Method of multiple scales for discrete time dynamical systems: a general approach and engineering applications

Abstract Bifurcation analysis is an essential tool for understanding nonlinear dynamical phenomena. It is particularly useful for the assessment of system robustness with regard to perturbations in the initial conditions. There are many methods and tools for bifurcation analysis in both continuous and discrete time dynamical systems. One popular approach in continuous time systems is the method of multiple scales. In this paper, we extend this method for discrete time dynamical systems, where the algorithm leads to difference equations only, based on Taylor-series expansion in the slow time-scale. The examples of digital force control and highly interrupted cutting demonstrate the efficiency of this approach for all the three elementary discrete bifurcation cases, that is, for fold, period-doubling and Neimark-Sacker types. Some further mathematical examples, including a codimension-2 bifurcation scenario, are presented to demonstrate further generalizations of the proposed method. The results are validated using both numerical and alternative analytical approaches.