Homoclinic Tangles Research Articles

Global bifurcation and chaos from automatic gain control loops

The dynamics of the automatic gain control (AGC) loops which employ variable-gain amplifiers with a square control law are studied. The selected loop filters are of second-order low-pass type. The existence of global homoclinic bifurcation and Smale horseshoe chaos are proved without the traditional small-signal model assumption. To investigate systems that are monitored during discrete-time intervals, sampled-data AGC loops are also considered. Under the same square gain-control law, the first-and second-order loop filters are studied. The period-doubling routes to chaos are verified for the first-order systems, and Hopf bifurcations for the second-order filters. Simulations are performed and the resultant homoclinic tangles, chaotic time waveforms, bifurcation diagrams, and Lyapunov exponents are illustrated to demonstrate the theoretical results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Resonant normal forms, interpolating Hamiltonians and stability analysis of area preserving maps

The geometrical and dynamical properties of area preserving maps in the neighborhood of an elliptic fixed point are analyzed in the framework of resonant normal forms. The interpolating flow is not obtained from a map tangent to the identity, but from the normal form of the given map and a time independent interpolating Hamiltonian H is introduced. On this Hamiltonian the local stability properties of the fixed point and the geometric structure of the orbits are transparent. Numerical agreement between the level lines of H and the orbits of the map suggests that the perturbative expansion of H is asymptotic. This is confirmed by a rigorous error analysis, based on majorant series: the error for the normal form expansion grows as n! while the truncation error for H also has a factorial growth and in a disc of radius r can be made exponentially small with 1/ r. The boundary of the global stability domain is considered; for the quadratic map the identification with the inner envelope of the homoclinic tangle of the hyperbolic fixed point is strongly suggested by numerical evidence.

Dynamics associated with a quasiperiodically forced Morse oscillator: Application to molecular dissociation.

The dynamics associated with a quasiperiodically forced Morse oscillator is studied as a classical model for molecular dissociation under external quasiperiodic electromagnetic forcing. The forcing entails destruction of phase-space barriers, allowing escape from bounded to unbounded motion. In contrast to the ubiquitous Poincar\'e map reduction of a periodically forced system, we derive a sequence of nonautonomous maps from the quasiperiodically forced system. We obtain a global picture of the dynamics, i.e., of transport in phase space, using a sequence of time-dependent two-dimensional lobe structures derived from the invariant homoclinic tangle of a persisting invariant saddle-type torus in a Poincar\'e section of an associated autonomous system phase space. Transport is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, and this provides the framework for studying basic features of molecular dissociation in the context of classical phase-space trajectories. We obtain a precise criterion for discerning between bounded and unbounded motion in the context of the forced problem. We identify and measure analytically the flux associated with the transition between bounded and unbounded motion, and study dissociation rates for a variety of initial phase-space ensembles, such as an even or weighted distribution of points in phase space, or a distribution on a particular level set of the unperturbed Hamiltonian (corresponding to a quantum state). A double-phase-slice sampling method allows exact numerical computation of dissociation rates. We compare single- and two-frequency forcing. Infinite-time average flux is maximal in a particular single-frequency limit; however, lobe penetration of the level sets of the unperturbed Hamiltonian can be maximal in the two-frequency case. The variation of lobe areas in the two-frequency problem gives one added freedom to enhance or diminish aspects of phase-space transport on finite time scales for a fixed infinite-time average flux, and for both types of forcing the geometry of lobes is relevant. The chaotic nature of the dynamics is understood in terms of a traveling horseshoe map sequence.

Quantum signatures of homoclinic tangles and separatrices.

