Heteroclinic Chains Research Articles

Applying Lin's method to constructing heteroclinic orbits near the heteroclinic chain

In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under ‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits and connected by three hyperbolic saddle points . Assume that the degeneracy of the unperturbed heteroclinic orbit is , the splitting index is . By applying Lin's method, we construct heteroclinic orbits connected and near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from to . Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices , and then obtain the existence of the heteroclinic orbits which backward asymptotic to and forward asymptotic to .

Transverse foliations in the rotating Kepler problem

We construct finite energy foliations and transverse foliations of neighbourhoods of the circular orbits in the rotating Kepler problem for all negative energies. This paper would be a first step towards our ultimate goal that is to recover and refine McGehee’s results on homoclinics [23] and to establish a theoretical foundation to the numerical demonstration of the existence of a homoclinic–heteroclinic chain in the planar circular restricted three-body problem [20], using pseudoholomorphic curves.

Bifurcation analysis of double cavity flows

The first few bifurcations in a two-dimensional incompressible double cavity flow are investigated using the linear stability analysis, the Floquet analysis, and the nonlinear direct numerical simulations (DNS). The prediction of the critical Reynolds number and the type of bifurcation (Hopf, pitchfork, inverse pitchfork, and Neimark–Sacker), which depend on cavity configuration, by the linear stability analysis and the Floquet analysis is consistent with nonlinear DNS. The nonlinear DNS results show that the state of the system passes through multiple intermediate (unstable) states before it reaches the stable attractor (heteroclinic chain), and the type of intermediate states depends on initial conditions. The intermediate states are reported as the asymptotic state in the literature for some flow conditions because it is not known a priori how long it will take to reach the asymptotic state in nonlinear simulations. The present study reports the actual asymptotic state for those flow conditions.

Torsional solutions of convection in rotating fluid spheres

A numerical study of the nonlinear torsional solutions of convection in rotating, internally heated, self-gravitating fluid spheres shows a sequence of transitions towards chaos in which heteroclinic chains connecting unstable periodic solutions can be recognized.

Non-transversal heteroclinic chain in thequintic nonlinear Schrödinger equation

Non-transversaldiffusion seems similar to the Arnold diffusion, but it is adifferent phenomenon since it could be applied to integrable systemsin contrast to the Arnold diffusion, which is a phenomenon that onlytakes place in non integrable systems. In this paper, the existenceof the non-transversal heteroclinic chain in the quintic nonlinearSchrodinger equation is studied. Based on a finite dimensionalsystem of ordinary differential equations, the Toy model systemassociated with the quintic nonlinear Schrodinger equation, weconstruct the non-transversal heteroclinic chain of the Toy modelsystem. The explicit expression for the dynamics of the heteroclinicis given.

Discrete Sequential Information Coding: Heteroclinic Cognitive Dynamics.

Discrete sequential information coding is a key mechanism that transforms complex cognitive brain activity into a low-dimensional dynamical process based on the sequential switching among finite numbers of patterns. The storage size of the corresponding process is large because of the permutation capacity as a function of control signals in ensembles of these patterns. Extracting low-dimensional functional dynamics from multiple large-scale neural populations is a central problem both in neuro- and cognitive- sciences. Experimental results in the last decade represent a solid base for the creation of low-dimensional models of different cognitive functions and allow moving toward a dynamical theory of consciousness. We discuss here a methodology to build simple kinetic equations that can be the mathematical skeleton of this theory. Models of the corresponding discrete information processing can be designed using the following dynamical principles: (i) clusterization of the neural activity in space and time and formation of information patterns; (ii) robustness of the sequential dynamics based on heteroclinic chains of metastable clusters; and (iii) sensitivity of such sequential dynamics to intrinsic and external informational signals. We analyze sequential discrete coding based on winnerless competition low-frequency dynamics. Under such dynamics, entrainment, and heteroclinic coordination leads to a large variety of coding regimes that are invariant in time.

Robust sequential working memory recall in heterogeneous cognitive networks.

