Order in chaos. (English) Zbl 0574.73062
This paper is intended as a tutorial survey on a rather new field of research which can have important applications in mechanical engineering. It is concerned with chaos in the technical sense of the word, i.e., more precisely speaking, with deterministic chaos. Two classes of problems are dealt with: (1) Study of deterministic chaos by means of model equations and in particular, how to characterize chaos. (2) How to determine the characteristic features of chaos from experimental data. As model equations the already mentioned Lorenz equations have been used which are autonomous. Examples of nonautonomous equations giving rise to chaos are the Helmholtz and the Duffing equations. For small driver amplitude A these equations can be solved by methods of perturbation theory, whereas in the chaotic regime a numerical integration is required.
MSC:
| 74H55 | Stability of dynamical problems in solid mechanics |
| 70K20 | Stability for nonlinear problems in mechanics |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 76E25 | Stability and instability of magnetohydrodynamic and electrohydrodynamic flows |
Keywords:
discrete maps; period doubling; intermittency; quasiperiodicity; Helmholtz equation; Lyapunov exponents; fractal dimension; strange attractor; Poincare return map; slaving principle; tutorial survey; deterministic chaos; characteristic features of chaos; Lorenz equations; autonomous; nonautonomous equations; Duffing equations; numerical integrationReferences:
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