The influence of homoclinic tangles and separatrices on the eigenstates and quasienergies of a quantized, nonintegrable, piecewise-linear standard map is investigated both analytically and numerically. The strength of scarring on hyperbolic fixed points is found to depend strongly on the phase-space area covered by the corresponding homoclinic tangles. It is particularly pronounced when the tangle degenerates to an unbroken separatrix and is seen to decrease with decreasing {h bar}. In the case of the unbroken separatrix, an interesting resonance phenomenon, which occurs for certain values of {h bar}, is also discussed. This effect is related to a degeneracy in the quasienergy differences for a triplet of states which lie energetically below, on, and above the saddle. This degeneracy is shown to give rise to periodic recurrences for a coherent state initially placed at the hyperbolic fixed point. The relevant time scale {ital T} is calculated semiclassically and exhibits the asymptotic behavior {ital T}{similar to}ln{h bar}. The implications of these results to other systems is also discussed.

Singularities in vibro-impact dynamics

The non-differentiable nature of vibro-impact dynamics can lead to breakdown of the global stable manifold theorem applicable to smooth dynamical systems. In one dimensional periodically excited system, the breakdown leads to the “shredding” of the stable manifolds, and this process is analyzed in detail. Bodily translation of filaments of stable manifolds can lead to homoclinic intersections, and this topic is discussed in the context of a new bifurcation due to “re-entrance”. Shredding leads to a strong similarity between the geometries of the singular subspaces and the geometries of homoclinic tangles and strange attracting sets.

Lobe dynamics and the escape from a potential well

Periodically-forced nonlinear oscillators that permit escape from a potential well frequently possess unstable periodic orbits whose invariant manifolds are homoclinically tangled. The trellises formed by such overlapping manifolds separate the Poincaré plane into regions (calledlobes) whose fates at successive time periods are amenable to analysis. The lobe configurations observed in simple escape systems are discussed, together with their interrelation with the location of periodic orbits and other invariant sets. Three applications of this procedure are illustrated: (i) a numerical technique for localizing the non-wandering set; (ii) a demonstration that the existence of a saddle possessing aboundedhomoclinically tangled unstable manifold branch implies the existence of safe open regions in the global basin of attraction; (iii) the proposal of definitions of global integrity and escape times which may be used to relate safe basin erosion to the evolution of a homoclinic tangle.

Homoclinic chaos for ray optics in a fiber

In the geometrical optics approximation, stable and unstable manifolds of periodic orbits, invariant tori, and hyperbolic invariant manifolds are shown to exist and produce trapping of bundles of light rays near the axis of a translation-invariant, axisymmetric optical fiber whose squared refractive index is a parabolic function of squared radius. Periodic symmetry-breaking perturbations in the refractive index are shown to destroy this ray trapping and to produce homoclinic tangles, through which nearby trapped rays may escape, and untrapped rays may become at least temporarily trapped. Melnikov's technique is used to prove that the perturbations cause the stable and unstable manifolds of the unperturbed periodic orbits, invariant tori, and hyperbolic invariant manifolds to develop transverse intersections and therefore, to form homoclinic tangles. These tangles imply either homoclinic chaos by the Smale horseshoe mechanism and the Poincaré-Birkhoff-Smale theorem, or Arnol'd diffusion. In both situations, lobe dynamics will dominate phase space transport, seen here as a flux of (initially) untrapped rays passing through the trapping region.

Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems

The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles.

Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory

The universal description of orbits in the domain swept by a slowly varying separatrix is provided through a symplectic map derived by means of an extension of classical adiabatic theory. This map connects action-angle-like variables of an orbit when far from the instantaneous separatrix to time-energy variables at a reference point of the orbit very close to the corresponding separatrix. When the separatrix pulsates periodically with a small frequency epsilon , the authors combine this map with WKB theory to obtain a description of the structure underlying chaos: the homoclinic tangle related to the hyperbolic fixed point whose separatrix is pulsating. For each extremum of the area within the pulsating separatrix, an initial branch of length O(1/ epsilon ) of the stable manifold is explicitly constructed, and makes O(1/ epsilon ) transverse homoclinic intersections with a similar branch of the unstable manifold.

Transport through chaos

Certain orbits of area preserving maps of the plane appear to wander randomly and to densely fill regions of the plane having positive Lebesgue measure. These regions are called ergodic zones. Trellises (or homoclinic tangles) embedded within these zones are shown to guide the transport of ensembles of points. Trellises or tangles can be localized within what are called resonance zones. Transport through resonance zones is calculated using the MacKay-Meiss-Percival action principle.