Psychiatric disorders are often caused by partial heterogeneous disinhibition in cognitive networks, controlling sequential and spatial working memory (SWM). Such dynamic connectivity changes suggest that the normal relationship between the neuronal components within the network deteriorates. As a result, competitive network dynamics is qualitatively altered. This dynamics defines the robust recall of the sequential information from memory and, thus, the SWM capacity. To understand pathological and non-pathological bifurcations of the sequential memory dynamics, here we investigate the model of recurrent inhibitory-excitatory networks with heterogeneous inhibition. We consider the ensemble of units with all-to-all inhibitory connections, in which the connection strengths are monotonically distributed at some interval. Based on computer experiments and studying the Lyapunov exponents, we observed and analyzed the new phenomenon—clustered sequential dynamics. The results are interpreted in the context of the winnerless competition principle. Accordingly, clustered sequential dynamics is represented in the phase space of the model by two weakly interacting quasi-attractors. One of them is similar to the sequential heteroclinic chain—the regular image of SWM, while the other is a quasi-chaotic attractor. Coexistence of these quasi-attractors means that the recall of the normal information sequence is intermittently interrupted by episodes with chaotic dynamics. We indicate potential dynamic ways for augmenting damaged working memory and other cognitive functions.

Collapsed heteroclinic snaking near a heteroclinic chain in dragged meniscus problems.

A liquid film is studied that is deposited onto a flat plate that is inclined at a constant angle to the horizontal and is extracted from a liquid bath at a constant speed. We analyse steady-state solutions of a long-wave evolution equation for the film thickness. Using centre manifold theory, we first obtain an asymptotic expansion of solutions in the bath region. The presence of an additional temperature gradient along the plate that induces a Marangoni shear stress significantly changes these expansions and leads to the presence of logarithmic terms that are absent otherwise. Next, we numerically obtain steady solutions and analyse their behaviour as the plate velocity is changed. We observe that the bifurcation curve exhibits collapsed (or exponential) heteroclinic snaking when the plate inclination angle is above a certain critical value. Otherwise, the bifurcation curve is monotonic. The steady profiles along these curves are characterised by a foot-like structure that is formed close to the meniscus and is preceded by a thin precursor film further up the plate. The length of the foot increases along the bifurcation curve. Finally, we prove with a Shilnikov-type method that the snaking behaviour of the bifurcation curves is caused by the existence of an infinite number of heteroclinic orbits close to a heteroclinic chain that connects in an appropriate three-dimensional phase space the fixed point corresponding to the precursor film with the fixed point corresponding to the foot and then with the fixed point corresponding to the bath.

Recent developments concerning generic spacelike singularities

This is a review of recent progress concerning generic spacelike singularities in general relativity. For brevity the main focus is on singularities in vacuum spacetimes, although the connection with, and the role of, matter for generic singularity formation is also commented on. The paper describes recent developments in two areas and show how these are connected within the context of the conformally Hubble-normalized state space approach. The first area is oscillatory singularities in spatially homogeneous cosmology and the connection between asymptotic behaviour and heteroclinic chains. The second area concerns oscillatory singularities in inhomogeneous models, especially spike chains and recurring spikes. The review also outlines some underlying reasons for why the structures that are the foundation for generic oscillatory behaviour exists at all, which entails discussing how underlying physical principles and applications of solution generating techniques yield hierarchical structures and connections between them. Finally, it is pointed out that recent progress concerning generic singularities motivates some speculations that suggest that a paradigm shift concerning their physical role, and what mathematical issues to address, might be in order.

Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a double-Hopf bifurcation

A methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, Newton-Krylov, and Arnoldi methods. An application to the study of a double-Hopf bifurcation of the basic state is shown for Ekman and Prandtl numbers $E={10}^{\ensuremath{-}4}$ and $\ensuremath{\sigma}=0.1$, respectively, radius ratios $\ensuremath{\eta}\ensuremath{\in}[0.32,0.35]$, Rayleigh numbers $R\ensuremath{\in}[1.8\ifmmode\times\else\texttimes\fi{}{10}^{5},6\ifmmode\times\else\texttimes\fi{}{10}^{5}]$, and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves.

Oscillatory Singularities in Bianchi Models with Magnetic Fields

An idea which has been around in general relativity for more than forty years is that in the approach to a big bang singularity solutions of the Einstein equations can be approximated by the Kasner map, which describes a succession of Kasner epochs. This is already a highly non-trivial statement in the spatially homogeneous case. There the Einstein equations reduce to ordinary differential equations and it becomes a statement that the solutions of the Einstein equations can be approximated by heteroclinic chains of the corresponding dynamical system. For a long time progress on proving a statement of this kind rigorously was very slow but recently there has been new progress in this area, particularly in the case of the vacuum Einstein equations. In this paper we generalize some of these results to the Einstein-Maxwell equations. It turns out that this requires new techniques since certain eigenvalues are in a less favourable configuration in the case with a magnetic field. The difficulties which arise in that case are overcome by using the fact that the dynamical system of interest is of geometrical origin and thus has useful invariant manifolds.

Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles

We consider cosmological models of Bianchi type. In particular, we are interested in the alpha-limit dynamics near the Kasner circle of equilibria for Bianchi classes VIII and IX. They correspond to cosmological models close to the big-bang singularity. We prove the existence of a codimension-one family of solutions that limit, for t to negative infinity, onto a heteroclinic 3-cycle to the Kasner circle of equilibria. The theory extends to arbitrary heteroclinic chains that are uniformly bounded away from the three critical Taub points on the Kasner circle, in particular to all closed heteroclinic cycles of the Kasner map.

Heteroclinic orbits and heteroclinic chains for a discrete Hamiltonian system

In the present work we prove some existence results of heteroclinic orbits and heteroclinic chains for a second order discrete Hamiltonian system of the form Δ2q(t − 1) + V′(q(t)) = 0, t ∈ ℤ. The methods we use are variational in nature. Our results show that under general conditions, for each maximum point β of V, the above system possesses multiple heteroclinic orbits joining β and some other maximum points of V. We also prove that for any pair of distinct maximum points η and ξ of V, there exists at least one heteroclinic chain from η to ξ.

Universality of Crystallographic Pinning

We study traveling waves for reaction diffusion equations on the spatially discrete domain \({\mathbb Z^2}\). The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. In [19] it was shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth, which itself was considered in [6]. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region.

Lin's method for heteroclinic chains involving periodic orbits

We present an extension of the theory known as Lin's method to heteroclinic chains that connect hyperbolic equilibria and hyperbolic periodic orbits. Based on the construction of a so-called Lin orbit, that is a sequence of continuous partial orbits that only have jumps in a certain prescribed linear subspace, estimates for these jumps are derived. We use the jump estimates to discuss bifurcation equations for homoclinic orbits near heteroclinic cycles between an equilibrium and a periodic orbit (EtoP cycles).

Cocoon bifurcation in three-dimensional reversible vector fields

The cocoon bifurcation is a set of rich bifurcation phenomena numerically observed by Lau (1992 Int. J. Bifurc. Chaos 2 543–58) in the Michelson system, a three-dimensional ODE system describing travelling waves of the Kuramoto–Sivashinsky equation. In this paper, we present an organizing centre of the principal part of the cocoon bifurcation in more general terms in the setting of reversible vector fields on . We prove that in a generic unfolding of an organizing centre called the cusp-transverse heteroclinic chain, there is a cascade of heteroclinic bifurcations with an increasing length close to the organizing centre, which resembles the principal part of the cocoon bifurcation.We also study a heteroclinic cycle called the reversible Bykov cycle. Such a cycle is believed to occur in the Michelson system, as well as in a model equation of a Josephson Junction (van den Berg et al 2003 Nonlinearity 16 707–17). We conjecture that a reversible Bykov cycle is, in its unfolding, an accumulation point of a sequence of cusp-transverse heteroclinic chains. As a first result in this direction, we show that a reversible Bykov cycle is an accumulation point of reversible generic saddle-node bifurcations of periodic orbits, the main ingredient of the cusp-transverse heteroclinic chain.

A note on the construction of Hamiltonian trajectories along heteroclinic chains

We provide a short, simple proof of the existence of Hamiltonian trajectories arbitrarily close to a given chain of heteroclinic orbits connecting ''codimension-one, KAM, whiskered tori''.

Hamilton-Jacobi equation, heteroclinic chains and Arnol'd diffusion in three time scale systems

Interacting systems consisting of two rotators and a point mass near a hyperbolic fixed point are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of quasi periodic motions in phase space is studied via the Hamilton-Jacobi equation. The main result, a high density theorem of invariant tori, is derived by the classical canonical transformation method extending previous results. As an application the existence of long heteroclinic chains (and of Arnol'd diffusion) is proved for systems interacting through a trigonometric polynomial in the angle variables.

A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems

Interacting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasiperiodic motions in phase space is studied via the Lindstedt series regarded as a sum of Feynman graphs and studied with renormalization group techniques based on Eliasson’s work on KAM tori. The result is a strong improvement, compared to our previous results, on the domain of validity of bounds that imply existence of invariant tori, large homoclinic angles, long heteroclinic chains, and drift-diffusion in phase space.

On the Stability of Double Homoclinic Loops

We consider 2-degrees of freedom Hamiltonian systems with an involutive symmetry and a pair of orbits bi-asymptotic (homoclinic) to a saddle-center equilibrium (related to pairs of pure real, ±ν, and pure imaginary eigenvalues, ±ω i). We show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one-dimensional scattering problem and ω/ν. We also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop.