Construction of a natural partition of incomplete horseshoes

We present a method for constructing a partition of an incomplete horseshoe in a Poincare map. The partition is based only on the unstable manifolds of the outermost fixed points and eventually their limits. Consequently, this partition becomes natural from the point of view of asymptotic scattering observations. The symbolic dynamics derived from this partition coincides with the one derived from the hierarchical structure of the singularities of the scattering functions.     

Symbolic Dynamics of the Forced Brusselator

The symbolic dynamics of the periodically forced Brusselator for certain parameters is established, based on the binary partition determined from primary tangencies between forward and backward foliations of the Poincaré map. The ordering rule for forward and backward sequences is obtained, and the admissibility condition for allowed sequences is discussed. Some allowed periodic orbits are determined and numerically verified. A method to construct allowed chaotic sequences is proposed.     

Symbolic Analysis of NMR-Laser Chaos

For the extended Bloch-type model of the NMR laser, a binary partition is determined directly from tangencies between forward and backward foliations of the Poincar6 map. Both forward and backward foliations are well ordered according to their symbolic sequences as those of the H6non map with a positive Jacobian. Based on the symbolic dynamics of the H6non map the admissibility condition for allowed sequences is derived and admissible periods are obtained, among which some were formally regarded as forbidden. The allowed periodic orbits are numerically verified. Allowed chaotic sequences are also constructed.     

A Geometric Criterion for Positive Topological Entropy¶II: Homoclinic Tangencies

In a series of important papers [GS1,GS2] Gavrilov and Shilnikov established a topological conjugacy between a surface diffeomorphism having a dissipative hyperbolic periodic point with certain types of quadratic homoclinic tangencies and the full shift on two symbols, thus exhibiting horseshoes near a tangential homoclinic point. In this note, which should be viewed of as an addendum to [BW] we extend this result by showing that such a diffeomorphism with a one-sided isolated homoclinic tangency having any order contact, possible with infinite order contact, possesses a horseshoe near the homoclinic point. Received: 2 March 1999 / Accepted: 14 May 1999     

Determination of Partition Lines from Dynamical Foliations for the Hénon Map

The binary partition lines for the Hhnon map are numerically constructed from tangencies between the contracting and expanding foliations. The ordering of foliations acording to their symbolic sequences are examined.     

Truncated horseshoes and formal languages in chaotic scattering

In this paper we study parameter families of truncated horseshoes as models of multiscattering systems which show a transition to chaos without losing hyperbolicity, so that the topological features of the transition are completely describable by a parametrized family of symbolic dynamics. At a fixed parameter value the corresponding horseshoe represents the set of orbits trapped in the scattering region. The bifurcations are a pure boundary effect and no other bifurcations such as saddle center bifurcations occur in this transition scenario. Truncated horseshoes actually arise in concrete potential scattering under suitable conditions. It is shown that a simple scattering model introduced earlier can realize this scenario in a certain parameter range (the "truncated sawshoe"). For this purpose, we solve the inverse scattering problem of finding the central potential associated to the sawshoe model. Furthermore, we review classification schemes for the transition to chaos of truncated horseshoes originating from symbolic dynamics and formal language theory and apply them to the truncated double horseshoe and the truncated sawshoe.     

Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system

Border-collision bifurcations arise when the periodic trajectory of a piecewise-smooth system under variation of a parameter crosses into a region with different dynamics. Considering a three-dimensional map describing the behavior of a DC/DC power converter, the Letter discusses a new type of border-collision bifurcation that leads to the birth of a “bilayered torus”. This torus consists of the union of two saddle cycles, their unstable manifolds, and a stable focus cycle. When changing the parameters, the bilayered torus transforms through a border-collision bifurcation into a resonance torus containing the stable cycle and a saddle. The Letter also presents scenarios for torus destruction through homoclinic and heteroclinic tangencies.     

Dynamics of three-tori in a periodically forced navier-stokes flow

Three-tori solutions of the Navier-Stokes equations and their dynamics are elucidated by use of a global Poincare map. The flow is contained in a finite annular gap between two concentric cylinders, driven by the steady rotation and axial harmonic oscillations of the inner cylinder. The three-tori solutions undergo global bifurcations, including a new gluing bifurcation, associated with homoclinic and heteroclinic connections to unstable solutions (two-tori). These unstable two-tori act as organizing centers for the three-tori dynamics. A discrete space-time symmetry influences the dynamics.     

