Showing papers by "Edward Ott published in 1989"
TL;DR: It is found that the set in this parameter plane for which the system exhibits strange nonchaotic attractors has Cantor-like structure and is enclosed between two critical curves, suggesting a possible route to chaos in two-frequency quasiperiodically forced systems.
Abstract: As a model displaying typical features of two-frequency quasiperiodically forced systems, we discuss the circle map with quasiperiodic coupling We present numerical and analytical evidence for the existence of strange nonchaotic attractors, and we use examples to illustrate various types of dynamical behavior that can arise in typical quasiperiodically forced systems We investigate the behavior of the system in the two-dimensional parameter plane of nonlinearity strength versus one of the driving frequencies We find that the set in this parameter plane for which the system exhibits strange nonchaotic attractors has Cantor-like structure and is enclosed between two critical curves One of these curves marks the transition from three-frequency quasiperiodic attractors to strange nonchaotic attractors; the other marks the transition from strange nonchaotic attractors to chaotic attractors This suggests a possible route to chaos in two-frequency quasiperiodically forced systems: (three-frequency quasiperiodicity)-->(strange nonchaotic behavior)-->(chaos)
141 citations
TL;DR: The onset of chaotic behavior in a class of classical scattering problems is shown to occur in two possible ways, one is abrupt and is related to a change in the topology of the energy surface and the other arises as a result of a complex sequence of saddle-node and period doubling bifurcations.
Abstract: The onset of chaotic behavior in a class of classical scattering problems is shown to occur in two possible ways. One is abrupt and is related to a change in the topology of the energy surface. The other arises as a result of a complex sequence of saddle-node and period doubling bifurcations. The abrupt bifurcation represents a new generic route to chaos and yields a characteristic scaling of the fractal dimension associated with the scattering function as (ln({ital E}{sub {ital c}}{minus}E){sup {minus}1}){sup {minus}1}, for particle energies {ital E} near the critical value {ital E}{sub {ital c}} at which the scattering becomes chaotic.
113 citations
TL;DR: Critical behavior associated with intermittent temporal bursting accompanying the sudden widening of a chaotic attractor was observed and investigated experimentally in a gravitationally buckled, parametrically driven, magnetoelastic ribbon.
Abstract: Critical behavior associated with intermittent temporal bursting accompanying the sudden widening of a chaotic attractor was observed and investigated experimentally in a gravitationally buckled, parametrically driven, magnetoelastic ribbon. As the driving frequency, f, was decreased through the critical value, ${f}_{c}$, we observed that the mean time between bursts scaled as \ensuremath{\Vert}${f}_{c}$-f${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$. .AE
79 citations
TL;DR: In this paper, the capacity dimension of these attractors is two while their information dimension is one, and it is shown that the capacity and information dimension of the attractors are equal.
73 citations
TL;DR: The present paper relates the stretching properties of the flow to the resulting fractal dimension spectrum by exploiting the kinematic magnetic dynamo problem and recent experiments which demonstrate the possibility of measuring the fractaldimension of the gradient squared of convected passive scalars.
Abstract: The passive convection of vector fields and scalar functions by a prescribed incompressible fluid flow v(x,t) is considered for the case where v(x,t) is chaotic. By chaotic v(x,t) it is meant that typical nearby fluid elements diverge from each other exponentially in time. It is shown that in such cases, as time increases, a convected vector field and the gradient of a convected scalar will generally concentrate on a set which is fractal. The present paper relates the stretching properties of the flow to the resulting fractal dimension spectrum. Motivation for these considerations is provided by the kinematic magnetic dynamo problem (in the vector case) and (in the scalar case) by recent experiments which demonstrate the possibility of measuring the fractal dimension of the gradient squared of convected passive scalars.
70 citations
TL;DR: Etude d'une chaine monodimensionnelle d'oscillateurs non lineaires forces presente le comportement typique de systemes non lineaire, spatialement etendus et forces periodiquement.
