Heteroclinic Tangles Research Articles

Exploration of non-resonant divertor features on the Compact Toroidal Hybrid

Non-resonant divertors (NRDs) separate the confined plasma from the surrounding plasma facing components (PFCs). The resulting striking field line intersection pattern on these PFCs is insensitive to plasma equilibrium effects. However, a complex scrape-off layer (SOL), created by chaotic magnetic topology in the plasma edge, connects the core plasma to the PFCs through varying magnetic flux tubes. The Compact Toroidal Hybrid (CTH) serves as a test-bed to study this by scanning across its inductive current. Simulations observe a significant change of the chaotic edge structure and an effective distance between the confined plasma and the instrumented wall targets. The intersection pattern is observed to be a narrow helical band, which we claim is a resilient strike line pattern. However, signatures of finger-like structures, defined as heteroclinic tangles in chaotic domains, within the plasma edge connect the island chains to this resilient pattern. The dominant connection length field lines intersecting the targets are observed via heat flux modeling with EMC3-EIRENE. At low inductive current levels, the excursion of the field lines resembles a limited plasma wall scenario. At high currents, a private flux region is created in the area where the helical strike line pattern splits into two bands. These bands are divertor legs with distinct SOL parallel particle flow channels. The results demonstrate the NRD strike line pattern resiliency within CTH, but also show the underlying chaotic edge structure determining if the configuration is diverted or limited. This work supports future design efforts for a mechanical structure for the NRD.

Chaotic advection in a recirculating flow: Effect of a fluid-flexible-solid interaction.

The present work is on laminar recirculating flow-induced deformation as well as motion of a neutrally buoyant flexible elliptical solid, resulting in Lagrangian chaos in a two-dimensional lid-driven cavity flow. Using a fully Eulerian and monolithic approach-based single-solver for the fluid flow and flexible-solid deformation, a chaotic advection study is presented for various aspect ratios β ( =0.5-1.0) and a constant volume fraction Φ=10% of an elliptical solid at a constant Ericksen number Er=0.05 and Reynolds number Re=100. Our initial analysis reveals maximum chaotic advection at β=0.5 for which a comprehensive nonlinear dynamical analysis is presented. The Poincaré map revealed elliptic islands and chaotic sea in the fluid flow. Three large elliptic islands, apart from certain smaller islands, were identified near the solid. Periodic point analysis revealed the lowest order hyperbolic/elliptic periodic points to be three. Adaptive material tracking gave a physical picture of a deforming material blob revealing its exponential stretch along with steep folds and demonstrated unstable/stable manifolds corresponding to lowest order hyperbolic points. Furthermore, adaptive material tracking demonstrates heteroclinic connections and tangles in the system that confirm the existence of chaos. For the transient as compared to the periodic flow, adaptive material tracking demonstrates a larger exponential increase of the blob's interfacial area. The finite-time Lyapunov exponent field revealed attracting/repelling Lagrangian coherent structures and entrapped fluid zones. Our work demonstrates an immersed deformable solid-based onset of chaotic advection, for the first time in the literature, which is relevant to a wide range of applications.

Rank-one strange attractors <i>versus</i> heteroclinic tangles

<p style='text-indent:20px;'>We present two different mechanisms for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. The unfolding of each parameter is associated to the emergence of different dynamical scenarios (rank-one attractors and heteroclinic tangles). These relative positions may be determined using the Melnikov method.</p><p style='text-indent:20px;'>Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "lar-ge" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and non-uniformly hyperbolic attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a <i>prevalent</i> phenomenon.</p>

Localization and snaking in axially compressed and internally pressurized thin cylindrical shells

Abstract This paper uncovers new manifestations of the homoclinic snaking mechanism in the post-buckling regime of a pressurized thin cylindrical shell under axial load. These new forms tend to propagate either wholly or partially in a direction that is orthogonal to the direction of the applied load and so, unlike earlier forms in Woods & Champneys (1999, Heteroclinic tangles in the unfolding of a degenerate Hamiltonian Hopf bifurcation. Phys. D, 129, 147–170), are fundamentally 2D in nature. The main effect of internal pressurization on the snaking mechanism is firstly to transition the circumferential multiplication of buckles from a one-tier pattern to a three-tier pattern. Secondly, internal pressurization can induce oblique snaking, whereby the sequential multiplication of buckles occurs in a helical pattern across the cylinder domain. For low levels of internal pressure, the single dimple remains—as in the unpressurized case—the unstable edge state that forms the smallest energy barrier around the stable pre-buckling equilibrium. For greater levels of pressure, the edge state changes to a single dimple surrounded by four smaller dimples. By tracing the limit point that denotes the onset of these edge states in the parameter space of internal pressure and axial load, we justify and validate the empirically derived design guideline for buckling of pressurized cylinders proposed by Fung & Sechler (1957, Buckling of thin-walled circular cylinders under axial compression and internal pressure. J. Aeronaut. Sci., 24, 351–356).

