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Determining Lyapunov exponents from a time series

Measuring the Strangeness of Strange Attractors

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.

Ergodic theory of chaos and strange attractors

Systems that are competitive and possess symmetric interactions admit a global Lyapunov function. However, a global Lyapunov function whose equilibrium set can be effectively analyzed has not yet been discovered. It remains an open question whether the Lyapunov function approach, which requires a study of equilibrium points, or an alternative global approach, such as the Lyapunov functional approach, which sidesteps a direct study of equilibrium points will ultimately handle all of the physically important cases.

Absolute stability of global pattern formation and parallel memory storage by competitive neural networks

This comprehensive text and reference work on numerical weather prediction, first published in 2002, covers not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. It incorporates all aspects of environmental computer modeling including an historical overview of the subject, equations of motion and their approximations, a modern and clear description of numerical methods, and the determination of initial conditions using weather observations (an important science known as data assimilation). Finally, this book provides a clear discussion of the problems of predictability and chaos in dynamical systems and how they can be applied to atmospheric and oceanic systems. Professors and students in meteorology, atmospheric science, oceanography, hydrology and environmental science will find much to interest them in this book, which can also form the basis of one or more graduate-level courses.

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Atmospheric Modeling, Data Assimilation and Predictability

Chaotic behavior of multidimensional difference equations

The liapunov dimension of strange attractors

This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value which is the first value for which the system possesses a homoclinic orbit. The arguments are similar to Smale's “horseshoe”.

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Preturbulence: A Regime Observed in a Fluid Flow Model of Lorenz*

Ordinary differential equations which yield periodic solutions of differential delay equations

Some infectious diseases have an incidence which is periodic in time. Contact rates may vary greatly during a year due to seasonal factors. In this paper we study the scalar delay integral equation x(t)= ʃ t t-τ f (s,x(s))ds , where f(t,x) is a continuous function which is periodic in t, f(t,0)=0, and where τ is a positive constant. This equation can be interpreted as a model for some infectious diseases with periodic contact rate, or as a growth equation for a single species population when the birth rate varies seasonally. The principal result is a threshold theorem: it is proved that if τ is small enough, all nonnegative solutions approach 0 as t→∞, whereas if τ is large enough, there exists a positive periodic solution with period equal to the period of f. Estimates on the critical size of τ are given in terms of ∂f ∂x (t,0) . A numerical example is given.

James L. Kaplan

Papers

Chaotic behavior of multidimensional difference equations

The liapunov dimension of strange attractors

Preturbulence: A Regime Observed in a Fluid Flow Model of Lorenz*

Ordinary differential equations which yield periodic solutions of differential delay equations

A periodicity threshold theorem for epidemics and population growth