Homoclinic Connections Research Articles

Chaotic spiking and incomplete homoclinic scenarios in semiconductor lasers with optoelectronic feedback

We demonstrate experimentally and theoretically the existence of slow chaotic spiking sequences in the dynamics of a semiconductor laser with ac-coupled optoelectronic feedback. The timescale of these dynamics is fully determined by the high-pass filter in the feedback loop and their erratic, though deterministic, nature is evidenced by means of the interspike interval (ISI) probability distribution. We eventually show that this regime is the result of an incomplete homoclinic scenario to a saddle-focus, where an exact homoclinic connection does not occur.

Bifurcations of mixed-mode oscillations in a stellate cell model

Experimental recordings of the membrane potential of stellate cells within the entorhinal cortex show a transition from subthreshold oscillations (STOs) via mixed-mode oscillations (MMOs) to relaxation oscillations under increased injection of depolarizing current. Acker et al. introduced a 7D conductance based model which reproduces many features of the oscillatory patterns observed in these experiments. For the first time, we present a comprehensive bifurcation analysis of this model by using the software package AUTO. In particular, we calculate the stable MMO branches within the bifurcation diagram of this model, as well as other MMO patterns which are unstable. We then use geometric singular perturbation theory to demonstrate how the bifurcations are governed by a 3D reduced model introduced by Rotstein et al. We extend their analysis to explain all observed MMO patterns within the bifurcation diagram. A key role in this bifurcation analysis is played by a novel homoclinic bifurcation structure connecting to a saddle equilibrium on the unstable branch of the corresponding critical manifold. This type of homoclinic connection is possible due to canards of folded node (folded saddle-node) type.

Foliation-Based Parameter Tuning in a Model of the GnRH Pulse and Surge Generator

We investigate a model of the GnRH pulse and surge generator, with the definite aim of constraining the model GnRH output with respect to a physiologically relevant list of specifications. The alternating pulse and surge pattern of secretion results from the interaction between a GnRH secreting system and a regulating system exhibiting slow-fast dynamics. The mechanisms underlying the behavior of the model are reviewed from the study of the Boundary-Layer System according to the “dissection method” principle. Using singular perturbation theory, we describe the sequence of bifurcations undergone by the regulating (FitzHugh–Nagumo) system, encompassing the rarely investigated case of homoclinic connection. Based on pure dynamical considerations, we restrict the space of parameter search for the regulating system and describe a foliation of this restricted space, whose leaves define constant duration ratios between the surge and the pulsatility phase in the whole system. We propose an algorithm to fix the parameter values also to meet the other prescribed ratios dealing with amplitude and frequency features of the secretion signal. We finally apply these results to illustrate the dynamics of GnRH secretion in the ovine species and the rhesus monkey.

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

We extend Lin’s method for use in the setting of parameter-dependent nonlinear functional dierential equations of mixed type (MFDEs). We show that the presence of M-homoclinic and M-periodic solutions that bifurcate from a prescribed homoclinic connection can be detected by studying a nite dimensional bifurcation equation. As an application, we describe the codimension two orbit-ip bifurcation in the setting of MFDEs.

Homoclinic and heteroclinic orbits for the 0² or 0²𝑖𝜔 singularity in the presence of two reversibility symmetries

This paper is devoted to the study of the dynamics of reversible vector fields close to an equilibrium when a 0 2 0^2 or a 0 2 i ω 0^2i\omega resonance occurs in the presence of two symmetries of reversibility. In the presence of a unique reversibility symmetry the existence of a homoclinic connection to 0 is known for the 0 2 0^2 resonance whereas for the 0 2 i ω 0^2i\omega resonance there is generically no homoclinic connection to 0 but there is always a homoclinic connection to an exponentially small periodic orbit. In the presence of a second symmetry of reversibility, the situation is more degenerate. Indeed, because of the second symmetry the quadratic part of the normal forms vanishes, and so the dynamics of the normal forms is governed by the cubic part. For the 0 2 0^2 resonance we prove the existence of homoclinic connections to 0 and of heteroclinic orbits. For the 0 2 i ω 0^2i\omega resonance we prove that in most of the cases the second symmetry induces the existence of homoclinic connections to 0 and of heteroclinic orbits whereas with a unique symmetry there is generically no homoclinic connection to 0. Such a reversible vector field with two reversibility symmetries occurs for instance after center manifold reduction when studying 2-dimensional waves in NLS type systems with one-dimensional potential or when studying localized waves in nonlinear chains of coupled oscillators. It also occurs when studying localized buckling in rods with noncircular cross section.