Experimental inference of flux tunneling between magnetic island chains in tokamaks

We report confinement bifurcations due to flux tunneling in magnetic islands in toroidal plasmas. This occurs in 2/1 islands, triggered by decoupling of non-overlapping adjacent 5/2 islands. While subject to electron cyclotron current drive (ECCD), the temperature at the 2/1 O-point is flat before but peaks after decoupling. This is shown to be due to a bifurcation from stochastic to nested magnetic structure, caused by the removal of flux tunneling through intersecting manifolds of overlapping heteroclinic tangles. 2/1 island stability correlates with coupling events in an ITER baseline scenario plasma where ECCD can not stabilize the coupled islands, showing the critical impact of flux tunneling on disruption avoidance in tokamaks.

The Role of the Saddle-Foci on the Structure of a Bykov Attracting Set

We consider a one-parameter family $$(f_\lambda )_{\lambda \, \geqslant \, 0}$$ of symmetric vector fields on the three-dimensional sphere whose flows exhibit a heteroclinic network between two saddle-foci inside a global attracting set. More precisely, when $$\lambda = 0$$, there is an attracting heteroclinic cycle between the two equilibria which is made of two 1-dimensional connections together with a 2-dimensional sphere which is both the stable manifold of one saddle-focus and the unstable manifold of the other. After slightly increasing the parameter while keeping the 1-dimensional connections unaltered, the two-dimensional invariant manifolds of the equilibria become transversal, and thereby create homoclinic and heteroclinic tangles. It is known that these newborn structures are the source of a countable union of topological horseshoes, which prompt the coexistence of infinitely many sinks and saddle-type invariant sets for many values of $$\lambda $$. We show that, for every small enough positive parameter $$\lambda $$, the stable and unstable manifolds of the saddle-foci and those infinitely many horseshoes are contained in the global attracting set of $$f_\lambda $$; moreover, the horseshoes belong to the heteroclinic class of the equilibria. In addition, we show that the set of chain-accessible points from either of the saddle-foci is chain-stable and contains the closure of the invariant manifolds of the two equilibria.

Dynamics of heteroclinic tangles with an unbroken saddle connection

This paper studies a second-order differential equation with two heteroclinic solutions to two saddle fixed points. When an equation is periodically perturbed, one heteroclinic solution generates tangle while the other remains unbroken. We illustrate chaotic dynamics in the sense of Smale horseshoes and Henon-like attractors with SRB measures. More explicitly, we obtain three different dynamical phenomena, namely the transient heteroclinic tangles containing no physical measures, heteroclinic tangles dominated by sinks representing stable dynamical behavior, and heteroclinic tangles with Henon-like attractors admitting SRB measures representing chaos. We also demonstrate that three types of phenomena repeat periodically as the forcing magnitude goes to zero.

Heteroclinic tangles in time-periodic equations

In this paper we prove that, when a heteroclinic loop is periodically perturbed, three types of heteroclinic tangles are created and they compete in the space of μ where μ is a parameter representing the magnitude of the perturbations. The three types are (a) transient heteroclinic tangles containing no Gibbs measures; (b) heteroclinic tangles dominated by sinks representing stable dynamical behavior; and (c) heteroclinic tangles with strange attractors admitting SRB measures representing chaos. We also prove that, as μ→0, the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections. The theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical Birkhoff–Melnikov–Smale method.

