Multiple timing and spatial scaling for bifurcations

Abstract

Scaling 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.