Symmetric Homoclinic Orbits Research Articles

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.

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.

Global dynamics and performance of vibration reduction for a new vibro-impact bistable nonlinear energy sink

A new vibro-impact bistable nonlinear energy sink (VI-BNES) with relatively light weight attached to a damped linear oscillator (LO) is proposed to form a two-degree-of-freedom nonlinear mechanical system with bilateral rigid constraints. The state of the system after instantaneous impact is described by the simplified shock theory and the momentum conservation. When the LO subjected to impulse excitations with different levels is considered as the driver of the VI-BNES, a two-dimensional non-smooth reduced impact system with a harmonic perturbation and an irrational nonlinear restoring force is obtained to study the threshold for the homoclinic bifurcations and chaotic responses of the VI-BNES. Bistability of this mechanical model makes the reduced unperturbed system have a pair of symmetric homoclinic orbits as the pseudo-separatrix for in-well and cross-well oscillations. The Melnikov method developed by us for non-smooth planar systems with bilateral rigid constraints is firstly employed to detect the threshold of the amplitude of excitations of the LO for the onset of chaotic oscillations. Furthermore, cubic polynomial approximation of the restoring force on the premise of maintaining the equilibria of the system is carried out such that the analytical Melnikov analysis for the global dynamics becomes possible. The performance of the VI-BNES is also studied and compared with other NESs in detail through numerical analysis and experiments to show the high efficiency of vibration reduction of the VI-BNES due to the dual beneficial role of the bistability and the impacts with the bilateral rigid constraints.

Analytic and Algebraic Conditions for Bifurcations of Homoclinic Orbits II: Reversible Systems

Following Part I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under some conditions. We next extend the Melnikov method of Part I to reversible systems and obtain some theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones.

Bifurcations of radially symmetric solutions in a coupled elliptic system with critical growth in [formula omitted] for d = 3,4

We consider a system of coupled elliptic partial differential equations with critical growth in Rd for d=3,4 and study bifurcations of three families of radially symmetric, bounded solutions. We reduce the problems of the three families to those of three symmetric homoclinic orbits in a four-dimensional reversible system of ordinary differential equations and show that transcritical or pitchfork bifurcations of the three families occur at infinitely many parameter values. Numerical computations are also given to demonstrate our theoretical results.

Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation u⁗+βu″+eu−1=0 for all parameter values β∈[0.5,1.9]. For each β, a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.

Symmetric Homoclinic Orbits at the Periodic Hamiltonian Hopf Bifurcation

We prove the existence of symmetric homoclinic orbits to a saddle-focus symmetric periodic orbit that appears in a generic family of reversible three degrees of freedom Hamiltonian system due to periodic Hamiltonian Hopf bifurcation, if some coefficient A of the normal form of the fourth order is positive. If this coefficient is negative, then for the opposite side of the bifurcation parameter value, we prove the existence of symmetric homoclinic orbits to saddle invariant 2-tori.

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L 3, as an example of a center × saddle equilibrium point in a Hamiltonian with two degrees of freedom. We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L 3, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ 1, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ 1, corresponding to m-round SHO. Some comments on related analytical results are also made.

Bifurcation of degenerate homoclinic orbits to saddle-center in reversible systems

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoclinic orbits near the primary homoclinic orbits is developed. Some known results are extended.

Modelling gap junctions in a neural field model

We study a nonlinear, one-dimensional neural field model based upon a partial integro-differential equation, that is used to model spatial patterns in working memory. Through the application of Fourier transforms to the PIDE, steady states of spatially localised areas of high activity can be represented by solutions of a fourth order ODE. Recent research has shown a high density of gap junctions in areas of the brain that experience epileptic events. We extend the model by including a diffusion-like term to model gap junctions and derive a sixth order ODE which we use to investigate changes in the dynamics of spatially localised solutions. We find that symmetric homoclinic orbits to a zero steady state exist for a wide area of parameter space. Numerical work shows families of solutions are destroyed as the strength of the term modelling gap junctions increases.

GLUING BIFURCATIONS IN CHUA OSCILLATOR

Gluing bifurcation in a modified Chua's oscillator is reported. Keeping other parameters fixed when a control parameter is varied in the modified oscillator model, two symmetric homoclinic orbits to saddle focus at origin, which are mirror images of each other, are glued together for a particular value of the control parameter. In experiments, two asymmetric limit cycles are homoclinic to the saddle focus origin for different values of the control parameter. However, imperfect gluing bifurcation has been observed, in experiments, when one stable and unstable limit cycles merge to the saddle focus origin via saddle-node bifurcation.

