Static and dynamic bifurcation analysis of autonomous ODE systems using reductive pertubation method

Abstract

Most chemical or biochemical systems exhibit features like multiple steady states, oscillatory steady states, etc., due to their nonlinear nature. Though many analytical techniques have been in use, few can deal with an irreducible n-equation system comprehensively. Reductive Perturbation Method (RPM) seems to address this issue having already proven very effective for an analytical treatment of Simple Zero Eigenvalue (SZE) and Hopf Point (HP). In this work, we apply the existing RPM results to some biochemical systems and develop extensions of RPM to address additional static and dynamic problems in autonomous ODE systems. We show that the analysis of SZE using fractional orders produces no new branching patterns and is in principle same as that of integer orders. Double Zero Eigenvalue (DZE) gives new theoretical results, but needs higher dimensional systems to illustrate the same. We analyze HP for various perturbation orders. We develop an algorithm which combines local analytical results of bifurcation points (predicted by RPM) to construct a global bifurcation diagram for any given system. We have done some preliminary work on the analysis of bifurcation of T-periodic solution.