Chaotic Dynamics

This is the transcript of a lecture given at UMass-Lowell in which I compare and contrast the work of Godel and of Turing and my own work on incompleteness. I also discuss randomness in physics vs randomness in pure mathematics.

Experimental data from a turbulent jet flow is analysed in terms of an additive, continuous stochastic process where the usual time variable is replaced by the scale. We show that the energy transfer through scales is well described by a linear Langevin equation, and discuss the statistical properties of the corresponding random force in detail. We find that the autocorrelation function of the random force decays rapidly: the process is therefore Markov for scales larger than Kolmogorov's dissipation scale η . The corresponding autocorrelation scale is identified as the elementary step of the energy cascade. However, the probability distribution function of the random force is both non-Gaussian and weakly scale-dependent.

It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.

A statistical model of self-organization in a generic class of one-dimensional nonlinear Schrodinger (NLS) equations on a bounded interval is developed. The main prediction of this model is that the statistically preferred state for such equations consists of a deterministic coherent structure coupled with fine-scale, random fluctuations, or radiation. The model is derived from equilibrium statistical mechanics by using a mean-field approximation of the conserved Hamiltonian and particle number for finite-dimensional spectral truncations of the NLS dynamics. The continuum limits of these approximated statistical equilibrium ensembles on finite-dimensional phase spaces are analyzed, holding the energy and particle number at fixed, finite values. The analysis shows that the coherent structure minimizes total energy for a given value of particle number and hence is a solution to the NLS ground state equation, and that the remaining energy resides in Gaussian fluctuations equipartitioned over wavenumbers. Some results of direct numerical integration of the NLS equation are included to validate empirically these properties of the most probable states for the statistical model. Moreover, a theoretical justification of the mean-field approximation is given, in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit.

A consistent description of simultaneous heat and particle transport, including cross effects, and the associated entropy balance is given in the framework of a deterministic dynamical system. This is achieved by a multibaker map where, besides the phase-space density of the multibaker, a second field with appropriate source terms is included in order to mimic a spatial temperature distribution and its time evolution. Conditions are given to ensure consistency in an appropriately defined continuum limit with the thermodynamic entropy balance. They leave as the only free parameter of the model the entropy flux let directly into a surroundings. If it vanishes in the bulk, the transport properties of the model are described by the thermodynamic transport equations. Another choice leads to a uniform temperature distribution. It represents transport problems treated by means of a thermostatting algorithm, similar to the one considered in non-equilibrium molecular dynamics.

Many systems can display a very short, rapid changing stage (quasi-discontinuous region) inside a relatively very long and slowly changing process. A quantitative definition for the "quasi-discontinuity" in these systems has been introduced. We have shown by a simplified model that extra-large Feigenbaum constants can be found inside some period-doubling cascades due to the quasi-discontinuity. As an example, this phenomenon has also been observed in Rose-Hindmash model describing neuron activities.

This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter α ; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order α . The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnold's theorem), but with respect to a right invariant H 1 metric instead of the L 2 metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, {\it even in the presence of boundaries}.

We present a measure for characterizing statistical relationships between two time sequences. In contrast to commonly used measures like cross-correlations, coherence and mutual information, the proposed measure is non-symmetric and provides information about the direction of interdependence. It is closely related to recent attempts to detect generalized synchronization. However, we do not assume a strict functional relationship between the two time sequences and try to define the measure so as to be robust against noise, and to detect also weak interdependences. We apply our measure to intracranially recorded electroencephalograms of patients suffering from severe epilepsies.

We use a recently found parametrization of the solutions of the inverse Frobenius-Perron problem within the class of complete unimodal maps to develop a Monte-Carlo approach for the construction of one-dimensional chaotic dynamical laws with given statistical properties, i.e. invariant density and autocorrelation function. A variety of different examples are presented to demonstrate the power of our method.

We propose a method that is able to analyze chaotic time series, gained from exp erimental data. The method allows to identify scalar time-delay systems. If the dynamics of the system under investigation is governed by a scalar time-delay differential equation of the form dy(t)/dt=h(y(t),y(t− τ 0 )) , the delay time τ 0 and the functi on h can be recovered. There are no restrictions to the dimensionality of the chaotic attractor. The method turns out to be insensitive to noise. We successfully apply the method to various time series taken from a computer experiment and two different electronic oscillators.