Heteroclinic Orbits Research Articles

Dynamics in diffusive Leslie–Gower prey–predator model with weak diffusion

This paper is concerned with the diffusive Leslie–Gower prey–predator model with weak diffusion. Assuming that the diffusion rates of prey and predator are sufficiently small and the natural growth rate of prey is much greater than that of predators, the diffusive Leslie–Gower prey–predator model is a singularly perturbed problem. Using travelling wave transformation, we firstly transform our problem into a multiscale slow-fast system with two small parameters. We prove the existence of heteroclinic orbit, canard explosion phenomenon and relaxation oscillation cycle for the slow-fast system by applying the geometric singular perturbation theory. Thus, we get the existence of travelling waves and periodic solutions of the original reaction–diffusion model. Furthermore, we also give some numerical examples to illustrate our theoretical results.

Melnikov-type method for a class of planar hybrid piecewise-smooth systems with impulsive effect and noise excitation: Heteroclinic orbits.

The classical Melnikov method for heteroclinic orbits is extended theoretically to a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation. We assume that the unperturbed system is a piecewise Hamiltonian system with a pair of heteroclinic orbits. The heteroclinic orbit transversally jumps across the first switching manifold by an impulsive effect and crosses the second switching manifold continuously. In effect, the trajectory of the corresponding perturbed system crosses the second switching manifold by applying the reset map describing the impact rule instantaneously. The random Melnikov process of such systems is then derived by measuring the distance of perturbed stable and unstable manifolds, and the criteria for the onset of chaos with or without noise excitation is established. In this derivation process, we overcome the difficulty that the derivation method of the corresponding homoclinic case cannot be directly used due to the difference between the symmetry of the homoclinic orbit and the asymmetry of the heteroclinic orbit. Finally, we investigate the complicated dynamics of a particular piecewise-smooth system with and without noise excitation under the reset maps, impulsive effect, and non-autonomous periodic and damping perturbations by this new extended method and numerical simulations. The numerical results verify the correctness of the theoretical results and demonstrate that this extended method is simple and effective for studying the dynamics of such systems.

A Cubic Memristive System with Two Twin Rössler-Type Chaotic Attractors Symmetrical About an Invariant Plane

Memristive circuits and systems have been widely studied in the last years due to their potential applications in several technological areas. They are capable of producing nonlinear periodic and chaotic oscillations, due to their locally-active characteristics. In this paper, we consider a cubic four-parameter differential system which models a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and a locally-active current-controlled generic memristor. This system has a saddle-focus equilibrium point at the origin, whose global stable and unstable manifolds are, respectively, the [Formula: see text]-axis and the plane [Formula: see text], which are invariant sets where the dynamic is linear. We show that this structure can generate two twin Rössler-type chaotic attractors symmetrical with respect to the plane [Formula: see text]. We describe the mechanism of creation of these chaotic attractors, showing that, although being similar to the Rössler attractor, the twin attractors presented here have simpler structural mechanism of formation, since the system has no homoclinic or heteroclinic orbits to the saddle-focus, as presented by the Rössler system. The studied memristive system has the rare property of having chaotic dynamics and an invariant plane with linear dynamic, which is quite different from other chaotic systems presented in the literature that have invariant surfaces filled by equilibrium points. We also present and discuss the electronic circuit implementation of the considered system and study its dynamics at infinity, via the Poincaré compactification, showing that all the solutions, except the ones contained in the plane [Formula: see text], are bounded and cannot escape to infinity.

Mechanism Analysis of Hidden Chaos in a Generalized Vijayakumar System

In order to further discover the hidden chaotic attractor and its generating mechanism in the Vijayakumar system, we give a generalized system showing hidden chaotic attractors which are not from homoclinic orbit or heteroclinic orbit and consider Hopf bifurcations (codimension one and two) by first Lyapunov coefficient and second Lyapunov coefficient. The existence of periodic orbits is strictly proved theoretically. We have considered the problem of Hopf bifurcation in the chaotic system with hidden attractors, which will be helpful to reveal the intrinsic relationship between the local stability of equilibria and global complex dynamical behaviors of the chaotic system. Finally, numerical simulations are obtained for showing the correctness of theoretical results.

Nonlinear dynamical analysis of drift ion acoustic shock waves in Electron-Positron-Ion plasma with adiabatic trapping

Propagation of nonlinear coupled drift ion acoustic shock wave is investigated in an electron–positron-ion (epi) plasma in the presence of fully degenerate adiabatically trapped electrons. A novel complex nonlinear equation with fractional nonlinear terms is formulated. Phase plane theory of planar dynamical system is employed to investigate this nonlinear equation of epi plasma. The variations in nonlinear periodic orbits (NPO), nonlinear heteroclinic orbits (NHO), time series plots and shock profiles are demonstrated by varying the values of different controlling parameters such as the ratio of positron to electron number density, collisional frequency, magnetic field, drift velocity and angle of propagation. It is seen that a pair of shock structures are obtained – which is the most significant result of this work.

