Multimode Character of Dynamical Systems As a Cause of Their Complex ("Chaotic") Behavior

Abstract eng:
This study puts under consideration dynamic systems, whose visible complex (“chaotic”) behavior is caused by transition from one motion mode to another. Such motion modes may be represented in particular by some stable in the small periodic motions with different periods. Two types of such systems are singled out. In the first case a number of motion regimes with closely located domains of attraction coexist in the phase space of the system at certain values of parameters. The transition from one stable regime of motion to another is due to the inaccuracy of computer calculations and variation of parameters in the corresponding physical system. In the second case a systematic transition from one regime of motion (which is not necessarily stable) to another occurs due to the internal properties of the system. As an example is the situation, when one of the phase variables may be considered as a relatively slowly varying parameter passing through the existence and steadiness domains of various regimes. The behavior of a particle over a horizontal vibrating plane and emergence of a turbulent surface layer in liquid placed in a vibrating vessel are considered in this paper as illustrative examples. The resemblance of this process to that of the thermoconvection is pointed out. The other example is an analogue of the Lorenz oscillator marked by a phase pattern projection similar to the well-known ”butterfly“ pattern. Some thoughts corroborating Landau turbulence theory are suggested. It is noted that the complex motion under consideration is characteristic for sufficiently wide scope of dynamic systems such as a pendulum with a vibrating axis, the self-synchronizing oscillating and rotating objects, the systems with period doubling, the parametrically excited distributed systems. The author is grateful to A.L. Fradkov for the discussion and comments.

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