On The Fundamental Issues In Nonstationary Dynamics And Chaos


In nonstationary (NS) systems some control parameters (CPs) have the following forms: CP(t) = CP0+ψ(t), where CP0 = const. and ψ(t) are arbitrary functions of t. Other arbitrary functions which play a pivotal role in the NS systems are the parameter functions φ(CP) = 0, CP = {CP1, CP2...CPn}. While the functions ψ(t) determine the time directions of the NS dynamical behavior, the functions φ(CP) = 0 determine the paths for the CPs to follow. The NS processes are permanently transient due to the functions ψ(t) and/or φ(CP) = 0, and for that reason, they can be extremely complex. Clearly then, it is essential to address the fundamental problems of cohesion and definiteness of these processes. Using select examples, these issues have been studied in this presentation and have been resolved in positive. Specifically, the following have been demonstrated: (1) convergence (definitiveness) of the NS logistic map and the softening Duffing oscillator to an NS limit motion (2) the appearance of a sequence of similar attractors for different NS bifurcations (3) the effects of different parameter paths, φ(CP) = 0, in the period doubling region of the Duffing oscillator (4) the effects of linear and cyclic paths in transition through the Ueda bifurcation regions. The results obtained show considerable complexity of the NS dynamic and chaotic responses (5) for exponential ψ(t), the Lorenz 'weather' three-term model exhibit a periodic 'window' in the chaotic range for an extended value of t. (6) the effects of different ψ(t) in the typical codimension one bifurcations (7) the ST chaos may be created or annihilated by injection of NS inputs (8) an efficient and fast stabilization, i.e., reduction of ST vibration to near the static equilibrium in a short time, can be accomplished by NS changes of the parameters of the system.

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American Society of Mechanical Engineers, Design Engineering Division (Publication) DE



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Article; Proceedings Paper



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0027803623 (Scopus)

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