Modern Asymptotic Approaches in Nonlinear Dynamics - CANCELLED

April 11, 2011 — April 15, 2011

Coordinator:

  • Igor V. Andrianov (RWTH Aachen University, Aachen, Germany)

Recent advances in nonlinear theories have led to substantial progress in Physics, Solid and Fluid Mechanics, Biology, Material Sciences, Engineering, and many multidisciplinary areas dealing with nonlinear phenomena. This likely happened due to dramatic jumps in power of computational tools accompanied by essentially new developments in analytical methods, first of all – asymptotic analysis. Such a viewpoint will be illustrated through the present lecture series based on different examples from theoretical and applied sections of Classic and Quantum Mechanics. The lecturers intend to discuss the core ideas with clear illustrations provided by mathematical formulations sufficient for practical use.

In particular, several lectures will be devoted to analytical techniques for systematic reduction of exponentially small residuals of asymptotic expansions. These will cover approximations of integrals and the Berry-Dingle theory, matching asymptotic approximations in the complex plane, weakly nonlocal solitary waves, quantum scattering ‘above the barrier’, high-degree Fourier and Chebyshev spectral coefficients, as well as resonances in nonlinear waves and dynamical systems.

Summation methods for perturbation series that extend their range of applicability on non-local effects will be introduced. The concept of asymptotically equivalent functions, homotopy perturbation approaches, and different continualization procedures that take into account long-range interactions between structural subcomponents of the discrete media will be discussed.

A new version of the multiple scales perturbation method for Ordinary Difference Equations (ODEs) will be presented and formulated completely in terms of difference operators. Also the recently developed technique on how to construct asymptotic approximations of first integrals or invariants for systems of Ordinary Differential Equations (ODEs) or ODEs will be presented.
An overview of different physical ideas and mathematical tools for implementing non-smooth and discontinuous substitutions in dynamical systems will be presented. A general purpose of such stage of modeling is to bring the differential equations of motion to the form, which is convenient for further use of either analytical or numerical methods of analyses.

Development of the asymptotic methods in application to discrete nonlinear systems will be described. The Fermi-Pasta-Ulam model and its generalizations on zigzag and helix oscillatory chains, and systems of weakly coupled 1D oscillatory chains will be analysed. It will be shown that degenerations of the linear spectrum of infinite and periodic oscillatory chains may lead to the occurrence of beating-like phenomena adequately described within the new concept of Limiting Phase Trajectories (LPT).
One of the challenging goals of mechanics of heterogeneous solids is identification of internal structures of materials based on test measurements of their macroscopic properties. This problem is of a great importance for various practical applications, such as non-destructive testing of composite materials, non-invasive diagnostic of live tissues, etc. The related ideas will be analysed by using the asymptotic homogenization method in combination with the boundary perturbation technique, singular perturbation approaches and Padé approximants.

The course is intended for doctoral and postdoctoral research-ers in Civil and Mechanical Engineering, Applied Mathematics and Physics, academic and industrial researchers, which are interested in conducting research in the topic.

KEYWORDS: Nonlinear dynamics, Asymptonic analysis, Perturbation methods, Waves in solids, Composite materials.

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