoclinic orbits and bifurcations of three-dimensional piecewise affine systems

Global dynamics for impacting cantilever beam supported by oblique springs

The chaos and subharmonic bifurcation of a cantilever beam supported by oblique springs under bilateral asymmetric rigid constraints are investigated in this paper. It is difficult to investigate analytically the chaos and subharmonic bifurcation of the system because the stiffness term of the oblique spring support structure is a transcendental function. Firstly, the stiffness term of the system is fitted by the approximation method, and the homoclinic orbit and its internal orbits of the approximate system are compared with the orbits of the original system. Secondly, the threshold conditions for subharmonic bifurcation and homoclinic chaos are presented by applying the Melnikov method to the non-smooth impacting cantilever beam system. Moreover, the stability of impacting orbits is analyzed by combining characteristic multipliers of smooth manifolds with impact function, and the relationship between subharmonic bifurcation and chaos is investigated. Finally, the effects of damping, excitation frequency, excitation amplitude and impact coefficient of restitution on chaos and subharmonic bifurcation are investigated based on threshold conditions, which further verifies the theoretical analysis.

Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback.

We study the dynamics and bifurcations of temporal dissipative solitons in an excitable system under time-delayed feedback. As a prototypical model displaying different types of excitability, we use the Morris-Lecar model. In the limit of large delay, soliton like solutions of delay-differential equations can be treated as homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of pulse solutions and to explain their dependence on the system parameters. In particular, we show how a homoclinic orbit flip of a single-pulse soliton leads to the destabilization of equidistant multi-pulse solutions and to the emergence of stable pulse packages. It turns out that this transition is induced by a heteroclinic orbit flip in the system without feedback, which is related to the excitability properties of the Morris-Lecar model.

Single- and Double-Peak Solitary Waves of Two-Component Drinfel’d–Sokolov–Wilson System with Kuramoto–Sivashinsky Perturbation

Solitary wave solutions of two-component Drinfel’d–Sokolov–Wilson system with Kuramoto–Sivashinsky perturbation are considered. We first employ geometric singular perturbation theory to reduce the higher-dimensional system of equations to the perturbed planar system. We then further exploit the Melnikov method to explore the persistence of one homoclinic orbit, and the generation of a new homoclinic orbit, indicating the existence of single- and double-peak solitary waves. Of particular interest is the appearance of the double-peak solitary wave solution. Finally, we include the numerical simulations to verify the theoretical results.