Premium
On the (in)stability of quasi‐static paths of smooth systems: definitions and sufficient conditions
Author(s) -
Martins J. A. C.,
Rebrova N. V.,
Sobolev V. A.
Publication year - 2006
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.707
Subject(s) - mathematics , perturbation (astronomy) , singular perturbation , stability (learning theory) , property (philosophy) , quasistatic process , instability , attractiveness , mathematical analysis , control theory (sociology) , computer science , mechanics , control (management) , psychology , philosophy , physics , epistemology , quantum mechanics , machine learning , artificial intelligence , psychoanalysis
A concept of stability of quasi‐static paths is discussed in this paper that takes into account the existence of fast (dynamic) and slow (quasi‐static) time scales in the evolution of many mechanical systems. The proposed concept is essentially a continuity property with respect to the smallness of the initial perturbations (as in Lyapunov stability) and the smallness of the quasi‐static loading rate (that plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Several examples illustrate the relevance of the definitions. Sufficient conditions for attractiveness or for instability of quasi‐static paths of smooth systems are proved. Copyright © 2005 John Wiley & Sons, Ltd.