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Nonlinear dynamical systems play an important role in a number of disciplines. The physical, biological, economic and even sociological worlds are comprised of com plex nonlinear systems that cannot be broken down into the behavior of their con stituents and then reassembled to form the whole. The lack of a superposition principle in such systems has challenged researchers to use a variety of analytic and numerical methods in attempts to understand the interesting nonlinear interactions that occur in the World around us. General relativity is a nonlinear dynamical theory par excellence. Only recently has the nonlinear evolution of the gravitational field described by the theory been tackled through the use of methods used in other disciplines to study the importance of time dependent nonlinearities. The complexity of the equations of general relativity has been (and still remains) a major hurdle in the formulation of concrete mathematical concepts. In the past the imposition of a high degree of symmetry has allowed the construction of exact solutions to the Einstein equations. However, most of those solutions are nonphysical and of those that do have a physical significance, many are often highly idealized or time independent.
List of contents
A Brief Review of "Deterministic Chaos in General Relativity".- Mathematical Preliminaries.- to Dynamical Systems.- A Short Course on Chaotic Hamiltonian Systems.- Geometry of Perturbation Theory.- Between Integrability and Chaos.- On Defining Chaos in the Absence of Time.- On the Dynamics of Generators of Cauchy Horizons.- Compact Relativistic Systems.- Chaos in the Case of Two Fixed Black Holes.- Particle Motion Around Perturbed Black Holes: The Onset of Chaos.- Critical Behaviour in Scalar Field Collapse.- Cosmological Systems.- Relativistic Cosmologies.- Homoclinic Chaos in Relativistic Cosmology.- Mixing Properties of k = ?1 FLRW Models.- Classical and Quantum Chaos in Robertson-Walker Cosmologies.- Relativistic Fractal Cosmologies.- Self-Similar Asymptotic Solutions of Einstein's Equations.- Nonlinearly Interacting Gravitational Waves in the Gowdy T3 Cosmology.- Bianchi IX (Mixmaster) Dynamics.- The Mixmaster Cosmological Metrics.- The Belinskii-Khalatnikov-Lifshitz Discrete Evolution as an Approximation to Mixmaster Dynamics.- How Can You Tell if the Bianchi IX Models Are Chaotic?.- The Chaoticity of the Bianchi IX Cosmological Model.- Chaos in the Einstein Equations-Characterization and Importance?.- Integrability of the Mixmaster Universe.- Continuous Time Dynamics and Iterative Maps of Ellis-MacCallum-Wainwright Variables.- A Dynamical Systems Approach to the Oscillatory Singularity in Bianchi Cosmologies.- Participants.
