The development of complex states of fluid motion is illustrated by reviewing a series of experiments, emphasizing film flows, surface waves, and thermal convection. In one dimension, cellular patterns bifurcate to states of spatiotemporal chaos. In two dimensions, even ordered patterns can be surprisingly intricate when quasiperiodic patterns are included. Spatiotemporal chaos is best characte...

A quantum characteristic exponent may be defined, with the same operational meaning as the classical Lyapunov exponent when the latter is expressed as a functional of densities. Existence conditions and supporting measure properties are discussed as well as the problems encountered in the numerical computation of the quantum exponents. Although an example of true quantum chaos may be exhibited,...

Following a brief historical introduction of the notions of chaos in dynamical systems, we will present recent developments that attempt to profit from the rich structure and complexity of the chaotic dynamics. In particular, we will demonstrate the ability to control chaos in realistic complex environments. Several applications will serve to illustrate the theory and to highlight its advantage...

An immensely influential paper in the history of the study of chaos was the work of Ruelle and Takens [1], later extended with Newhouse [2] on the robustness of quasiperiodic dynamics with 3 or more incommensurate frequencies. This addressed the question of whether complexity in dynamics was likely to occur through the accumulation of Hopf-like bifurcations adding additional frequencies (and pr...

A new methodology for a robust analysis of uncertain nonlinear dynamic systems is presented in this paper. The originality of the method proposed lies in the combination of the centre manifold theory with the polynomial chaos approach. The first one is known to be a powerful tool for model reduction of nonlinear dynamic systems in the neighbourhood of the Hopf bifurcation point while the polyno...

This paper describes an investigation into possible chaotic behavior in a coupled-oscillator system and the possible control of this behavior for communications. The established mathematical models for these oscillator arrays are demonstrated to exhibit chaos when the coupling strength between oscillators is below the range for phase locking. The complexity and predictability of the array dynam...

In this paper we study mathematically the eigenvalue problem for stochastic elliptic partial differential equation of Wick type. Using the Wick-product and the Wiener-Itô chaos expansion, the stochastic eigenvalue problem is reformulated as a system of an eigenvalue problem for a deterministic partial differential equation and elliptic partial differential equations by using the Fredholm altern...

Previous studies on heart rate variability (HRV) using chaos theory, fractal scaling analysis, and many other methods, while fruitful in many aspects, have produced much confusion in the literature. Especially the issue of whether normal HRV is chaotic or stochastic remains highly controversial. Here, we employ a new multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), t...

Many systems in the natural world exhibit chaos or non-linear behavior, the complexity of which is so great that they appear to be random. Identification of chaos in experimental data is essential for characterizing the system and for analyzing the predictability of the data under analysis. The Lyapunov exponents provide a quantitative measure of the sensitivity to initial conditions and are th...

Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method ...