General relativity exhibits a unique feature not represented in standard examples of chaotic systems; it is a spacetime diffeomorphism invariant theory. Thus many characterizations of chaos do not work. It is therefore necessary to develop a definition of chaos suitable for application to general relativity. This presentation will present results towards this goal.

Bi-instability, in contrast to bistability, is shown to generate unstable chaotic saddles prior to the onset of chaos. The theory and numerics are applied to a CO2 laser model with modulated losses where unstable pairs of saddles coexist, form heteroclinic connections, and allow mixing between local chaotic attractors to produce global mixed-mode chaos.

It is a major source of contention in brain dynamics as to whether the electrical rhythms of the brain show signs of chaos. Here we discuss evidence for the existence of chaos in a theory of brain electrical activity and provide unique depictions of the dynamics of this model. r 2002 Elsevier Science Ltd. All rights reserved.

the term paradigm hase penetrated in the web of many fields form natural to social sciences. therefore, for each researcher, understanding it's concept and dimensions is necessary. it is become to a common term in philsophy of science. there are many definitions of paradigm, but we can define it as: a set of believes and basic assumptions that are reflected in the ontology, epistemology an...

A positive Lyapunov exponent is one practical deÞnition of chaos. We develop a formal test for chaos in a noisy system based on the consistent standard errors of the nonparametric Lyapunov exponent estimators. When our procedures are applied to international real output series, the hypothesis of the positive Lyapunov exponent is signiÞcantly rejected in many cases. One possible interpretation o...

This paper studies the problem of controlling chaos in Lorenz system. By dividing Lorenz system into two subsystems, the small gain theorem is used to design linear feedback controllers to make the equilibrium points of the controlled system globally asymptotically stable. Thus chaos in Lorenz system disappears under the controller.

It has been pointed out that a chaos control technique wherein a weak excitation is added to the system could be useful for the control of friction at the nanoscale. We examine the claim that this technique can achieve chaos control even though the added excitation is weak. We show that this claim is only valid for the particular case of systems with very shallow potential wells. However, the a...

In this paper we propose a necessary condition for the existence of chaos in delay differential equations of fractional order. To explain the proposed theory, we discuss fractional order Liu system and financial system involving delay. Keywords—Caputo derivative, delay, stability, chaos.

Nowadays, chaos theory and nonlinear dynamics lack research focuses. Here we mention a few major open problems: 1. an effective description of chaos and turbulence, 2. rough dependence on initial data, 3. arrow of time, 4. the paradox of enrichment, 5. the paradox of pesticides, 6. the paradox of plankton.

This paper considers the problem of the chaos suppression for the Permanent Magnet Synchronous Motor (PMSM) system via the finite-time control. Based on Lyapunov stability theory and the finite-time controller are developed such that the chaos behaviors of PMSM system can be suppressed. The effectiveness and accuracy of the proposed methods are shown in numerical simulations.