Existence of Traveling Wave Fronts for a Generalized Nonlinear Schrodinger Equation

In the presented paper, a generalized nonlinear Schr o dinger equation without delay convolution kernel and with special delay convolution kernel is investigated. By using the geometric singular perturbation theory, the existence of traveling wave fronts is proved. Firstly, we show that such traveling wave fronts exist without delay by non-Hamiltonian qualitative analysis. Then, for the generalized nonlinear Schr o dinger equation with a special local strong delay convolution kernel, the desired heteroclinic orbit is obtained by using the Fredholm theory.

Sturm attractors for fully nonlinear parabolic equations

We explicitly construct global attractors of fully nonlinear parabolic equations in one spatial dimension. These attractors are decomposed as equilibria (time independent solutions) and heteroclinic orbits (solutions that converge to distinct equilibria backwards and forwards in time). In particular, we state necessary and sufficient conditions for the occurrence of heteroclinics between hyperbolic equilibria, which is accompanied by a method that computes such conditions.

Traveling pulses with oscillatory tails, figure-eight-like stack of isolas, and dynamics in heterogeneous media

The interplay between 1D traveling pulses with oscillatory tails (TPOs) and heterogeneities of the bump type is studied to formulate a generalized three-component FitzHugh–Nagumo system of equations. We present that stationary pulses with oscillatory tails (SPOs) form a “snaky” structure in homogeneous spaces and then TPO branches form a “figure-eight-like stack of isolas” located adjacent to the snaky structure of the SPO. Here, we adopt input resources such as the voltage-difference as a bifurcation parameter. A drift bifurcation from the SPO to TPO is observed by introducing another parameter at which these two solution sheets merge. In contrast to the monotone tail case, in the heterogeneous problem, a nonlocal interaction appears between the TPO and the heterogeneity, creating infinitely many saddle solutions and finitely many stable stationary solutions distributed across the line. The response of the TPO shows various dynamics including pinning and depinning processes along with penetration (PEN) and rebound (REB). The stable/unstable manifolds of these saddles interact with the TPO in a complex manner, creating a subtle dependence on the initial condition, and a difficulty to predict the behavior after collision even in 1D space. For the 1D case, a systematic global exploration of solution branches induced by heterogeneities (heterogeneity-induced-ordered patterns; HIOPs) reveals that HIOPs contain all the asymptotic states after collision to facilitate the prediction of the solution results without solving the PDEs. The reduction method of finite-dimensional system enables the clarification of the detailed mechanism of the transitions from PEN to pinning and pinning to REB based on a dynamical system perspective. Consequently, the basin boundary between two distinct outputs against the heterogeneities yields infinitely many successive reconnections of heteroclinic orbits among those saddles when the strength of heterogeneity increases. This causes the aforementioned subtle dependence of initial condition.

Slow localized patterns in singularly perturbed two-component reaction–diffusion equations

Localized patterns in singularly perturbed reaction–diffusion equations typically consist of slow parts, in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system, alternated by fast excursions, in which the solution jumps from one slow manifold to another, or back to the original slow manifold. In this paper we consider the existence and stability of stationary and travelling localized patterns that do not exhibit such jumps, i.e. that are completely embedded in a slow manifold of the singularly perturbed spatial dynamical system. These ‘slow patterns’ have rarely been considered in the literature, for two reasons: (i) in the classical Gray–Scott/Gierer–Meinhardt type models that dominate the literature, the flow on the slow manifold is typically linear and thus cannot exhibit homoclinic pulse or heteroclinic front solutions; (ii) the slow manifolds occurring in the literature are typically ‘vertical’, i.e. given by u ≡ u 0, where u is the fast variable, so that the stability problem is determined by a simple (decoupled) scalar equation. The present research concerns a general system of singularly perturbed reaction–diffusion equations and is motivated by several explicit ecosystem models that do give rise to non-vertical normally hyperbolic slow manifolds on which the flow may exhibit both homoclinic and heteroclinic orbits that correspond to either stationary or travelling localized slow patterns. The associated spectral stability problems are at leading order given by a nonlinear, but scalar, eigenvalue problem with Sturm–Liouville characteristics and we establish that homoclinic pulse patterns are typically unstable, while heteroclinic fronts can either be stable or unstable. However, we also show that homoclinic pulse patterns that are asymptotically close to a heteroclinic cycle may be stable. This result is obtained by explicitly determining the leading order approximations of four critical asymptotically small eigenvalues. By this analysis, that involves several orders of magnitude in the small parameter, we also obtain full control over the nature of the bifurcations—saddle-node, Hopf, global, etc—that determine the existence and stability of the (stationary and/or travelling) heteroclinic fronts and/or homoclinic pulses. Finally, we show that heteroclinic orbits may correspond to stable (slow) interfaces in two-dimensional space, while the homoclinic pulses must be unstable as localized stripes, even when they are stable in one space dimension.