Transversality in scalar reaction–diffusion equations on a circle

We prove that stable and unstable manifolds of hyperbolic periodic orbits for general scalar reaction–diffusion equations on a circle always intersect transversally. The argument also shows that for a periodic orbit there are no homoclinic connections. The main tool used in the proofs is Matano's zero number theory dealing with the Sturm nodal properties of the solutions.

Breaking of Symmetrical Periodic Solutions in a Singularly Perturbed KDV Model

There are several recent developments in the well-known problem of breaking of homoclinic orbits (splitting of separatrices) of a system that undergoes a singular perturbation. First, survival of a homoclinic orbit is an exceptional situation that can be linked to triviality of the Stokes phenomenon of the underlying “truncated” equation. Second, homoclinic connections to exponentially small periodic orbits survive the perturbation in the generic case. In this paper we consider a different problem: we study deformations of “genuine” periodic orbits of the second order equation $y”=y+y^2$ that undergoes the singular perturbation $\varepsilon^2y””+(1-\varepsilon^2)y”=y+y^2$, where $\varepsilon>0$ is a small parameter. We prove that if the period and the constant of motion do not change too rapidly (in $\varepsilon$), a genuine (nontrivial) periodic solution does not survive the perturbation.

Complex dynamics of femtosecond terawatt laser pulses in air

Complex dynamics of femtosecond terawatt laser pulses in air is investigated theoretically and numerically by considering both χ(5) susceptibility and multiphoton ionization. Our investigation shows that these two high-order nonlinearities, acting as a Hamiltonian perturbation, can destroy homoclinic orbit connection of phase trajectory and result in spatial chaos and complex pattern of laser wave field. Due to the chaotic behavior of laser field, laser filamentation appears when laser intensity increases close to the ionization threshold.

Analysis of the T-point-Hopf bifurcation

In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-2 heteroclinic cycle occurs. If the parameter space is three-dimensional, such a bifurcation is located generically on a curve. A more degenerate scenario appears when this curve reaches a surface of Hopf bifurcations of one of the equilibria involved in the heteroclinic cycle. We are interested in the analysis of this codimension-3 bifurcation, which we call T-point-Hopf. In this work we propose a model, based on the construction of a Poincaré map, that describes the global behavior close to a T-point-Hopf bifurcation. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved. The predictions deduced from this model strongly agree with the numerical results obtained in a modified van der Pol–Duffing electronic oscillator.

Symmetric homoclinic solutions to the periodic orbits in the Michelson system

The Michelson system (6) x'''+x'+0.5x2 = c2 for the parame- ter value c = 1 is investigated. It was proven in (8) that the system possesses two odd periodic solutions. We shall show that there exist infinitely many homoclinic and heteroclinic connections between them. Moreover, we shall show that the family of homoclinic solutions contains a countable set of odd homoclinic solutions.

OPEN-TO-CLOSED CURVES OF SADDLE-NODE BIFURCATIONS OF PERIODIC ORBITS NEAR A NONTRANSVERSAL T-POINT IN CHUA'S EQUATION

In this work, the evolution of a family of open-to-closed curves of saddle-node bifurcations of periodic orbits is numerically studied in Chua's equation. This family is organized by open and closed curves of Shil'nikov homoclinic connections in the presence of a nontransversal T-point. We remark the important role of cusp catastrophes of periodic orbits in the complete mechanism of creation/destruction of such closed curves of saddle-node bifurcations.

Invariant manifolds of L3 and horseshoe motion in the restricted three-body problem

In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (HPOs) (which surround three equilibrium points called L3, L4 and L5), when the mass parameter μ is positive and small; we describe the structure of such families from the two-body problem (μ = 0). On the other hand, the region of existence of HPOs for any value of μ ∊ (0, 1/2] implies the understanding of the behaviour of the invariant manifolds of L3. So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L3 and on the simple infinite and double infinite period homoclinic phenomena are analysed. Finally, the relationship between the horseshoe homoclinic orbits and the HPO is considered in detail.

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in ${\Bbb R}^{3}.$ This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit between two critical points along the z axis. We prove that, generically, when the whole family is considered, this heteroclinic connection is destroyed. Moreover, we give an asymptotic formula of the distance between the stable and unstable manifolds when they meet the plane z = 0. This distance is exponentially small with respect to the unfolding parameter, and the main term is a suitable version of the Melnikov integral given in terms of the Borel transform of some function depending on the higher-order terms of the family. The results are obtained in a perturbative setting that does not cover the generic unfoldings of the Hopf singularity, which can be obtained as a singular limit of the considered family. To deal with this singular case, other techniques are needed. The reason to study the breakdown of the heteroclinic orbit is that it can lead to the birth of some homoclinic connection to one of the critical points in the unfoldings of the Hopf-zero singularity, producing what is known as a Shilnikov bifurcation.

