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Theoretical and Mathematical Physics, 130(2): 245255 (2002) DISSIPATIVE AND HAMILTONIAN SYSTEMS WITH
 

Summary: Theoretical and Mathematical Physics, 130(2): 245­255 (2002)
DISSIPATIVE AND HAMILTONIAN SYSTEMS WITH
CHAOTIC BEHAVIOR: AN ANALYTIC APPROACH
A. K. Abramyan1
and S. A. Vakulenko1
Some classes of dissipative and Hamiltonian distributed systems are described. The dynamics of these
systems is effectively reduced to finite-dimensional dynamics which can be "unboundedly complex" in
a sense. Yarying the parameters of these systems, we can obtain an arbitrary (to within the orbital
topological equivalence) structurally stable attractor in the dissipative case and an arbitrary polynomial
weakly integrable Hamiltonian in the conservative case. As examples, we consider Hopfield neural networks
and some reaction­diffusion systems in the dissipative case and a nonlinear string in the Hamiltonian case.
1. Introduction
We consider the problem of chaotic behavior for dissipative and Hamiltonian systems described by
systems of coupled oscillators and systems of nonlinear partial differential equations. Our objective is to
describe some physical and mechanical models in which the appearance of various types of structurally
stable periodic and chaotic behavior and the formation of space­time structures are observed. We consider
the dependence of this behavior on the system parameters. The suggested approach is purely analytic, and
the related investigation results cannot be obtained using computers, even in principle.
We begin by presenting the physical idea of our approach and then consider examples of fundamental
dissipative systems with chaotic behavior constructed in [1], [2]. We note that these results are analytic

  

Source: Abramian, Andrei K. - Institute for Problems in Mechanical Engineering, Russian Academy of Sciences

 

Collections: Engineering