Homoclinic Orbits Research Articles

Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs.

Non-smooth systems can generate dynamics and bifurcations that are drastically different from their smooth counterparts. In this paper, we study such homoclinic bifurcations in a piecewise-smooth analytically tractable Lorenz-type system that was recently introduced by Belykh et al. [Chaos 29, 103108 (2019)]. Through a rigorous analysis, we demonstrate that the emergence of sliding motions leads to novel bifurcation scenarios in which bifurcations of unstable homoclinic orbits of a saddle can yield stable limit cycles. These bifurcations are in sharp contrast with their smooth analogs that can generate only unstable (saddle) dynamics. We construct a Poincaré return map that accounts for the presence of sliding motions, thereby rigorously characterizing sliding homoclinic bifurcations that destroy a chaotic Lorenz-type attractor. In particular, we derive an explicit scaling factor for period-doubling bifurcations associated with sliding multi-loop homoclinic orbits and the formation of a quasi-attractor. Our analytical results lay the foundation for the development of non-classical global bifurcation theory in non-smooth flow systems.

A criterion for mixed dynamics in two-dimensional reversible maps.

We give conditions for non-conservative dynamics in reversible maps with transverse and non-transverse homoclinic orbits.

Dynamics of a plant-herbivore model with a chemically-mediated numerical response

A system of two ordinary differential equations is proposed to model chemically-mediated interactions between plants and herbivores by incorporating a toxin-modified numerical response. This numerical response accounts for the reduction in the herbivore's growth and reproduction due to chemical defenses from plants. It is shown that the system exhibits very rich dynamics including saddle-node bifurcations, Hopf bifurcations, homoclinic bifurcations and co-dimension 2 bifurcations. Numerical simulations are presented to illustrate the occurrence of multitype bistability, limit cycles, homoclinic orbits and heteroclinic orbits. We also discuss the ecological implications of the resulting dynamics.

Theoretical and experimental studies of global dynamics for a class of bistable nonlinear impact oscillators with bilateral rigid constraints

A new bistable impact oscillator with bilateral rigid constraints under periodic excitations is established and the global dynamics are studied in detail respectively by the extended analytical Melnikov method for non-smooth systems and dynamical experiments. Firstly, the Melnikov method is extended from a new viewpoint of geometry for an abstract non-smooth dynamical system denoting a class of bistable impact oscillators with bilateral rigid constraints. Secondly, the analytical expression of homoclinic orbits is difficult to obtain, a semi-analytical and semi-numerical method for calculating the Melnikov function is applied to obtain the threshold of parameters for the global bifurcations and chaotic oscillations of the established impact oscillator. Then, numerical simulations and dynamical experiments are carried out together to show intra-well periodic oscillations, inter-well periodic or chaotic oscillations under different amplitudes of periodic excitations, which furthermore validates the reliability of the extended Melnikov method for this class of non-smooth systems.

Homoclinic orbits of sub-linear Hamiltonian systems with perturbed terms

By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.

No Chaos in Dixon’s System

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon’s system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different regions in each of which the system exhibits qualitatively the same behavior. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist.

An approximate technique to test chaotic region in a rotating pendulum system with bistable characteristics

This paper is concerned with the chaotic dynamics of a rotating pendulum system with bistable characteristics subjected to a viscous damping and a harmonic forcing. As a prototype of the single-degree-of-freedom system with bistable characteristics, this pendulum system exhibits a transition from smooth to discontinuous dynamics by changing a geometrical parameter. The dynamic behaviors of the unperturbed system with irrational nonlinearity bear significant similarities to the coupling of a simple pendulum and the smooth and discontinuous (SD) oscillator with the coexistence of the standard homoclinic orbits of Duffing type and pendulum type and the coexistence of the nonstandard homoclinic orbits of SD type and pendulum type in the smooth and discontinuous case, respectively. For the perturbed smooth system, we present an approximate technique to analytically obtain the lower bound line for horseshoes chaos arising from the homoclinic orbits of Duffing-type and Pendulum-type tangling, which overcomes the natural difficulties of solving the analytical expression of the homoclinic orbits and calculating the complicated Melnikov integrals. The chaotic thresholds of the perturbed discontinuous system are calculated by applying the numerical technique due to its non-smooth feature. Numerical simulations are carried out to certify the chaotic thresholds, which show the efficiency of the proposed techniques and demonstrate the predicated chaotic motions. Finally, different types of chaotic motions are illustrated via the cylindrical phase portraits. The contribution of this study is also helpful for exploring the dynamical behaviors of the complex nonlinear dynamical system containing the standard homoclinic or heteroclinic orbit in terms of the quantitative calculation.