Homoclinic Connections of Unstable Plane Waves of the Long‐Wave–Short‐Wave Equations

A Bäcklund transformation is obtained for unstable plane wave solutions of the long‐wave–short‐wave resonance equations that appear in continuum mechanics. Explicit expressions for the periodic homoclinic connections that arise from the instabilities are constructed by evaluating the transform at double points of the Floquet spectral curve.

The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems

We consider families of one and a half degrees of freedom rapidly forced Hamiltonian systems which are perturbations of one degree of freedom Hamiltonians with a homoclinic connection. We derive the inner equation for this class of Hamiltonian system which is expressed as the Hamiltonian–Jacobi equation of a one a half degrees of freedom Hamiltonian. The inner equation depends on a parameter not necessarily small.We prove the existence of special solutions of the inner equation with a given behaviour at infinity. We also compute the asymptotic expression for the difference between these solutions. In some perturbative cases, this asymptotic expression is strongly related with the Melnikov function associated with our initial Hamiltonian.

Dressing procedure for some homoclinic connections of the Manakov system

The dressing procedure is used to obtain explicit expressions for homoclinic connections of unstable plane waves of the Manakov system of coupled nonlinear Schrödinger equations.

Heteroclinic orbits and rotation sets for twist maps

We show how the shadowing property can be used in connection torotation sets. We review the concept of periodic chains and theshadowing rotation property, and study a class of diffeomorphismswith invariant sets that have such property. In particular, weconsider invariant sets that arise from homoclinic andheteroclinic connections for twist maps in higher dimensions. As aconsequence, we can show the existence of a family of twist maps,each one with an open set of rotation vectors that are realized bypoints that are close to a fixed point.

Destroying ergodicity in geodesic flows on surfaces

On surfaces, metrics that contain focusing cap outside of which the curvature is non-positive give rise to ergodic, and indeed Bernouli, geodesic flow. There exist C∞ small perturbations of these metrics that destroy the closed geodesic in the focusing cap, thereby producing what we call partially focusing systems. We prove that arbitrarily close to the ergodic focusing system are non-ergodic partially focusing systems. The proof involves perturbing near a homoclinic connection so as to produce a one-parameter family of horseshoe maps which leads to the existence of elliptic periodic orbits.Dedicated to the memory of Vladimir F Lazutkin.

High-dimensional heteroclinic and homoclinic connections in odd point-vortex ring on a sphere

We consider the motion of the N-vortex points that are equally spaced along a line of latitude on a sphere with fixed pole vortices, called ‘N-ring’. We are especially interested in the case when the number of the vortex points is odd. Since the eigenvalues that determine the stability of the odd N-ring are double, each of the unstable and stable manifolds corresponding to them is two-dimensional. Hence, it is generally difficult to describe the global structure of the manifolds. In this paper, based on the linear stability analysis, we propose a projection method to observe the structure of the iso-surface of the Hamiltonian, in which the orbit of the vortex points evolves. Applying the projection method to the motion of the 3-ring and 5-ring, we characterize the complex evolution of the unstable odd N-ring from the topological structure of the iso-surface of the Hamiltonian.

Homoclinic connections and subcritical Neimark bifurcation in a duopoly model with adaptively adjusted productions

Abstract In this paper we study some global bifurcations arising in the Puu’s oligopoly model when we assume that the producers do not adjust to the best reply but use an adaptive process to obtain at each step the new production. Such bifurcations cause the appearance of a pair of closed invariant curves, one attracting and one repelling, this latter being involved in the subcritical Neimark bifurcation of the Cournot equilibrium point. The aim of the paper is to highlight the relationship between the global bifurcations causing the appearance/disappearance of two invariant closed curves and the homoclinic connections of some saddle cycle, already conjectured in [Agliari A, Gardini L, Puu T. Some global bifurcations related to the appearance of closed invariant curves. Comput Math Simul 2005;68:201–19]. We refine the results obtained in such a paper, showing that the appearance/disappearance of closed invariant curves is not necessarily related to the existence of an attracting cycle. The characterization of the periodicity tongues (i.e. a region of the parameter space in which an attracting cycle exists) associated with a subcritical Neimark bifurcation is also discussed.