Abstract: Etude d'une chaine monodimensionnelle d'oscillateurs non lineaires forces. Ce modele presente le comportement typique de systemes non lineaires, spatialement etendus et forces periodiquement
64 citations
TL;DR: In this article, it was shown that the torus is destroyed when the stable and unstable manifolds of an unstable orbit become tangent and no chaotic orbits lying on a torus were observed.
32 citations
TL;DR: In this article, a theory for first order phase transitions of multifractal chaotic attractors of nonhyperbolic two-dimensional maps is presented, and a complete picture of the behavior associated with the phase transition is obtained.
31 citations
TL;DR: In this article, it was shown that the attractors occur in a Cantor set of positive Lebesque measure in parameter space and have a distinctive frequency spectrum, thus making them potentially observable in experiments.
Abstract: Quasiperiodically forced nonlinear dynamical systems may exhibit strange nonchaotic attractors. We argue that these attractors occur in a Cantor set of positive Lebesque measure in parameter space. Furthermore, we show that strange nonchaotic attractors have a distinctive frequency spectrum thus making them potentially observable in experiments.
21 citations
TL;DR: Multifractal dimension spectra for the stable and unstable manifolds of invariant chaotic sets are studied for the case of invertible two-dimensional maps and it is demonstrated that dimension algorithms based on the Lyapunov partition functions are often very efficient.
Abstract: Multifractal dimension spectra for the stable and unstable manifolds of invariant chaotic sets are studied for the case of invertible two-dimensional maps. A dynamical partition-function formalism giving these dimensions in terms of local Lyapunov numbers is obtained. The relationship of the Lyapunov partition functions for stable and unstable manifolds to previous work is discussed. Numerical experiments demonstrate that dimension algorithms based on the Lyapunov partition functions are often very efficient. Examples supporting the validity of the approach for hyperbolic chaotic sets and for nonhyperbolic sets below the phase transition (q${q}_{T}$) are presented.
20 citations
TL;DR: Analysis of examples of steady dynamos supports the conjecture that, forsufficiently small resistivity, 2 dynamo action can occur in typical, smooth, steady, three-dimensional, chaotic fluid flows.
Abstract: This paper considers the question of the existense of a steady fast kinematic magnetic dynamo for a conducting fluid with a steady velocity field and vanishingly small electrical resistivity. The analysis of examples of steady dynamos, found by considering the zero-resistivity dynamics, indicated that, for sufficiently small resistivity, dynamo action can indeed occur in steady smooth three-dimensional chaotic fluid flows and that fast dynamos should consequently be a typical occurrence for such flows.
TL;DR: It is pointed out that the creation of fractal basin boundaries is a characteristic feature accompanying the intermittency transition to chaos, and the scaling of the dimension of basin boundaries near these intermittency transitions is examined.
Abstract: It is the purpose of this paper to point out that the creation of fractal basin boundaries is a characteristic feature accompanying the intermittency transition to chaos. (Here ''intermittency'' transition is used in the sense of Pomeau and Manneville (Commun. Math. Phys. 74, 189 (1980)); viz., a chaotic attractor is created as a periodic orbit becomes unstable.) In particular, we are here concerned with type-I and type-III intermittencies. We examine the scaling of the dimension of basin boundaries near these intermittency transitions. We find, from numerical experiments, that near the transition the dimension scales with a system parameter /ital p/ according to the power law /ital d//congruent//ital d//sub 0//minus/k/vert bar/p/minus/p/sub I//vert bar//sup /beta// with /beta/=1/2, where /ital d//sub 0/ is the dimension at the intermittency transition parameter value /ital p/=/ital p//sub I/ and /ital k/is a scaling constant. Furthermore, for type-I intermittency/ital d//sub 0//lt/D, while for type-III intermittency /ital d//sub 0/=D,where /ital D/ is the dimension of the space. Heuristic analytic argumentssupporting the above are presented.