Reggie Brown

A unifying definition of synchronization for dynamical systems

We propose a unifying definition for synchronization between stationary finite dimensional deterministic dynamical systems. By example, we show that the synchronization phenomena discussed in the dynamical systems literature fits within the framework of this definition, and discuss problems with previous definitions of synchronization. We conclude with a discussion of possible extensions of the definition to infinite dimensional systems described by partial differential equations and/or systems where noise is present. (c) 2000 American Institute of Physics.

LYAPUNOV EXPONENTS IN CHAOTIC SYSTEMS: THEIR IMPORTANCE AND THEIR EVALUATION USING OBSERVED DATA

We review the idea of Lyapunov exponents for chaotic systems and discuss their evaluation from observed data alone. These exponents govern the growth or decrease of small perturbations to orbits of a dynamical system. They are critical to the predictability of models made from observations as well as known analytic models. The Lyapunov exponents are invariants of the dynamical system and are connected with the dimension of the system attractor and to the idea of information generation by the system dynamics. Lyapunov exponents are among the many ways we can classify observed nonlinear systems, and their appeal to physicists remains their clear interpretation in terms of system stability and predictability. We discuss the familiar global Lyapunov exponents which govern the evolution of perturbations for long times and local Lyapunov exponents which determine the predictability over a finite number of time steps.

Variation of Lyapunov exponents on a strange attractor

SummaryWe introduce the idea of local Lyapunov exponents which govern the way small perturbations to the orbit of a dynamical system grow or contract after afinite number of steps,L, along the orbit. The distributions of these exponents over the attractor is an invariant of the dynamical system; namely, they are independent of the orbit or initial conditions. They tell us the variation of predictability over the attractor. They allow the estimation of extreme excursions of perturbations to an orbit once we know the mean and moments about the mean of these distributions. We show that the variations about the mean of the Lyapunov exponents approach zero asL → ∞ and argue from our numerical work on several chaotic systems that this approach is asL−v. In our examplesv ≈ 0.5–1.0. The exponents themselves approach the familiar Lyapunov spectrum in this same fashion.

Local Lyapunov exponents computed from observed data

SummaryWe develop methods for determining local Lyapunov exponents from observations of a scalar data set. Using average mutual information and the method of false neighbors, we reconstruct a multivariate time series, and then use local polynomial neighborhood-to-neighborhood maps to determine the phase space partial derivatives required to compute Lyapunov exponents. In several examples we demonstrate that the methods allow one to accurately reproduce results determined when the dynamics is known beforehand. We present a new recursive QR decomposition method for finding the eigenvalues of products of matrices when that product is severely ill conditioned, and we give an argument to show that local Lyapunov exponents are ambiguous up to order 1/L in the number of steps due to the choice of coordinate system. Local Lyapunov exponents are the critical element in determining the practical predictability of a chaotic system, so the results here will be of some general use.

Synchronization of chaotic systems: Transverse stability of trajectories in invariant manifolds

We examine synchronization of identical chaotic systems coupled in a drive/response manner. A rigorous criterion is presented which, if satisfied, guarantees that synchronization to the driving trajectory is linearly stable to perturbations. An easy to use approximate criterion for estimating linear stability is also presented. One major advantage of these criteria is that, for simple systems, many of the calculations needed to implement them can be performed analytically. Geometrical interpretations of the criterion are discussed, as well as how they may be used to investigate synchronization between mutual coupled systems and the stability of invariant manifolds within a dynamical system. Finally, the relationship between our criterion and results from control theory are discussed. Analytical and numerical results from tests of these criteria on four different dynamical systems are presented. (c) 1997 American Institute of Physics.