Normal forms of double Hopf bifurcation for a reaction-diffusion system with delay and nonlocal spatial average and applications

In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained.

Complex dynamics of perturbed solitary waves in a nonlinear saturable medium: A Melnikov approach

Objective:To investigate the nonlinear dynamics of a periodically perturbed second-order ordinary differential equation obtained by using traveling wave variables in the model of pulse propagation in a nonlinear medium with saturation. Method:The Melnikov function of the investigated system along its homoclinic and heteroclinic orbits is constructed. It is established that the necessary condition for the occurrence of Melnikov chaos is always met. By analogy with the well-known Duffing equation, a damping term is added to the system to control chaos. Using the numerical calculation of the Melnikov integrals, conditions are found on the parameters of the new system for which the Melnikov chaos takes place. To verify the results obtained by the Melnikov method, attraction basins of the system are constructed. Results:The results obtained by the Melnikov method go in agreement with the structure of the constructed basin boundaries.

Traveling waves of an FKPP-type model for self-organized growth

We consider a reaction–diffusion system of densities of two types of particles, introduced by Hannezo et al. (Cell 171(1):242–255.e27, 2017). It is a simple model for a growth process: active, branching particles form the growing boundary layer of an otherwise static tissue, represented by inactive particles. The active particles diffuse, branch and become irreversibly inactive upon collision with a particle of arbitrary type. In absence of active particles, this system is in a steady state, without any a priori restriction on the amount of remaining inactive particles. Thus, while related to the well-studied FKPP-equation, this system features a game-changing continuum of steady state solutions, where each corresponds to a possible outcome of the growth process. However, simulations indicate that this system self-organizes: traveling fronts with fixed shape arise under a wide range of initial data. In the present work, we describe all positive and bounded traveling wave solutions, and obtain necessary and sufficient conditions for their existence. We find a surprisingly simple symmetry in the pairs of steady states which are joined via heteroclinic wave orbits. Our approach is constructive: we first prove the existence of almost constant solutions and then extend our results via a continuity argument along the continuum of limiting points.

Bifurcations in the diffusive Bazykin model

Dangerous tipping points and catastrophic transitions in ecosystems have recently been popular for detecting early warning signals in ecology. B-tipping is induced by bifurcation such as spatial pattern formation resulting from Turing instability. As one of the most important models in predator-prey interactions, we extend the Bazykin model to incorporate diffusive movement under homogeneous Neumann boundary conditions. For the local model, we provide some preliminary analysis on stability and Hopf bifurcation. For the reaction-diffusion model, we first improve some sufficient conditions for the local and global stability of a semi-trivial constant steady state or a unique positive constant steady state in Du and Lou (2001) [11]. Next we obtain the sufficient and necessary conditions for Turing instability, show the existence of Turing bifurcation, Hopf bifurcation, Turing-Turing bifurcation, Turing-Hopf bifurcation and Turing-Turing-Hopf bifurcation, and the nonexistence of triple-Turing bifurcation. Our results reveal that the model can exhibit complex spatial, temporal and spatiotemporal patterns, including complex regime shifts and critical transitions at bifurcation points, transient states (spatially inhomogeneous periodic solutions), tristability (a pair of non-constant steady states and a spatially homogeneous periodic solution), heteroclinic orbits (connecting a spatially inhomogeneous periodic solution to a non-constant steady state or a spatially homogeneous periodic solution, connecting a spatially homogeneous periodic solution to non-constant steady states and vice versa). Finally, numerical simulations illustrate complex dynamics and verify our theoretical results.

Canard Cycles and Relaxation Oscillations in a Singularly Perturbed Leslie–Gower Predator–Prey Model with Allee Effect

In this paper, we investigate the dynamics of a modified Leslie–Gower predator–prey model with Allee effect on prey. And the Holling type II functional response is considered in this model. When prey reproduces much faster than predator, by combining the normal form theory of slow–fast systems and the geometric singular perturbation theory, we observe much richer new dynamical phenomena than the existing ones. In the case of strong Allee effect, we prove the existence of canard cycles, homoclinic orbits and heteroclinic orbits. Furthermore, we focus on the case of weak Allee effect. In addition to the similar dynamics of that exhibited by the system with strong Allee effect, we also demonstrate the occurrence of relaxation oscillations created by entry-exit function. Moreover, the existence of canard explosion is further explained analytically and numerically with the help of sophisticated slow–fast techniques.