Dynamics of subthreshold and suprathreshold resonance modulated by hyperpolarization-activated cation current in a bursting neuron

Resonance of both subthreshold and suprathreshold membrane potentials has been found in brain or motor neurons involving in the information processing or locomotion control via frequency response, which is mainly related to the hyperpolarization-activated cation current ( $$I_{{\text {h}}}$$ ). In the present paper, the modulations of $$I_{{\text {h}}}$$ current on resonance are acquired in a bursting neuron model, which closely match experimental observations. Firstly, the increase of $$I_{{\text {h}}}$$ current induces stable node changed to stable focus related to Hopf bifurcation, and to bursting via complex bifurcations including the big homoclinic orbit related to type I excitability. Secondly, the subthreshold resonance can be evoked by small stimulation from focus instead of node, resembling the experimental observation that subthreshold resonance appears at strong ( $$I_{{\text {h}}}$$ ) current. Thirdly, the modulation regularity of $$I_{{\text {h}}}$$ current on frequency and amplitude of resonance is simulated, closely matching the experimental observations, which can be well explained with the changing trends of imaginary and real part of eigenvalues of the focus. Last, the suprathreshold bursting is induced by strong stimulation from the stable focus near big homoclinic orbit, resembling that observed in the experiment. The bursting exhibits “suprathreshold resonance” similar to those evoked near Hopf bifurcation corresponding to type II excitability. A novel condition for “suprathreshold resonance” and the underlying codimension-2 bifurcations are acquired. The results present the comprehensive viewpoints on the subthreshold resonance and extend the condition for the “suprathreshold resonance”, which are helpful for deep understanding resonance in the brain or motor neurons with $$I_{{\text {h}}}$$ current.

Dynamical modeling and analysis of hyperelastic spherical shells under dynamic loads and structural damping

This paper investigates the effects of dynamic loads and structural damping on the nonlinear behaviors of incompressible hyperelastic spherical shells modeled by the Yeoh strain energy function. Firstly, the dynamical modeling is formulated for the nonlinear behaviors of the shells by the variational principle, and the second order nonlinear ordinary differential equation describing the radially symmetric motion is obtained. Then, the dynamic behaviors, such as periodic, quasiperiodic and chaotic motions, are discussed under different loading types. Particularly, for constant loads, the first integral of the integrable Hamiltonian system without damping is given and it is numerically proved that there exists an asymmetric "∞" homoclinic orbit for the prescribed material parameters obtained in experiments; moreover, it is found that for different prestretches, the structural parameter has a completely different role on the nonlinearity of the system, and the basins of attraction are given with the structural damping. For periodic loads, there exist some interesting dynamic phenomena, i.e., quasiperiodic oscillation in the approximately integrable Hamiltonian system, limit cycles and chaos with the damping. The criterion for chaos is discussed by the Melnikov method combined with the numerical calculation and the chaos is further analyzed with the Poincaré section and the phase plane.

Smooth travelling wave solutions in a generalized Degasperis–Procesi equation

In this paper, by using a dynamical system-based approach, namely, the singular travelling wave method, we classify all the travelling wave solutions including periodic wave solutions, periodic peakon solutions, solitary wave solutions and compacton solutions etc. in the generalized Degasperis–Procesi (gDP) equation. All these travelling waves are smooth and classical solutions. The parameter conditions for the existence of these travelling waves are also obtained. Then we combine geometric singular perturbation theory with an “explicit”Melnikov method to analyze the persistence of solitary wave solutions under singular perturbation, i.e., the existence of solitary wave solutions in a perturbed gDP equation. By calculating the representations of the (unperturbed) homoclinic orbits and the associated Melnikov integral explicitly, the persistence of solitary wave solutions is shown and the speed of the wave is determined (to leading order).

