Mathematical Sciences: Chaos-Integrability Transition in Nonlinear Dynamical Systems: Exponental Asymptotics Approach

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9500644 Tovbis The objective of the project is to study the transition from integrable (predictable) to nonintegrable (chaotic) dynamics in perturbed nonlinear systems. This transition plays a fundamental role in a very wide variety of physical, biological, chemical, etc. problems (turbulence in incompressible fluids, scattering of atoms from metal surfaces, host - parasitoid models in population dynamics, complex formation in atom-diatom collisions, etc.). It is also crucial in understanding the nature of numerical instabilities (numerical chaos) in computational problems. The proposed approach to the problem combines classical analytic and modern asymptotic technique (exponential asymptotics). In our initial studies the model example shows an excellent agreement between the "theoretical" results and numerical simulations. %%% The objective of the project is to study the transition from integrable to nonintegrable (chaotic) dynamics in singularly perturbed nonlinear systems. This transition plays a fundamental role in a very wide variety of physical, biological, chemical, etc. problems. It is also crucial in understanding the nature of numerical instabilities (numerical chaos) il computational problems. Of our particular interest are situations when the effect of perturbation is exponentially small in the small parameter of the problem. We propose to apply the technique of exponential asymptotics to the study of the analytical mechanism of the onset of chaos in perturbed integrable systems and to approximate the dynamics of perturbed systems. This approach has proven fruitful in our initial investigation. ***