Abstract
AbstractScaling time and spatial domains often seem natural in ODEs, but unfortunately, its useful explicit form does not always agree with intuition. After reviewing the well-known methods, multiple timing and averaging, we will show that algebraic timelike variables may play a part in bifurcations. Part of this discussion is tied in with the problem of structural stability in the analysis of matrices, and another part is determined by bifurcation theory of nonlinear systems. Secondly, the theory of resonance manifolds for higher dimensional problems involving at least 2 angles will show the presence of unexpected small spatial domains that may emerge involving long timescales and containing interesting phenomena. A number of examples from mechanics are presented to demonstrate the theory.
Publisher
Springer Science and Business Media LLC
Subject
Electrical and Electronic Engineering,Applied Mathematics,Mechanical Engineering,Ocean Engineering,Aerospace Engineering,Control and Systems Engineering
Reference30 articles.
1. Bakri, T., Verhulst, F.: From A Tondl’s Dutch contacts to Neimark-Sacker-bifurcation. Appl. Comput. Mech. (2022). https://doi.org/10.24132/acm.2022.770
2. Chen, Lin-Yuan., Goldenfeld, N., Oono, Y.: Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory. Phys. Rev. 54, 376–394 (1996)
3. Chiba, H.: Extension and unification of singular perturbation methods for ODEs based on the renormalization group method. SIAM J. Appl. Dyn. Syst. 8, 1066–1115 (2009)
4. Cochran, J.A.: Problems in singular perturbation theory. Stanford University (1962)
5. Contopoulos, G.: Resonance phenomena and the non-applicability of the “third” integral, Coll. Besançon 1966 (F.Nahon and M. Hénon, eds.), Ed. du CNRS VII (1967)
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