James B. Kadtke

chaotic advection, diffusion, and reactions in open flows

We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 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.

Point vortex dynamics: recent results and open problems

Abstract he concept of point vortex motion, a classical model in the theory of two-dimensional, incompressible fluid mechanics, was introduced by Helmholtz in 1858. Exploration of the solutions to these equations has made fitful progress since that time as the point vortex model has been brought to bear on various physical situations: atomic structure, large scale weather patterns, “vortex street” wakes, vortex lattices in superfluids and superconductors, etc. The point vortex equations also provide an interesting example of transition to chaotic behavior. We give a brief historical introduction to these topics and develop two of them in particular to the point of current understanding: (i) Steadily moving configurations of point vortices; and (ii) Collision dynamics of vortex pairs.

CLASSIFICATION OF HIGHLY NOISY SIGNALS USING GLOBAL DYNAMICAL MODELS

Abstract We present a classification scheme for time series with nonlinear correlations, using global models of chaotic dynamical systems theory. We demonstrate classification in high-noise regimes, and argue that classification probabilities can be directly computed from ensemble statistics in the model coefficient space. We also develop a modification for nonstationary signals.

GLOBAL DYNAMICAL EQUATIONS AND LYAPUNOV EXPONENTS FROM NOISY CHAOTIC TIME SERIES

We discuss the extraction of few-parameter, global dynamical models from noisy time series of chaotic systems. In particular, we consider the class of models which are approximations to sets of dynamical equations in the reconstructed phase space. We show that certain numerical methods significantly improve the quality of the resulting models, and central to these methods is the idea of eliminating model terms which are “dynamically insignificant” and add only numerical noise. For the purposes of the paper, we quantify model quality by the rather strict measure of its ability to recover the dynamical invariants of the original system, in particular, the Lyapunov spectrum. Consequently, we also postulate that by first extracting a global model, the Lyapunov spectrum of a generating system can be recovered from time series whose noise levels are much higher than current algorithms would allow. We present several numerical examples to demonstrate the above ideas.

Fractality, chaos, and reactions in imperfectly mixed open hydrodynamical flows

We investigate the dynamics of tracer particles in time-dependent open flows. If the advection is passive the tracer dynamics is shown to be typically transiently chaotic. This implies the appearance of stable fractal patterns, so-called unstable manifolds, traced out by ensembles of particles. Next, the advection of chemically or biologically active tracers is investigated. Since the tracers spend a long time in the vicinity of a fractal curve, the unstable manifold, this fractal structure serves as a catalyst for the active process. The permanent competition between the enhanced activity along the unstable manifold and the escape due to advection results in a steady state of constant production rate. This observation provides a possible solution for the so-called “paradox of plankton”, that several competing plankton species are able to coexists in spite of the competitive exclusion predicted by classical studies. We point out that the derivation of the reaction (or population dynamics) equations is analog to that of the macroscopic transport equations based on a microscopic kinetic theory whose support is a fractal subset of the full phase space.

Estimating statistics for detecting determinism using global dynamical models

Abstract Classification of time series using a dynamical system ansatz is potentially powerful, however assessing performance for noisy experimental data is problematic. Here, we develop a rigorous statistical framework for calculating classification probabilities using global dynamical models, and analytically derive some asymptotic properties. We illustrate the method numerically by attempting to detect “determinism” in a noisy data set.

Chaotic capture of vortices by a moving body. I. The single point vortex case.

The study of the dynamical properties of vortex systems is an important and topical research area, and is becoming of ever increasing usefulness to a variety of physical applications. In this paper, we present a study of a model of a rotational singularity which obeys a logarithmic potential interacting with a bluff body in a uniform inviscid laminar flow, e.g., a line vortex interacting with a cylinder in three dimensions or a point vortex with a circular boundary in two dimensions. We show that this system is Hamiltonian and simple enough to be solved analytically for the stagnation points and separatrices of the flow, and a bifurcation diagram for the relevant parameters and classification of the various types of motion is given. We also show that, by introducing a periodic perturbation to the body, chaotic motion of the vortex can be readily generated, and we present analytic criteria for the generation of chaos using the Poincare-Melnikov-Arnold method. This leads to an important dynamical effect for the model, i.e., that the possibility exists for the vortex to be chaotically captured around the body for periods of time which are extremely sensitive to initial conditions. The basic mechanism for this capture is due to the chaotic dynamics and is similar to that of other chaotic scattering phenomena. We show numerically that cases exist where the vortex can be captured around an elliptic point external to (and possibly far from) the body, and the existence of other very complicated motions are also demonstrated. Finally, generalizations of the problem of the vortex-body interaction are indicated, and some possible applications are postulated such as the interaction of line vortices with aircraft wings.

Adaptive methods for chaotic communication systems

We describe several existing and two new nonlinear dynamics (NLD)-based communication schemes, addressing the shortcomings and advantages of each. One new method is based on modulating the parameters of one or more chaotic generating systems with the signal(s) of interest. The output of the resulting nonstationary chaotic system is transmitted across the channel, and the signals are extracted via a continuous adaptation of system coefficients. The other novel method involves injecting the signal of interest as dynamic or feedback noise into the chaotic generating system. The noisy output is transmitted, and the dynamics of the generating system are estimated at the receiver. These dynamics are used to generate short term prediction errors, which are proportional to the original dynamic noise if the recovered system coefficients are reasonably accurate. Both methods have particular application to radio frequency (rf) communications, where spectral noise and frequency selective fading of signal power are real problems.

Estimating dynamical models using generalized correlation functions

Abstract We develop a method for estimating closed-form nonlinear dynamical models from observed time series, which expresses the unknown coefficients as functions of generalized higher-order data correlations. Besides robust numerical properties, this method often yields analytic coefficient representations which provide theoretical insight into general model properties.