Prediction and system identification in chaotic nonlinear systems: Time series with broadband spectra

Abstract We consider the problem of prediction and system identification for time series having broadband power spectra which arise from the intrinsic nonlinear dynamics of the system. We view the motion of the system in a reconstructed phase space which captures the attractor (usually strange) on which the system evolves, and give a procedure for constructing parameterized maps which evolve points in the phase space into the future. The predictor of future points in the phase space is a combination of operation on past points by the map and its iterates. Thus the map is regarded as a dynamical system, not just a fit to the data. The invariants of the dynamical system — the Lyapunov exponents and aspects of the invariant density on the attractor — are used as constraints on the choice of mapping parameters. The parameter values are chosen through a least-squares optimization procedure. The method is applied to “data” from the Henon map and shown to be feasible. It is found that the parameter values which minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system. The maps which do reproduce the values of the invariants are not optimum in the least-squares sense, yet still are excellent predictors. We discuss several technical and general problems associated with prediction and system identification on strange attractors. In particular, we consider the matter of the evolution of points that are off the attractor (where little or no data is available), onto the attractor, where long-term motion takes place.

The goodness of ergodic adiabatic invariants

For a “slowly” time-dependent Hamiltonian system exhibiting chaotic motion that ergodically covers the energy surface, the phase space volume enclosed inside this surface is an adiabatic invariant. In this paper we examine, both numerically and theoretically, how the error in this “ergodic adiabatic invariant” scales with the slowness of the time variation of the Hamiltonian. It is found that under certain circumstances, the error is diffusive and scales likeT−1/2, whereT is the characteristic time over which the Hamiltonian changes. On the other hand, for other cases (where motion in the Hamiltonian has a long-time 1/t tail in a certain correlation function), the error scales like [T−1 ln(T)]1/2. Both of these scalings are verified by numerical experiments. In the situation where invariant tori exist amid chaos, the motion may not be fully ergodic on the entire energy surface. The ergodic adiabatic invariant may still be useful in this case and the circumstances under which this is so are investigated numerically (in particular, the islands have to be small enough).

Symbol statistics and spatio-temporal systems

Abstract We consider the problem of estimating parameters from time-series observations of spatio-temporal systems. Two types of models are considered: (a) a one-dimensional coupled map lattice with nearest neighbor diffusive coupling; and (b) the complex Ginzburg-Landau equation in one and two spatial dimensions. Model parameters are to be estimated using time-series observations from only a few sites. A symbolic partition of the time series is introduced and the probabilities of observing various symbol sequences in the data are measured. The parameter fitting is accomplished by adjusting parameters of the model until it produces time series whose symbol sequences have the same probabilities as the data. We show that it is possible to reliably estimate the parameters from a single time series when the spatio-temporal dynamics is “turbulent”, i.e. it displays a wide range of space and time scales with no discernible patterns.

Hamiltonian formulation of inviscid flows with free boundaries

The formulation of the Hamiltonian structures for inviscid fluid flows with material free surfaces is presented in both the Lagrangian specification, where the fundamental Poisson brackets are canonical, and in the Eulerian specification, where the dynamics is given in noncanonical form. The noncanonical Eulerian brackets are derived explicitly from the canonical Lagrangian brackets. The Eulerian brackets are, with the exception of a single term at each material free surface separating flows in different phases, identical to those for isentropic flow of a compressible, inviscid fluid. The dynamics of the free surface is located in the Hamiltonian and in the definition of the Eulerian variables of mass density, ρ(x, t), momentum density, M(x,t) [which is ρ times the fluid velocity v(x,t)], and the specific entropy, σ(x,t). The boundary conditions for the Eulerian variables and the evolution equations for the free surfaces come from the Euler equations of the flow. This construction provides a unified treatment of inviscid flows with any number of free surfaces.

Modeling and synchronizing chaotic systems from experimental data

Abstract The inverse problem of extracting evolution equations from chaotic time series measured from continuous systems is considered. The resulting equations of motion form an autonomous system of nonlinear ordinary differential equations (ODEs). The vector fields are modeled in the manner of implicit Adams integration using a basis set of polynomials that are constructed to be orthonormal on the data. The fitting method uses the Rissanen minimum description length (MDL) criterion to determine the optimal polynomial vector field. It is then demonstrated that one can synchronize the model to an experimentally measured time series. In this case synchronization is used as a nontrivial test for the validity of the models.