Showing papers by "Edward Ott published in 1994"
TL;DR: In this paper, the authors consider situations where a nonlinear dynamical system possesses a smooth invariant manifold and show that for parameter values p less than a critical value p, a blowout bifurcation occurs, in which the former attraction to the manifold changes to repulsion, and the chaotic set in the manifold ceases to be an attractor of the system.
404 citations
TL;DR: This paper analyzes the different types of grazing bifurcations that can occur in a simple sinusoidally forced oscillator system in the presence of friction and a hard wall with which the impacts take place.
Abstract: Impact oscillators demonstrate interesting dynamical features. In particular, new types of bifurcations take place as such systems evolve from a nonimpacting to an impacting state (or vice versa), as a system parameter varies smoothly. These bifurcations are called grazing bifurcations. In this paper we analyze the different types of grazing bifurcations that can occur in a simple sinusoidally forced oscillator system in the presence of friction and a hard wall with which the impacts take place. The general picture we obtain exemplifies universal features that are predicted to occur in a wide variety of impact oscillator systems.
291 citations
TL;DR: A general criterion for the occurrence of border-collision bifurcations is given and illustrative numerical results, including transitions to chaotic attractors, are presented.
Abstract: Recently physical and computer experiments involving systems describable by continuous maps that are nondifferentiable on some surface in phase space have revealed novel bifurcation phenomena. These phenomena are part of a rich new class of bifurcations which we call border-collision bifurcations. A general criterion for the occurrence of border-collision bifurcations is given. Illustrative numerical results, including transitions to chaotic attractors, are presented. These border-collision bifurcations are found in a variety of physical experiments.
272 citations
TL;DR: The use of chaos to transmit information is demonstrated experimentally by use of extremely small perturbing current pulses in a chaotic electrical oscillator to carry a prescribed message.
Abstract: The use of chaos to transmit information is demonstrated experimentally. The symbolic dynamics of a chaotic electrical oscillator is controlled to carry a prescribed message by use of extremely small perturbing current pulses.
250 citations
Book•
01 Jan 1994
TL;DR: In this article, the authors present an analysis of data from a chaotic system, including prediction, filtering, control and communication in a cascade of periodic orbits and Symbolic Dynamics, and the theory of Stabilization of Unstable Orbits.
Abstract: BACKGROUND. Dimension. Symbolic Dynamics. Lyapunov Exponents and Entropy. The Theory of Embedding. ANALYSIS OF DATA FROM CHAOTIC SYSTEMS. The Practice of Embedding. Dimension Calculations. Calculation of Lyapunov Exponents. Periodic Orbits and Symbolic Dynamics. PREDICTION, FILTERING, CONTROL AND COMMUNICATION IN CHAOTIC SYSTEMS. Prediction. Noise Reduction. Control: Theory of Stabilization of Unstable Orbits. Control: Experimental Stabilization of Unstable Orbits. Control: Targeting and Goal Dynamics. Synchronism and Communication. Bibliography. Index.
156 citations
TL;DR: In this paper, the authors consider the dynamics near the transition to chaotic attractors and obtain the characteristic behaviors near this transition using a simple analyzable model, consistent with the conjecture that these results are universal for the class of systems considered.
150 citations
TL;DR: In this paper, the authors point out that exact synchronism may also occur for a large class of systems that are not replicas of part of the original system, and discuss the possibility of using this freedom to choose synchronizer systems with improved performance.
Abstract: Recent work has considered the situation where a state variable (or variables) of a chaotically evolving system is used as an input to a replica of part of the original system. It was found that the replica subsystem often synchronizes to the chaotic evolution of the original system, and it has been suggested that this phenomenon may be used for secure communications. In this paper we point out that exact synchronism may also occur for a large class of systems that are not replicas of part of the original system. This allows greater freedom in choosing synchronizer systems, and we discuss the possibility of using this freedom to choose synchronizer systems with improved performance. Two explicit examples illustrating this statement are given, one where the chaotic system consists of three autonomous differential equations, and the other where the chaotic system is a two-dimensional map.
61 citations
TL;DR: In this article, a method is proposed whereby the full state vector of a chaotic system can be reconstructed and tracked using only the time series of a single observed scalar, assuming that an accurate mathematical description of the system is available.
Abstract: A method is proposed whereby the full state vector of a chaotic system can be reconstructed and tracked using only the time series of a single observed scalar. It is assumed that an accurate mathematical description of the system is available. Noise effects on the procedure are investigated using as an example a kicked mechanical system which results in a four-dimensional dissipative map.
55 citations
TL;DR: In this article, the authors examine the problem of controlling a chaotic system embedded in a time varying environment, where the environmental variation may be of relatively large amplitude, and may have a fairly irregular nature.
17 citations
TL;DR: A control strategy, based on a previous method of chaos control, which can prevent failures of irregularly driven dynamical systems from taking place even when the temporal behavior of the environment has substantial irregularity is proposed.
Abstract: Responses of many man-made systems, such as ships or oil-drilling platforms, when subject to irregularly time varying environments, can be described by irregularly driven dynamical systems. Consequently, failures of such systems (e.g., capsize of a ship or collapse of a platform), under increasingly severe environmental conditions, come about when the system state escapes from a destroyed chaotic attractor located in some favorable region of the phase space. In this paper we propose a control strategy, based on a previous method of chaos control, which can prevent such failures from taking place. The key feature of our strategy is the incorporation of prediction of the evolution of the environment. This makes possible effective operation of the control even when the temporal behavior of the environment has substantial irregularity. We illustrate the ideas using ship capsizing as an example.
11 citations
TL;DR: In this article, the authors introduced a spectrum of cancellation exponents, which is called cancellation exponent spectrum, and discussed the relationship of Dq spectra with the signed generalized functions.
01 Jan 1994
TL;DR: In this paper, the authors investigate the tendency for global ergodic behavior of typical Hamiltonian systems as the number of degrees of freedom N is increased, and find that the fraction of phase space volume occupied by invariant tori decreases strongly.
Abstract: Numerical experiments on a symplectic coupled map system are performed to investigate the tendency for global ergodic behavior of typical Hamiltonian systems as the number of degrees of freedom N is increased As N increases, we find that the fraction of phase space volume occupied by invariant tori decreases strongly Nevertheless, due to observed very long time correlated behavior, a conclusion of effective gross ergodicity cannot be confirmed, even though extremely long numerical runs were employed
TL;DR: A brief review of chaotic dynamics is presented in this article, where basic concepts, recent developments, and applications are discussed, as well as applications and applications for chaotic dynamics in general.
Abstract: A brief review of chaotic dynamics is presented. Topics discussed include basic concepts, recent developments, and applications. >