Multiple Scale Bifurcation Analysis for Finite-Dimensional Autonomous Systems

Codimension-M bifurcations for general, finite-dimensional, autonomous, nonlinear dynamical systems are analyzed. Basic concepts of bifurcation analysis are first summarized. Some sample structures are then introduced as prototype systems for low-codimension bifurcations. Mechanical as well as geometrical aspects of the bifurcation phenomenon are discussed. Eigensolution Sensitivity Analysis is then illustrated for Jacobian matrices admitting complete (non-defective) or incomplete (defective) systems of eigenvectors. The use of integer and fractional power series of a perturbation parameter is discussed and a reconstitution procedure for eigenvalue sensitivity equations is suggested. The Multiple Scale Method of analyzing multiple bifurcations is then illustrated. Non-defective bifurcations are first considered under general conditions of resonance among the critical eigenvalues. The structure of the Amplitude Modulation Equations at various orders of the perturbation procedure is obtained and several algorithmic aspects discussed, namely: the search for steady-state solutions, expansion vs ordering of the bifurcation parameters, reconstitution of the Amplitude Equations and imperfections accounting. Defective bifurcations are then analyzed by exploiting a formal analogy with Sensitivity Analysis. By limiting the study to bifurcation points at which the Jacobian matrix contains a unique critical Jordan block, the cases of defective divergence and defective Hopf bifurcations are analyzed. In both cases the reconstitution procedure is applied and the structure of the relevant bifurcation equations is given. In particular, rules to obtain the equations at any order of approximation are furnished. The techniques illustrated are finally employed to study several low-codimension bifurcations. In particular, the post-critical behavior of the prototype systems previously introduced is analyzed and some results commented.