Showing papers by "Jean-Pierre Eckmann published in 1995"

Recurrence Plots of Dynamical Systems

Abstract: A new graphical tool for measuring the time constancy of dynamical systems is presented and illustrated with typical examples.

Ergodic theory of chaos and strange attractors

Abstract: Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

Spectral duality for planar billiards

Abstract: For a bounded open domain Ω with connected complement inR2 and piecewise smooth boundary, we consider the Dirichlet Laplacian-ΔΩ on Ω and the S-matrix on the complementΩc. We show that the on-shell S-matricesSk have eigenvalues converging to 1 ask↑k0 exactly when--ΔΩ has an eigenvalue at energyk02. This includes multiplicities, and proves a weak form of “transparency” atk=k0. We also show that stronger forms of transparency, such asSk0 having an eigenvalue 1 are not expected to hold in general.

Inside-outside duality for planar billiards: A numerical study

Abstract: This paper reports the results of extensive numerical studies related to spectral properties of the Laplacian and the scattering matrix for planar domains (called billiards). There is a close connection between eigenvalues of the billiard Laplacian and the scattering phases, which is, basically, that every energy at which a scattering phase is 2\ensuremath{\pi} corresponds to an eigenenergy of the Laplacian. Interesting phenomena appear when the shape of the domain does not allow an extension of the eigenfunction to the exterior. In this paper these phenomena are studied and illustrated from several points of view.

Transport properties of the Lorentz gas in terms of periodic orbits

Abstract: We establish a formula relating global diffusion in a space periodic dynamical system to cycles in the elementary cell which tiles the space under translations.

Phase Slips and the Eckhaus Instability

Abstract: We consider the Ginzburg-Landau equation, delta tu= delta x2u+u-u mod u mod 2, with complex amplitude u(x, t). We first analyse the phenomenon of phase slips as a consequence of the local shape of u. We next prove a global theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth-order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.

Scattering phases and density of states for exterior domains

Abstract: For a bounded open domain $\Omega\subset\mathbb R^2$ with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian $\Delta$ on $\Omega$ and the S-matrix on its complement. We obtain precise bounds on the total scattering phase and and a Krein spectral formula, which improve similar results found in the literature

Scattering Phases and Density of States for Exterior Domain

Abstract: For a bounded open domain $\Omega\in \real^2$ with connected complement and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\DO$ on $\Omega$ and the S-matrix on the complement $\Omega^c$. Using the restriction $A_E$ of $(-\Delta-E)^{-1}$ to the boundary of $\Omega $, we establish that $A_{E_0}^{-1/2}A_EA_{E_0}^{-1/2}-1$ is trace class when $E_0$ is negative and give bounds on the energy dependence of this difference. This allows for precise bounds on the total scattering phase, the definition of a $\zeta$-function, and a Krein spectral formula, which improve similar results found in the literature.

Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems

Abstract: We show that values of the correlation dimension estimated over a decade from the Grassberger-Procaccia algorithm cannot exceed the value 2 log10N if N is the number of points in the time series. When this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present. The estimation of Lyapunov exponents is also discussed.

Phase Slips and the Eckhaus Instability

Abstract: We consider the Ginzburg-Landau equation, $ \partial_t u= \partial_x^2 u + u - u|u|^2 $, with complex amplitude $u(x,t)$. We first analyze the phenomenon of phase slips as a consequence of the {\it local} shape of $u$. We next prove a {\it global} theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.

Chaos on demand

Abstract: Engineers have long known about the use of feedback in control loops, enabling a potentially unstable system to be maintained in a stable state or adjusted to another. In recent years a new dimension has been added to this discipline: the control of chaotic behaviour. With papers now emerging frequently, it is an appropriate time to take stock.