Dissecting the snake: Transition from localized patterns to spike solutions

An investigation is undertaken of coupled reaction–diffusion systems in one spatial dimension that are able to support, in different regions of their parameter space, either an isolated spike solution, or stable localized patterns with an arbitrary number of peaks. The distinction between the two cases is characterized through the behavior of the far field, where there is either an oscillatory or a monotonic decay. This transition is illustrated with two examples: a generalized Schnakenberg system that arises in cellular-level morphogenesis and a continuum model of urban crime spread. In each, it is found that localized patterns connected via a so-called homoclinic snaking curve in parameter space transition into a single spike solution as a second parameter is varied, via a change in topology of the snake into a series of disconnected branches. The transition is caused by a so-called Belyakov–Devaney transition between complex and real spatial eigenvalues of the far field of the primary pulse. A codimension-two problem is studied in detail where a non-transverse homoclinic orbit undergoes this transition. A Shilnikov-style analysis is undertaken which reveals the asymptotics of how the infinite family of folds of multi-pulse orbits are all destroyed at the same parameter value. The results are shown to be consistent with numerical experiments on the examples.

Critical homoclinics in a restricted four-body problem: numerical continuation and center manifold computations

The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four body problem. The problem can be viewed as a two parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for computing normal forms and center stable/unstable manifold parameterizations. After careful numerical study of a number of special cases we formulate several conjectures about the global bifurcations of the homoclinic families.

Saddle-center and periodic orbit: Dynamics near symmetric heteroclinic connection

An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian, and two non-symmetric heteroclinic orbits permuted by the involution. This is a codimension one structure; therefore, it can be met generally in one-parameter families of reversible Hamiltonian systems. There exist two possible types of such connections depending on how the involution acts near the equilibrium. We prove a series of theorems that show a chaotic behavior of the system and those in its unfoldings, in particular, the existence of countable sets of transverse homoclinic orbits to the saddle periodic orbit in the critical level, transverse heteroclinic connections involving a pair of saddle periodic orbits, families of elliptic periodic orbits, homoclinic tangencies, families of homoclinic orbits to saddle-centers in the unfolding, etc. As a by-product, we get a criterion of the existence of homoclinic orbits to a saddle-center.

Shilnikov Saddle-Focus Homoclinic Orbits from Numerics: Higher Dimensions

In a previous paper we studied parametrized autonomous systems and gave a computable criterion that an approximate orbit connecting hyperbolic equilibria is shadowed by a true connecting orbit. This criterion was used to give rigorously verified examples of Shilnikov saddle-focus homoclinic orbits in three dimensions. This involved verifying a condition on the eigenvalues of the linearization at the equilibrium. In dimensions greater than three, there are three more conditions which must be established: general position, asymptotic tangency and a transversality condition. In this paper we give computable criteria for verifying these three conditions. An example in four dimensions, in which detailed rigorous computations are carried out, is given.

Traveling pulses in a coupled FitzHugh–Nagumo equation

For a coupled FitzHugh–Nagumo (FHN) equation with a parameter a∈[0,1], when a∈(0,1∕2) the existence of traveling pulses of this equation was proved in 2013 by Holzer, Doelman and Kaper. Here we adopt a new approach to obtain the existence of the traveling pulse of the same equation for a∈[0,1∕2), which includes the degenerate case a=0, where the origin on the critical set loses normal hyperbolicity while it is for a∈(0,1). Also we show that the pulse does not exhibit an oscillatory tail at the homoclinic orbit when the time t→±∞, whereas the classical FHN equation could have an oscillatory tail of the traveling pulse depending on the choice of the parameters of the system. Finally we present an explanation on why traveling pulses cannot exist when a∈[1∕2,1].

Bifurcation phenomena in the peristaltic transport of non-Newtonian fluid with heat and mass transfer effects

An in-depth bifurcation analysis is carried out for the peristaltic transport of non-Newtonian fluid with heat and mass transfer through an axisymmetric channel. Based on the perturbation technique, analytical solutions for flow rate and stream function are presented. This function and its velocity fields build a nonlinear dynamic system in two spatial dimensions. We are then interested in identifying the global and local bifurcation of the invariant curves in which the recurrent fluid dynamics close to the stagnation points change qualitatively. For this purpose, a geometric description, analytical expressions and the development of a computational algorithm are provided to recognize the multiplicity of admissible/virtual stagnation points. In a variety of physical parameters, the analysis highlights the presence of several types of nonlinear phenomena, such as infinite/finite heteroclinic and homoclinic orbits, saddle-node and border-collision bifurcations. These results guide the streamline patterns for capturing novel complex behaviors such as multiple trapping phenomena and critical transition to distinguish between different flow regions.

