Contributions of plasma physics to chaos and nonlinear dynamics

This topical review focusses on the contributions of plasma physics to chaos and nonlinear dynamics bringing new methods which are or can be used in other scientific domains. It starts with the development of the theory of Hamiltonian chaos, and then deals with order or quasi order, for instance adiabatic and soliton theories. It ends with a shorter account of dissipative and high dimensional Hamiltonian dynamics, and of quantum chaos. Most of these contributions are a spin-off of the research on thermonuclear fusion by magnetic confinement, which started in the fifties. Their presentation is both exhaustive and compact. [15 April 2016

Stochastic stability of viscoelastic systems under Gaussian and Poisson white noise excitations

As the use of viscoelastic materials becomes increasingly popular, stability of viscoelastic structures under random loads becomes increasingly important. This paper aims at studying the asymptotic stability of viscoelastic systems under Gaussian and Poisson white noise excitations with Lyapunov functions. The viscoelastic force is approximated as equivalent stiffness and damping terms. A stochastic differential equation is set up to represent randomly excited viscoelastic systems, from which a Lyapunov function is determined by intuition. The time derivative of this Lyapunov function is then obtained by stochastic averaging. Approximate conditions are derived for asymptotic Lyapunov stability with probability one of the viscoelastic system. Validity and utility of this approach are illustrated by a Duffing-type oscillator possessing viscoelastic forces, and the influence of different parameters on the stability region is delineated

Stabilization of Stochastic Quantum Dynamics via Open and Closed Loop Control

In this paper we investigate parametrization-free solutions of the problem of quantum pure state preparation and subspace stabilization by means of Hamiltonian control, continuous measurement and quantum feedback, in the presence of a Markovian environment. In particular, we show that whenever suitable dissipative effects are induced either by the unmonitored environment or by non Hermitian measurements, there is no need for feedback control to accomplish the task. Constructive necessary and sufficient conditions on the form of the open-loop controller can be provided in this case. When open-loop control is not sufficient, filtering-based feedback control laws steering the evolution towards a target pure state are provided, which generalize those available in the literature

Nontrivial Dynamics in the Early Stages of Inflation

Inflationary cosmologies, regarded as dynamical systems, have rather simple asymptotic behavior, insofar as the cosmic baldness principle holds. Nevertheless, in the early stages of an inflationary process, the dynamical behavior may be very complex. In this paper, we show how even a simple inflationary scenario, based on Linde's ``chaotic inflation'' proposal, manifests nontrivial dynamical effects such as the breakup of invariant tori, formation of cantori and Arnol'd's diffusion. The relevance of such effects is highlighted by the fact that even the occurrence or not of inflation in a given Universe is dependent upon them.Comment: 26 pages, Latex, 9 Figures available on request, GTCRG-94-1

Chirikov Diffusion in the Asteroidal Three-Body Resonance (5,-2,-2)

The theory of diffusion in many-dimensional Hamiltonian system is applied to asteroidal dynamics. The general formulations developed by Chirikov is applied to the Nesvorn\'{y}-Morbidelli analytic model of three-body (three-orbit) mean-motion resonances (Jupiter-Saturn-asteroid system). In particular, we investigate the diffusion \emph{along} and \emph{across} the separatrices of the (5,-2,-2) resonance of the (490) Veritas asteroidal family and their relationship to diffusion in semi-major axis and eccentricity. The estimations of diffusion were obtained using the Melnikov integral, a Hadjidemetriou-type sympletic map and numerical integrations for times up to $10^{8}$ years.Comment: 27 pages, 6 figure