Summary: Formation of complex spatialtime structures for elastic systems with defects
A. K. Abramyan, S. A. Vakulenko
Institute of Mechanical Engineering Problems,
Russian Academy of Science, Saint Petersbourg
Given paper is focused on the problem of a complex large time behaviour and the structure formation
for nonlinear nondissipative and dissipative systems.
Our aim is to describe physical and mechanical models where we can observe many different kinds of
the large time behaviour and the spatial structures ( depending on the system parameters). Our approach
is analytic, results cannot be obtained by a computer (even very powerful).
For dissipative systems, the mathematical results [V1] [V2] were based on a physical idea: the com
plicated behaviour arises as a result of a nonlocal interaction of ''localized modes''.
Our goal is to show, by formal asymptotic, that such approach also is applicable to nondissipative
systems. First, let us recall the results for the dissipative case.
2. Some results for dissipative systems
Dissipative systems (for example, the reaction diffusion ones) describe a wide number of phenomena
in biology, ecology, physics and chemistry ( for example, [NP] [Hak] [Ku] among many others).
These results show the existence of finite dimensional global attractors for wide classes of dissipative
systems. Detailed investigation of the global attractor structure is possible only under some essential re
strictions. The attractor structure and the large time behaviour have been well studied for the quasilinear