From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow

AbstractWe study numerically a succession of transitions in pipe Poiseuille flow that lead from simple travelling waves to waves with chaotic time-dependence. The waves at the origin of the bifurcation cascade are twofold azimuthally periodic, shift–reflect symmetric, and have a non-dimensional axial wavelength of $1. 927$ diameters. As the Reynolds number is increased, successive transitions result in a wide range of time-dependent solutions that include spiralling, modulated travelling, modulated spiralling, doubly modulated spiralling and mildly chaotic waves. Numerical evidence suggests that the latter spring from heteroclinic tangles of the stable and unstable invariant manifolds of two shift–reflect symmetric, modulated travelling waves. The chaotic set thus produced is confined to a limited range of Reynolds numbers, bounded by the occurrence of manifold tangencies. The subspace of discrete symmetry to which the states studied here belong makes many of the bifurcation and path-following investigations presented readily accessible. However, we expect that most of the phenomenology carries over to the full state space, thus suggesting a mechanism for the formation and break-up of invariant states that can give rise to chaotic dynamics.

Partitioning two-dimensional mixed phase spaces

Hamiltonian systems typically exhibit a mixture of chaos and regularity, complicating any scheme to partition phase space and extract a symbolic description of the dynamics. In particular, the dynamics in the vicinity of stable islands can exhibit complicated topology that is qualitatively distinct from that away from the islands. We develop an approach to partition the chaotic phase space of a general dynamical system represented by a two-dimensional map (homeomorphism). This approach can accommodate mixed phase space structure with an arbitrarily high degree of accuracy. The partitioning scheme is built around networks of nested heteroclinic tangles—fundamental geometric objects that organize phase–space transport. These tangles can be used to progressively approximate the dynamics in the vicinity of stable island chains. The net result is a symbolic approximation to the dynamics in the chaotic sea, and an associated phase–space partition, which includes the influence of stable islands and which is approximately Markov.

Dynamics of a Hysteretic Relay Oscillator with Periodic Forcing

The dynamics of a hysteretic relay oscillator with simple harmonic forcing is studied in this paper. Even though there are no bounded solutions in the absence of forcing, periodic excitation gives rise to more complex responses including periodic, quasi-periodic, and chaotic behavior. A Poincaré map is introduced to facilitate mathematical analysis. Families of period-one solutions are determined as fixed points of the Poincaré map. These represent coexisting subharmonic responses. Conditions on the amplitude and frequency of the forcing for the existence of periodic solutions have been obtained. Linear stability analysis reveals that these solutions can be classified as centers or saddles. The presence of higher periodic or quasi-periodic motions together with homoclinic and heteroclinic tangles implies the existence of chaotic solutions.

The topology of nested homoclinic and heteroclinic tangles

Homoclinic and heteroclinic tangles are fundamental phase space structures that help in organizing the transport defined by a chaotic map on a two-dimensional phase space. Previous work introduced the technique of “homotopic lobe dynamics” to study the topology of a homoclinic tangle. The present work generalizes homotopic lobe dynamics to describe the topology of a network of nested homoclinic and/or heteroclinic tangles. Using this theory, we demonstrate that the topological analysis of a network of nested tangles can lead to an efficient symbolic description of a chaotic phase space, even in the vicinity of islands of stability. By increasing the amount of topological information used (e.g. by introducing additional tangles at finer scales), this technique suggests an efficient route for defining accurate symbolic dynamics for phase spaces that exhibit a mixture of chaos and regularity.

Novel routes to chaos through torus breakdown in non-invertible maps

The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.

Onset of chaotic advection in open flows

In this paper we investigate the transition to chaos in the motion of particles advected by open flows with obstacles. By means of a topological argument, we show that the separation points on the surface of the obstacle imply the existence of a saddle point downstream from the obstacle, with an associated heteroclinic orbit. We argue that as soon as the flow becomes time periodic, these orbits give rise to heteroclinic tangles, causing passively advected particles to experience transient chaos. The transition to chaos thus coincides with the onset of time dependence in open flows with stagnant points, in contrast with flows with no stagnant points. We also show that the nonhyperbolic nature of the dynamics near the walls causes anomalous scalings in the vicinity of the transition. These results are confirmed by numerical simulations of the two-dimensional flow around a cylinder.

Dynamical chaos: systems of classical mechanics

This article is a methodological manual for those who are interested in chaotic dynamics. An exposition is given on the foundations of the theory of deterministic chaos that originates in classical mechanics systems. Fundamental results obtained in this area are presented, such as elements of the theory of nonlinear resonance and the Kolmogorov–Arnol'd–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and the Mel'nikov method. Particular attention is given to the analysis of the phenomena underlying the self-similarity and nature of chaos: splitting of separatrices and homoclinic and heteroclinic tangles. Important properties of chaotic systems — unpredictability, irreversibility, and decay of temporal correlations — are described. Models of classical statistical mechanics with chaotic properties, which have become popular in recent years — billiards with oscillating boundaries — are considered. It is shown that if a billiard has the property of well-developed chaos, then perturbations of its boundaries result in Fermi acceleration. But in nearly-integrable billiard systems, excitations of the boundaries lead to a new phenomenon in the ensemble of particles, separation of particles in accordance their velocities. If the initial velocity of the particles exceeds a certain critical value characteristic of the given billiard geometry, the particles accelerate; otherwise, they decelerate.