Phase space structure and chaotic scattering in near-integrable systems

We investigate the bifurcation phenomena and the change in phase space structure connected with the transition from regular to chaotic scattering in classical systems with unbounded dynamics. The regular systems discussed in this paper are integrable ones in the sense of Liouville, possessing a degenerated unstable periodic orbit at infinity. By means of a McGehee transformation the degeneracy can be removed and the usual Melnikov method is applied to predict homoclinic crossings of stable and unstable manifolds for the perturbed system. The chosen examples are the perturbed radial Kepler problem and two kinetically coupled Morse oscillators with different potential parameters which model the stretching dynamics in ABC molecules. The calculated subharmonic and homoclinic Melnikov functions can be used to prove the existence of chaotic scattering and of elliptic and hyperbolic periodic orbits, to calculate the width of the main stochastic layer and of the resonances, and to predict the range of initial conditions where singularities in the scattering function are found. In the second example the value of the perturbation parameter at which channel transitions set in is calculated. The theoretical results are supplemented by numerical experiments.     

Wild hyperbolic sets,yet no chance for the coexistence of infinitely many KLUS-simple Newhouse attracting sets

The phenomenon of the coexistence of infinitely many sinks for two dimensional dissipative diffeomorphisms is a result due to Newhouse [Ne1, Ne2]. In fact, for each parameter value at which a homoclinic tangency is formed nondegenerately, there exist intervals in the parameter space containing dense sets of parameter values for which there are infinitely many coexisting sinks (Robinson [R]). The structure of the sinks constructed by Newhouse is limited. Simple Newhouse parameter values are values at which there are infinitely many sinks having some special well defined property concerning the structure. A result due to Tedeschini-Lalli and Yorke [TY] says that the Lebesgue measure of the set of simple Newhouse parameter values is zero when the tangencies are due to the standard affine horseshoe map. It is argued in [TY] and [PR] that a more general derivation of this measure zero result would be desirable. The main result of this paper is that the Lebesgue measure of the set of KLUS-simple parameter values (including the simple Newhouse parameter values) is zero for saddle hyperbolic basic sets forming tangencies.Research in Part supported by Gruppo Nazionale per a Fisica Matermatica, CNR     

Secondary homoclinic bifurcation theorems

We develop criteria for detecting secondary intersections and tangencies of the stable and unstable manifolds of hyperbolic periodic orbits appearing in time-periodically perturbed one degree of freedom Hamiltonian systems. A function, called the "Secondary Melnikov Function" (SMF) is constructed, and it is proved that simple (resp. degenerate) zeros of this function correspond to transverse (resp. tangent) intersections of the manifolds. The theory identifies and predicts the rotary number of the intersection (the number of "humps" of the homoclinic orbit), the transition number of the homoclinic points (the number of periods between humps), the existence of tangencies, and the scaling of the intersection angles near tangent bifurcations perturbationally. The theory predicts the minimal transition number of the homoclinic points of a homoclinic tangle. This number determines the relevant time scale, the minimal stretching rate (which is related to the topological entropy) and the transport mechanism as described by the TAM, a transport theory for two-dimensional area-preserving chaotic maps. The implications of this theory on the study of dissipative systems have yet to be explored. (c) 1995 American Institute of Physics.     

Estimating optimal partitions for stochastic complex systems

Partitions provide simple symbolic representations for complex systems. For a deterministic system, a generating partition establishes one-to-one correspondence between an orbit and the infinite symbolic sequence generated by the partition. For a stochastic system, however, a generating partition does not exist. In this paper, we propose a method to obtain a partition that best specifies the locations of points for a time series generated from a stochastic system by using the corresponding symbolic sequence under a constraint of an information rate. When the length of the substrings is limited with a finite length, the method coincides with that for estimating a generating partition from a time series generated from a deterministic system. The two real datasets analyzed in Kennel and Buhl, Phys. Rev. Lett. 91, 084102 (2003), are reanalyzed with the proposed method to understand their underlying dynamics intuitively.     

Globally enumerating unstable periodic orbits for observed data using symbolic dynamics

The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an observed time series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from observed time series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.     

