Noise Induced Dissipation in Discrete-Time Classical and Quantum Dynamical Systems

We introduce a new characteristics of chaoticity of classical and quantum dynamical systems by defining the notion of the dissipation time which enables us to test how the system responds to the noise and in particular to measure the speed at which an initially closed, conservative system converges to the equilibrium when subjected to noisy (stochastic) perturbations. We prove fast dissipation result for classical Anosov systems and general exponentially mixing maps. Slow dissipation result is proved for regular systems including non-weakly mixing maps. In quantum setting we study simultaneous semiclassical and small noise asymptotics of the dissipation time of quantized toral symplectomorphisms (generalized cat maps) and derive sharp bounds for semiclassical regime in which quantum-classical correspondence of dissipation times holds.Comment: PhD Dissertation, University of California at Davis, June 2004, LaTex, 195 page

Islands and Ellipses in 2D Dynamical Systems

Three main results are presented in this thesis. The first is a proof of the existence of absolutely continuous invariant measures (ACIMs) for two dimensional maps supported on islands, which are small, disjoint regions of R2. The proof is computer-assisted and uses both numerical evidence and a combinatorial method. We give examples of weak chaos for which ACIMs exist: within islands there is chaos, but from a distance orbits are periodic. The second main result is a geometrical proof of the asymptotic behavior of generalized tent maps with memory which we call elliptical maps. It is proved that for certain π-rational angles, all points in the domain except for (0, 0) fall into a polygonal region whose characteristics we determine. When the angles are π-irrational we prove that these points either fall in a unique ellipse, or accumulate on its boundary. The third result is a proof that ACIMs exist for a certain range of parameters in generalized β-tent maps with memory. The thesis begins with discussions about ACIMs and why they are interesting, followed by Tsujii’s theorem and other tools, notes on computer calculations and graphics, and on weak chaos. At the end, we highlight some unanswered questions and puzzling phenomena that we encountered during our research

Philosophical aspects of chaos: definitions in mathematics, unpredictability, and the observational equivalence of deterministic and indeterministic descriptions

This dissertation is about some of the most important philosophical aspects of chaos research, a famous recent mathematical area of research about deterministic yet unpredictable and irregular, or even random behaviour. It consists of three parts. First, as a basis for the dissertation, I examine notions of unpredictability in ergodic theory, and I ask what they tell us about the justification and formulation of mathematical definitions. The main account of the actual practice of justifying mathematical definitions is Lakatos's account on proof-generated definitions. By investigating notions of unpredictability in ergodic theory, I present two previously unidentified but common ways of justifying definitions. Furthermore, I criticise Lakatos's account as being limited: it does not acknowledge the interrelationships between the different kinds of justification, and it ignores the fact that various kinds of justification - not only proof-generation - are important. Second, unpredictability is a central theme in chaos research, and it is widely claimed that chaotic systems exhibit a kind of unpredictability which is specific to chaos. However, I argue that the existing answers to the question "What is the unpredictability specific to chaos?" are wrong. I then go on to propose a novel answer, viz. the unpredictability specific to chaos is that for predicting any event all sufficiently past events are approximately probabilistically irrelevant. Third, given that chaotic systems are strongly unpredictable, one is led to ask: are deterministic and indeterministic descriptions observationally equivalent, i.e., do they give the same predictions? I treat this question for measure-theoretic deterministic systems and stochastic processes, both of which are ubiquitous in science. I discuss and formalise the notion of observational equivalence. By proving results in ergodic theory, I first show that for many measure-preserving deterministic descriptions there is an observationally equivalent indeterministic description, and that for all indeterministic descriptions there is an observationally equivalent deterministic description. I go on to show that strongly chaotic systems are even observationally equivalent to some of the most random stochastic processes encountered in science. For instance, strongly chaotic systems give the same predictions at every observation level as Markov processes or semi-Markov processes. All this illustrates that even kinds of deterministic and indeterministic descriptions which, intuitively, seem to give very different predictions are observationally equivalent. Finally, I criticise the claims in the previous philosophical literature on observational equivalence

Chaos Characterization of Pulse-Coupled Neural Networks in Balanced State

In the present work the formalism of Ergodic theory, used for the statistical study of complex, nonlinear dynamical systems of N ≫ 1 dimensions in general, is applied to the time evolution of large-scale pulse-coupled neural networks in the so-called balanced state. The aim is to measure the ergodic properties of such systems, consider how they are related to the network parameters, and finally characterize dynamically the participation of individual network nodes (the “neurons”) in the collective dynamics