Observation of the heteroclinic tangles in the heat flux pattern of the ergodic divertor at TEXTOR

A fine structure of open chaotic field lines, namely, a heteroclinic tangle, in the ergodic divertor has been observed by measurements of heat deposition pattern on the divertor plates at TEXTOR. Calculations show that magnetic footprints on the divertor plates are formed by open field lines coming from the plasma along narrow stripe regions called fingers. The latter are determined by the structure of stable and unstable manifolds of the outermost resonant magnetic island. This fact is confirmed by observations of the bifurcations of the heat flux pattern on the divertor plates with changing edge safety factor.

Reconnection of Unstable/Stable Manifolds of the Harper Map: -- Asymptotics-Beyond-All-Orders Approach --

The Harper map is one of the simplest chaotic systems exhibiting reconnection of invariant manifolds. The method of asymptotics beyond all orders (ABAO) is used to construct unstable/stable manifolds of the Harper map. By enlarging the neighborhood of a singularity, the perturbative solution of the unstable manifold is expressed as a Borel summable asymptotic expansion in a sector including t = -∞ and is analytically continued to the other sectors, where the solution acquires new terms describing heteroclinic tangles. It is shown that when the parameter is changed, upon reaching the reconnection threshold, the unstable/stable manifolds acquire new oscillatory parts corresponding to the heteroclinic tangle after the reconnection.

Heteroclinic behavior in a dripping faucet experiment

Heteroclinic tangles are ubiquitous and responsible for a large number of complex behaviors, like catastrophic bifurcations. We have found evidences of the presence of a heteroclinic scenario in the dripping faucet experiment by applying topological analysis. We have measured the time ( T n ) between the nth and ( n+1)th drops, and we have reconstructed the phase space of the system by plotting T n+1 vs. T n . Two control parameters were used, the faucet opening and the inclination ( θ) of the faucet measured from vertical position. We have already shown, for θ=0 (vertical) by varying the faucet opening, a boundary crisis known as chaotic blue sky catastrophe in which there is a sudden change of the attractive basins. For values of θ≠0° the system symmetry is broken and we have observed, even for small angles ( θ∼5°), strong changes in the attractors topology when compared with those obtained with the nozzle in the vertical position. Instead of the catastrophe, a smooth transition from a chaotic attractor to another one takes place, and it is possible to observe the attraction basins merged. We also have shown evidences of connections between the manifolds of two saddle points, resulting in a heteroclinic scenario.

A simple feedback control system: Bifurcations of periodic orbits and chaos

We consider a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small time constant, so that the feedback control system can be approximated by a periodically forced oscillator. It was previously shown by Melnikov's method that transverse homoclinic and heteroclinic orbits exist and chaos may occur in certain parameter regions. Here we study local bifurcations of harmonics and subharmonics using the second-order averaging method and Melnikov's method. The Melnikov analysis was performed by numerically computing the Melnikov functions. Numerical simulations and experimental measurements are also given and are compared with the previous and present theoretical predictions. Sustained chaotic motions which result from homoclinic and heteroclinic tangles for not only single but also multiple hyperbolic periodic orbits are observed. Fairly good agreement is found between numerical simulation and experimental results.

Dynamos in weakly chaotic two-dimensional flows

Abstract The dynamo action of a time-periodic two-dimensional flow close to integrability is analyzed. At fixed Reynolds number R M and frequency ω, magnetic structures develop in the form of both eddies and filaments. The growth rate of the eddies appears to be the same for all frequencies and decreases with R M, while the growth rate of the filaments displays a strong co-dependence and, except in the limit of zero or infinite frequencies, converges to a non-zero value as R M → ∞. Magnetic filaments develop in the widest chaotic zones located near the homoclinic or heteroclinic tangles, and their growth rate is strongly influenced by the width of these zones which is estimated using Melnikov formalism. This study illustrates quantitatively that not only a local stretching but also a sizable chaotic zone is required for fast dynamo action.