Detection of symmetric homoclinic orbits to saddle-centres in reversible systems

We present a perturbation technique for the detection of symmetric homoclinic orbits to saddle-centre equilibria in reversible systems of ordinary differential equations. We assume that the unperturbed system has primary, symmetric homoclinic orbits, which may be either isolated or appear in a family, and use an idea similar to that of Melnikov’s method to detect homoclinic orbits in their neighbourhood. This technique also allows us to identify bifurcations of unperturbed or perturbed, symmetric homoclinic orbits. Our technique is of importance in applications such as nonlinear optics and water waves since homoclinic orbits to saddle-centre equilibria describe embedded solitons (ESs) in systems of partial differential equations representing physical models, and except for special cases their existence has been previously studied only numerically using shooting methods and continuation techniques. We apply the general theory to two examples, a four-dimensional system describing ESs in nonlinear optical media and a six-dimensional system which can possess a one-parameter family of symmetric homoclinic orbits in the unperturbed case. For these examples, the analysis is compared with numerical computations and an excellent agreement between both results is found.

A PHENOMENOLOGICAL APPROACH TO NORMAL FORM MODELING: A CASE STUDY IN LASER INDUCED NEMATODYNAMICS

An experimental setting for the polarimetric study of optically induced dynamical behavior in nematic liquid crystal films presented by G. Cipparrone, G. Russo, C. Versace, G. Strangi and V. Carbone allowed to identify most notably some behavior which was recognized as gluing bifurcations leading to chaos. This analysis of the data used a comparison with a model for the transition to chaos via gluing bifurcations in optically excited nematic liquid crystals previously proposed by G. Demeter and L. Kramer. The model of these last authors, relying on the original model for chaos by cascade of gluing bifurcations proposed by A. Arneodo, P. Coullet and C. Tresser about twenty years before, does not have the central symmetry which one would expect for minimal dimensional model for chaos in nematics in view of the time series near the gluing bifurcation. What we show here is that the simplest truncated normal forms for gluing with the appropriate symmetry and minimal dimension do exhibit time signals that are embarrassingly similar to the ones that could be found using the above-mentioned experimental settings. It so happens that the gluing bifurcation scenario itself is only visible in limited parameter ranges, and that a substantial aspect of the chaos that can be observed is due to other factors. First, out of the immediate neighborhood of the homoclinic curve, nonlinearity can produce expansion which easily produces chaos when combined with the recurrence induced by the homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme sensitivity to noise, so that when the noiseless approach to attracting homoclinic pairs contains a rich behavior, minute noise can transform the complex damping into sustained chaos. As Leonid Shilnikov has taught us, combining global considerations and local spectral analysis near critical points is crucial to understand the phenomenology associated to homoclinic bifurcations in dissipative systems. We see here on an example how this helps construct a phenomenological approach to modeling experiments in nonlinear dissipative contexts.

Symmetric Heteroclinic Connections in the Michelson System: A Computer Assisted Proof

In this paper we present a new technique of proving the existence of an infinite number of symmet- ric heteroclinic and homoclinic solutions. This technique combines the covering relations method introduced by Zgliczynski (Topol. Methods Nonlinear Anal., 8 (1996), pp. 169-177; Nonlinearity ,1 0 (1997), pp. 243-252) with symmetry properties of a dynamical system. As an example we present a computer assisted proof of the existence of an infinite number of heteroclinic connections be- tween equilibrium points in the Kuramoto-Sivashinsky ODE (D. Michelson, Phys. D, 19 (1986), pp. 89-111). Moreover, we present the proof of the existence of an infinite number of heteroclinic connections between periodic orbits and equilibrium points. 1. Introduction. The aim of this paper is to present a new method for proving the ex- istence of symmetric homoclinic or heteroclinic solutions in systems possessing the reversing symmetry property. In (5) (and references given there) a method of proving the existence of time reversing symmetric homoclinic and heteroclinic solutions for dynamical systems is presented and it is called the fixed set iteration method. It applies to dynamical systems with continuous and discrete time. The basic idea of such a method is to search for the points u which are invariant under the symmetry and whose trajectories converge to an equilibrium point or a periodic orbit. This allows us to conclude that the trajectories of the points u must be homoclinic or heteroclinic. Galias and Zgliczynski (3) presented the method for proving the existence of homoclinic and heteroclinic solutions for maps R 2 → R 2 . This result was applied to the planar circular restricted three body problem (1, 16), where the existence of an infinite number of homoclinic and heteroclinic connections between periodic orbits was shown. In this paper we demonstrate how to combine these two methods for proving the existence of symmetric homoclinic or heteroclinic orbits in systems possessing the reversing symmetry property. Moreover, we present some generalization of the Galias-Zgliczynski method. We show how to prove the existence of heteroclinic orbits between objects possessing unequal dimensions—for example, the equilibrium points and periodic orbits.

Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation

We consider reversible and -symmetric systems of ordinary differential equations (ODEs) that possess a symmetric homoclinic orbit to a degenerate equilibrium. The equilibrium is supposed to undergo a reversible pitchfork bifurcation, controlled by the system's parameter. It has been shown in Wagenknecht (Nonlinearity 15 2097–119) that a multitude of homoclinic orbits emerges in this bifurcation. In particular, if a coefficient in the normal form of the local bifurcation has the correct sign such that this bifurcation is of eye-type, then globally a reversible homoclinic pitchfork bifurcation can be observed. This means, that similar to the local bifurcation in which two new equilibria emerge, two-homoclinic orbits to these equilibria bifurcate from the primary homoclinic orbit. In this paper, we investigate the emergence of two-homoclinic and two-heteroclinic orbits, that is, orbits making two windings in a neighbourhood of the primary orbit, in this bifurcation. Using a combination of geometrical and analytical techniques we prove the emergence of a family of two-homoclinic orbits to periodic orbits and of a two-heteroclinic cycle between equilibria. The general analysis is illustrated by numerical results for an example system of two second order ODEs.

About a homoclinic pitchfork bifurcation in reversible systems with additional ℤ2-symmetry

This paper studies bifurcations from a homoclinic orbit to a degenerate fixed point. We consider reversible ℤ2-symmetric systems of ordinary differential equations (ODEs) and assume the existence of a symmetric homoclinic orbit to a fixed point which itself undergoes a pitchfork bifurcation. We are interested in bifurcations from the primary homoclinic orbit in an unfolding of the degenerate situation.The studies are motivated by numerical investigations on a model-system of second-order ODEs. There one finds a similar behaviour in the local and the global bifurcation. While locally two new fixed points are created, numerical computations show that at the same time two homoclinic orbits to these fixed points bifurcate from the primary orbit. We call the global scenario a reversible homoclinic pitchfork bifurcation.An analysis of this homoclinic bifurcation is performed in a general frame. Depending on the sign of a higher order coefficient in the normal form we distinguish two cases of the local pitchfork bifurcation: the eye case (which is the one encountered in the model-system) and the figure-eight case. Adopting Lin's method to the non-hyperbolic situation the bifurcation of one-homoclinic orbits to the local centre manifold of the fixed point is investigated. Rigorous existence results for homoclinic orbits to fixed points and periodic orbits are derived. The global bifurcation picture is found to depend crucially on the local bifurcation.

Tendril Perversion in Intrinsically Curved Rods

A straight elastic rod with intrinsic curvature under varying tension can undergo an instability and bifurcate to a filament made out of two helices with opposite handedness. This inversion of handedness, known as perversion, appears in a wide range of biological and physical systems and is investigated here within the framework of thin elastic rods described by the static Kirchhoff equations. In this context, a perversion is represented by a heteroclinic orbit joining asymptotically two fixed points representing helices with opposite torsion. A center manifold reduction and a normal form transformation for a triple zero eigenvalue reduce the dynamics to a third-order reversible dynamical system. The analysis of this reduced system reveals that the heteroclinic connection representing the physical solution results from the collapse of pairs of symmetric homoclinic orbits. Results of the normal form calculation are compared with numerical solutions obtained by continuation methods. The possibility of self-contact and the elastic characteristics of the perverted rod are also studied.

Symmetry-Breaking Homoclinic Bifurcations in Diffusively Coupled Equations

In this study we examine a symmetry-breaking bifurcation of homoclinic orbits in diffusively coupled ordinary differential equations. We prove that asymmetric homoclinic orbits can bifurcate from a symmetric homoclinic orbit when the equilibria to which the latter is homoclinic undergoes a pitchfork bifurcation. A condition which defines the direction of the bifurcation in a parameter space is given. All hypotheses of the main theorem are verified for a diffusively coupled logistic system and the twistedness of the bifurcating homoclinic orbits is computed for a range of coupling strengths.

A method for locating symmetric homoclinic orbits using symbolic dynamics

In this paper we present a method which can identify and locate symmetric homoclinic orbits in a homoclinic tangle formed by the intersecting stable and unstable manifolds of a symmetric 2D map. The method consists of a systematic search in parameter space and determination of the order in which these orbits arise using symbolic dynamics. Each orbit corresponds to a unique sequence and it is computed by iterating the map along the unstable manifold to match a specific symmetry at the middle of the orbit. An application of the method to the determination of multibreather solutions of 1D lattices is discussed.

Symmetrically Homoclinic Orbits for Symmetric Hamiltonian Systems

In this paper, we study the existence of symmetric homoclinic orbits for first order and second order Hamiltonian systems with some symmetric Hamiltonian functions.