Classical and quantal morphology of a piecewise-linear standard map.

We consider a standard map with a continuous, piecewise-linear forcing term. The piecewise linearity allows for an extremely accurate calculation of the classical manifolds and associated homoclinic tangles. A simple coding for the periodic orbits is another useful property of the map. The quantized map is then studied and the resolution of stable and unstable invariant classical structures by the quantum eigenstates is assessed.

Geometry and dynamics of stable and unstable cylinders in Hamiltonian systems

Stable and unstable manifolds in phase space with an S 1 × R 1 (cylindrical) geometry are shown to exist for certain two degree of freedom Hamiltonian systems. Specific attention is given to Hamiltonian systems with potential barriers, although the concepts developed are more general. The existence of these cylinders is independent of the nature of the Hamiltonian dynamics (i.e. regular or chaotic). A detailed discussion is given where we show that appropriate Poincaré sections of the cylinders yield a map structure (which we denote as “reactive islands”) that is distinct from the usual homoclinic tangle. The cylinders have the physical property that all motion across a barrier must occur through the interior of these surfaces. The cylinders thus mediate the reaction dynamics. A kinetic mechanism based upon the properties of the cylinders is developed and tested against several numerical simulations of the reaction dynamics of a model Hamiltonian system. The threshold limiting form of the standard theory of microcanonical reaction rates is derived.

New fixed point of the renormalisation operator associated with the recurrence of invariant circles in generic Hamiltonian maps

The phenomenon of 'recurrent KAM tori' in twist maps is studied from a renormalisation point of view. A fixed point of the renormalisation operator for invariant circles, related to this phenomenon, is discussed. Connected to the fixed point are a three-cycle and a six-cycle. It is shown that for both the fixed point and the six-cycle there are homoclinic tangles giving rise to a basin of attraction of the simple fixed point with a very complicated shape. A piecewise-linear map is also studied. Even though the behaviour here seems related to the analytic case, it is found that in this case there is a fixed point, a family of two-cycles and a three-cycle for the renormalisation operator. The crossover behaviour is studied.

Fractal control boundaries of driven oscillators and their relevance to safe engineering design

When a metastable, damped oscillator is driven by strong periodic forcing, the catchment basin of constrained motions in the space of the starting conditions { x (0), ẋ (0)} develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds. The four-dimensional basin in the phase-control space spanned by { x (0), ẋ )(0), F , ω }, where F is the magnitude and ω the frequency of the excitation, will likewise acquire a fractal boundary, and we here explore the engineering significance of the control cross section corresponding, for example, to x (0) = ẋ (0) = 0. The fractal boundary in this section is a failure locus for a mechanical or electrical system subjected, while resting in its ambient equilibrium state, to a sudden pulse of excitation. We assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well. Absolute and transient basins are examined, giving control-space maps analogous to familiar pictures of the Mandelbrot set. Generalizing from this prototype study, it is argued that in engineering design, against boat capsize or earthquake damage, for example, a study of safe basins should augment, and perhaps entirely replace, conventional analysis of the steady-state attracting solutions.

Integrity measures quantifying the erosion of smooth and fractal basins of attraction