Double-zero degeneracy and heteroclinic cycles in a perturbation of the Lorenz system

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed.

Maslov index for heteroclinic orbits of non-Hamiltonian systems on a two-dimensional phase space

Maslov index for heteroclinic orbits of non-Hamiltonian systems on a two-dimensional phase space

On the Birth of Discrete Lorenz Attractors Under Bifurcations of 3D Maps with Nontransversal Heteroclinic Cycles

Lorenz attractors are important objects in the modern theory of chaos. The reason, on the one hand, is that they are encountered in various natural applications (fluid dynamics, mechanics, laser dynamics, etc.). On the other hand, Lorenz attractors are robust in the sense that they are generally not destroyed by small perturbations (autonomous, nonautonomous, stochastic). This allows us to be sure that the object observed in the experiment is exactly a chaotic attractor rather than a long-time periodic orbit. Discrete-time analogs of the Lorenz attractor possess even more complicated structure — they allow homoclinic tangencies of invariant manifolds within the attractor. Thus, discrete Lorenz attractors belong to the class of wild chaotic attractors. These attractors can be born in codimension-three local and certain global (homoclinic and heteroclinic) bifurcations. While various homoclinic bifurcations leading to such attractors have been studied, for heteroclinic cycles only cases where at least one of the fixed points is a saddle-focus have been considered to date. In the present paper the case of a heteroclinic cycle consisting of saddle fixed points with a quadratic tangency of invariant manifolds is considered. It is shown that, in order to have three-dimensional chaos such as the discrete Lorenz attractors, one needs to avoid the existence of lower-dimensional global invariant manifolds. Thus, it is assumed that either the quadratic tangency or the transversal heteroclinic orbit is nonsimple. The main result of the paper is the proof that the original system is the limiting point in the space of dynamical systems of a sequence of domains in which the diffeomorphism possesses discrete Lorenz attractors.

Global dynamics in nonconservative nonlinear Schrödinger equations

In this paper, we study the global dynamics of a class of nonlinear Schrödinger equations using perturbative and non-perturbative methods. We prove the semi-global existence of solutions for initial conditions close to constant. That is, solutions will exist for all positive time or all negative time. The existence of an open set of initial data which limits to zero in both forward and backward time is also demonstrated. This result in turn forces the non-existence of any real-analytic conserved quantities. For the quadratic case, we prove the existence of two (infinite) families of nontrivial unstable equilibria and prove the existence of heteroclinic orbits limiting to the nontrivial equilibria in backward time and to zero in forward time. By a time reversal argument, we also obtain heteroclinic orbits limiting to the nontrivial equilibria in forward time and to zero in backward time. The proofs for the quadratic equation are computer-assisted and rely on three separate ingredients: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of the flow (starting from the unstable manifold) and a proof that the solution enters a validated stable set (hence showing convergence to zero).

A Geometric Criterion for the Existence of Chaos Based on Periodic Orbits in Continuous-Time Autonomous Systems

A new geometric criterion is derived for the existence of chaos in continuous-time autonomous systems in three-dimensional Euclidean spaces, where a type of Smale horseshoe in a subshift of finite type exists, but the intersection of stable and unstable manifolds of two points on a hyperbolic periodic orbit does not imply the existence of a Smale horseshoe of the same type on any cross section of these two points. This criterion is based on the existence of a hyperbolic periodic orbit, differing from the classical equilibrium-based Shilnikov criterion and the condition of transversal homoclinic or heteroclinic orbit of a Poincaré map.

Dynamic Stability Analysis of Size-Dependent Viscoelastic/Piezoelectric Nano-Beam

This paper analyzes the dynamic stability of an isotropic viscoelastic Euler–Bernoulli nano-beam using piezoelectric materials. For this purpose, the size-dependent theory was used in the framework of the modified couple stress theory (MCST) for piezoelectric materials. In order to capture the geometrical nonlinearity, the von Karman strain displacement relation was applied. Hamilton’s principle was also employed to obtain the governing equations. Furthermore, the Galerkin method was used in order to convert the governing partial differential equations (PDEs) to a nonlinear second-order ordinary differential one. Dynamic stability analysis was performed and the effects of such parameters as viscoelastic coefficients, size effect, and piezoelectric coefficient were investigated. The results showed that in this system, saddle points, central points, Hopf bifurcation points, and fork bifurcation points could be created, and the phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, and homoclinic orbits.