Validity of threshold-crossing analysis of symbolic dynamics from chaotic time series

A practical and popular technique to extract the symbolic dynamics from experimentally measured chaotic time series is the threshold-crossing method, by which an arbitrary partition is utilized for determining the symbols. We address to what extent the symbolic dynamics so obtained can faithfully represent the phase-space dynamics. Our principal result is that such a practice can lead to a severe misrepresentation of the dynamical system. The measured topological entropy is a Devil's staircase-like, but surprisingly nonmonotone, function of a parameter characterizing the amount of misplacement of the partition.     

Symbolic dynamics and topological entropy at the onset of pruning

The topological entropy and pruning rules are investigated for two-dimensional smooth maps at the onset of pruning. Typically the difference of the parameter-dependent topological entropy from its maximum value increases with a power law. Superimposed on this decrease, there are periodic or quasiperiodic oscillations on a logarithmic scale. Both, the scaling exponent and the periodicity are determined by the Lyapunov exponents of the first pruned orbit and the minimal number of letters in the alphabet of the symbolic dynamics. If, at the onset of pruning, the averaged Lyapunov exponent is sufficiently large and the first pruned orbit is homoclinic, the entropy function of area-preserving maps exhibits a series of plateaux. On the plateaux, the symbolic dynamics can be described by finitely many finite forbidden words. There is a series of plateaux which, in different systems, can be described by the same type of forbidden words.     

Periodic Orbit Quantization: How to make Semiclassical Trace Formulae Convergent

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of the periodic orbit signal are the semiclassical eigenvalues, and are determined by either linear predictor, Padé approximant, or signal diagonalization. The second method is based upon the direct application of the Padé approximant to the periodic orbit sum. The Padé approximant allows the resummation of the, typically exponentially, divergent periodic orbit terms. Both techniques do not depend on the existence of a symbolic dynamics, and can be applied to bound as well as to open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard.     

Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems

Nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated. We study the bifurcation sequences of the mixed-mode oscillations generated by the homoclinicity, which are shown to belong to two different types, depending on the nature of the Liapunov numbers of the basic periodic orbit. A detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation. Chaotic cool flame oscillations are also observed in the same model. When the control parameter crosses a critical value, this chaotic set of trajectories becomes globally unstable and forms a Cantor-like hyperbolic repellor, and the ignition mechanism generates ahomoclinic tangency to the Cantor set of trajectories. The complex bifurcation diagram may be globally reconstructed from a one-dimensional dynamical system, thanks to the strong contractivity of thermokinetics. It is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system.     

Routes to bursting in a periodically driven oscillator

The dynamical behaviors of a periodic excited oscillator with multiple time scales in the form that order gap exists between the frequency of the excitation and the natural frequency, are investigated in this Letter. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamics. Different types of bursting phenomena, such as fold/Hopf bursting, fold/Hopf/homoclinic bursting and Hopf/homoclinic bursting, are presented, the mechanism of which is obtained based on the bifurcations of the generalized autonomous system as well as the introduction of the so-called transformed phase portraits. Furthermore, the evolution of the bursting is discussed in details, in which one may find that when the two limit cycles caused by the Hopf bifurcations of the two related equilibrium points interact with each other, homoclinic bifurcation may occur, leading to the merge of the two cycles to form a large amplitude cycle. The homoclinic bifurcation may cause the two asymmetric bursters to merge into a symmetric enlarged burster, in which the large amplitude of the spiking state agrees well with the amplitude of the cycle caused by the homoclinic bifurcation.     

Multi-pulse chaotic dynamics in non-planar motion of parametrically excited viscoelastic moving belt

This paper investigates the multi-pulse global bifurcations and chaotic dynamics for the nonlinear, non-planar oscillations of the parametrically excited viscoelastic moving belt using an extended Melnikov method in the resonant case. Using the Kelvin-type viscoelastic constitutive law and Hamilton's principle, the equations of motion are derived for the viscoelastic moving belt with the external damping and parametric excitation. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:1 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics. The paper demonstrates how to employ the extended Melnikov method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear non-planar oscillations of the viscoelastic moving belt, the Shilnikov-type multi-pulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that the chaos for the Smale horseshoe sense in motion exists.