Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

Adaptive Neuro-Genetic Control of Chaos applied to the Attitude Control Problem

Conventional adaptive control techniques have, for the most part, been based on methods for linear or weakly non-linear systems. More recently, neural network and genetic algorithm controllers have started to be applied to complex, non-linear dynamic systems. The control of chaotic dynamic systems poses a series of especially challenging problems. In this paper, an adaptive control architecture using neural networks and genetic algorithms is applied to a complex, highly nonlinear, chaotic dynamic system: the adaptive attitude control problem (for a satellite), in the presence of large, external forces (which left to themselves led the system into a chaotic motion). In contrast to the OGY method, which uses small control adjustments to stabilize a chaotic system in an otherwise unstable but natural periodic orbit of the system, the neuro-genetic controller may use large control adjustments and proves capable of effectively attaining any specified system state, with no a prioriknowledge of the dynamics, even in the presence of significant noise

Genetic Programming for Prediction and Control

The relatively ‘new’ field of genetic programming has received a lot of attention during the last few years. This is because of its potential for generating functions which are able to solve specific problems. This paper begins with an extensive overview of the field, highlighting its power and limitations and providing practical tips and techniques for the successful application of genetic programming in general domains. Following this, emphasis is placed on the application of genetic programming to prediction and control. These two domains are of extreme importance in many disciplines. Results are presented for an oral cancer prediction task and a satellite attitude control problem. Finally, the paper discusses how the convergence of genetic programming can be significantly speeded up through bulk synchronous model parallelisation