Homoclinic orbits for first-order Hamiltonian system with local super-quadratic growth condition

In this paper we study the first-order Hamiltonian system Moreover precisely, assuming that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition, we obtain new existence results of ground state homoclinic orbits and infinitely many geometrically distinct homoclinic orbits by using a variational method. An interesting problem is that the nonlinearity may be super-quadratic on some domains and asymptotically quadratic on other domains under local super-quadratic condition. Since we are without more global information on the nonlinearity, we apply a perturbation approach and some special techniques in the proofs.

Global Effect of Non-Conservative Perturbations on Homoclinic Orbits

We study the effect of time-dependent, non-conservative perturbations on the dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. We assume that the unperturbed system is Hamiltonian, and the normally hyperbolic invariant manifold is parametrized via action-angle coordinates. The homoclinic excursions can be described via the scattering map, which gives the future asymptotic of an orbit as a function of its past asymptotic. We provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map expressed in action-angle coordinates. We illustrate these formulas for perturbations of both uncoupled and coupled rotator-pendulum systems.

Multi-pass stable periodic points of diffeomorphism of a plane with a homoclinic orbit

A diffeomorphism of a plane into itself with a fixed hyperbolic point and a nontransversal point homoclinic to it is studied. There are various ways of touching a stable and unstable manifold at a homoclinic point. Periodic points whose trajectories do not leave the vicinity of the trajectory of a homoclinic point are divided into a countable set of types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns that lie outside a sufficiently small neighborhood of the hyperbolic point. Earlier in the articles of Sh. Newhouse, L. P. Shil’nikov, B. F. Ivanov and other authors, diffeomorphisms of the plane with a nontransversal homoclinic point were studied, it was assumed that this point is a tangency point of finite order. In these papers, it was shown that in a neighborhood of a homoclinic point there can be infinite sets of stable two-pass and three-pass periodic points. The presence of such sets depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point with a finite order of tangency of a stable and unstable manifold. It is shown in the paper that for any fixed natural number n, a neighborhood of a nontransversal homolinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents separated from zero.

Subharmonic resonance bifurcation and chaos of simple pendulum system with vertical excitation and horizontal constraint

In order to improve the working performance and optimize the working parameters of the typical engineering pendulum of a typical system that it is abstracted as a physical simple pendulum model with vertical excitation and horizontal constraint. The dynamical equation of the system with vertical excitation and horizontal constraint is established by using Lagrange equation. The multiple-scale method is used to analyze the subharmonic response characteristics of the system. The amplitude-frequency response equation and the phase-frequency response equation are obtained through calculation. The effects of the system parameters on the amplitude resonance bandwidth and variability are clarified. According to the singularity theory and the universal unfolding theory, the bifurcation topology structure of the subharmonic resonance of the system is obtained. The Melnikov function is applied to the study of the critical conditions for the chaotic motion of the system. The parameter equation of homoclinic orbit motion is obtained through calculation. The threshold conditions of chaos in the sense of Smale are analyzed by solving the Melnikov function of the homoclinic motion orbit. The dynamic characteristics of the system, including single-parameter bifurcation, maximum Lyapunov exponent, bi-parameter bifurcation, and manifold transition in the attraction basin, are analyzed numerically. The results show that the main path of the system entering into the chaos is an almost period doubling bifurcation. Complex dynamical behaviors such as periodic motion, period doubling bifurcation and chaos are found. The bi-parameter matching areas of the subharmonic resonance bifurcation and chaos of the system are clarified. The results reveal the global characteristics of the system with vertical excitation and horizontal constraint, such as subharmonic resonance bifurcation, periodic attractor multiplication, and the coexistence of periodic and chaotic attractors. The results further clarify the mechanism of the influence of system parameters change on the movement form transformation, energy distribution and evolution law of the system. The mechanism of the influence of relevant parameters on the performance of the engineering system with vertical excitation and horizontal constraint is also obtained. The results of this research provide theoretical bases for adjusting the parameters of working performances of this typical physical system in engineering domain and the vibration reduction and suppression of the system in actual working conditions.