The phase portraits of a dissipative dynamical system are characterized by the existence of one or more stable attractors, which typically include point equilibria, periodic oscillations (harmonic and subharmonic), quasi-periodic solutions and chaotic attractors. Each attractor is embedded in its own domain or basin of attraction, bounded by a separatrix' associated with an unstable saddle solution. Under the variation of a control parameter, as the attractors move and bifurcate, the basins also undergo corresponding changes and metamorphoses. These changes in size and shape are usually continuous but can be discontinuous as when an attractor vanishes, along with its basin, at a saddle-node bifurcation. Associated with the homoclinic tangling of the invariant manifolds of the saddle solution, basin boundaries can also change in nature from smooth to fractal. In this paper, the escape of a driven oscillator from a cubic potential well, as an archetypal example, is used to explore the engineering significance of the basin erosions that occur under increased forcing. Various measures of engineering integrity of the constrained attractor are introduced: a global measure assesses the overall basin area; a local measure assesses the distance from the attractor to the basin boundary; and a velocity measure is related to the size of impulse that could be sustained without failure. Since engineering systems may be subjected to pulse loads of finite duration, attention is given to both the absolute and transient basin boudaries. The significant erosion of these at homoclinic tangencies is particularly highlighted in the present study, the fractal basins having a severely reduced integrity under all three criteria.

Order and Chaos in a Discrete Duffing Oscillator: Implications on Numerical Integration

In this paper, we study the dynamics of some two-dimensional mappings which arise when standard numerical integration schemes are applied to an unforced oscillator with a cubic stiffness nonlinearity, i.e., the Duffing equation. While the continuous time problem is integrable and is solved analytically in terms of Jacobi elliptic functions, the discrete versions of this simple system arising from standard integration schemes exhibit very complicated dynamics due to the presence of homoclinic tangles. We present an alternative scheme for discretizing the nonlinear term which preserves the integrable dynamics of the continuous system and derive analytic expressions for the orbits and invariant curves of the resulting mapping.

Chaotic phenomena triggering the escape from a potential well

This paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold. The aim is to show how the well-known atoms of dissipative dynamics (saddle-node folds, period-doubling flips, cascades to chaos, boundary crises, etc.) assemble to form molecules of overall response (hierarchies of cusps, incomplete Feigenbaum trees, etc.). Particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle, the latter being accurately predicted by a Melnikov analysis. After escape, chaotic transients are shown to conform to recent scaling laws. Analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible.

Basin boundary metamorphoses in the canonical escape equation

A study is made of the escape boundary, in the space of the starting values of displacement and velocity, for a sinusoidally driven nonlinear oscillator. The oscillator has constant inertia and linear viscous damping, with a restoring force that corresponds to a cubic potential well. This cubic form of the total potential energy has the universal canonical form encountered by all metastable mechanical systems as they approach a generic fold catastrophe under a single control parameter. The basin boundary metamorphoses are observed, and an engineering integrity diagram is proposed, quantifying the rapid erosion of area that is triggered under increasing excitation by a homoclinic tangency. The pictures show clearly the fractal basin associated with the developed homoclinic tangle. The boundary crisis, at which the final chaotic attractor loses its stability, is examined in terms of an accessible saddle point, which is here a directly unstable subharmonic of order six.

Forced oscillations of a self-oscillating bimolecular surface reaction model

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-periodic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations.

Chaotic dynamics of a whirling pendulum

A physical system is considered consisting of a rigid frame which is free to rotate about a vertical axis and to which is attached a planar simple pendulum. This system has “one and a half” degrees of freedom due to the fact that the frame and pendulum may freely rotate about the vertical axis, i.e., conservation of angular momentum holds for the “ideal”, or unperturbed, system. Using a Hamiltonian formulation we reduce the unperturbed equations of motion to a conservative planar system in which the constant angular momentum plays the role of a parameter. This system is shown to possess one or two sets of homoclinic motions depending on the level of the angular momentum. When this system is perturbed by external excitations and dissipative forces these homoclinic motions can break into homoclinic tangles providing the conditions for chaotic motions of the horseshoe type to exist. The criteria for this to occur can be formulated using a variation of Melnikov's method developed for slowly varying oscillators [1, 2]. For the present problem, the angular momentum becomes a slowly varying parameter upon addition of the disturbances. These ideas are used to rigorously prove the existence of chaotic motions for this system and to compute, to first order, global bifurcation parameter conditions. Since two types of homoclinic motions can occur, two different chaotic modes of motion can result and physical interpretations of these motions are given. In addition, a limiting case is considered in which the system becomes a single degree of freedom oscillator with parametric excitation.