Computational aspects of normal form perturbation expansions

The method of normal forms is used to develop analytic solutions to weakly nonlinear ordinary differential equations about an equilibrium solution of the system. Such equations arise in a broad spectrum of areas where one models vibrations and oscillations of mechanical systems, oscillations and feedback in biological and ecological systems, tracking of particles in an accelerator and long-time planetary motion in astronomy. An approximation to the true solution is sought that is valid for a long time with a prescribed error. It is constructed by means of a near-identity transformation from the original system to a nearby one. The transformation is determined by a perturbation expansion as a power series in a small parameter. The method of normal forms, introduced by Poincare in his Ph.D. thesis, was further expanded by Bruno, Arnold, Kummer and others who have emphasized the nonuniqueness of the transformation. Recently, Kahn and Zarmi (1991) developed the method of minimal normal forms (MNF) which uses the nonuniqueness to terminate the normal form equation in an early order of the expansion. In the past, investigators may have missed exploiting this "freedom" because traditionally calculations were carried only to first or second order and for Hamiltonian systems this nonuniqueness is fixed when one requires that the transformation be canonical.;The work presented in this thesis centers on three main aspects: (1) establishing the numerical efficacy of MNF for conservative planar systems (Kahn, Murray and Zarmi (1993)), (2) displaying the computational and conceptual simplicity of MNF for dissipative systems (Murray 1994a)), (3) and demonstrating the utility of computer algebra programs in performing very high-order calculations (Forest and Murray) and in exploring the different dynamical features of a system (Kahn, Murray and Zarmi (1994a and b)). In sections 4-11 we have included many examples in order to illustrate the wide applicability of the method of MNF to a broad spectrum of mostly planar problems. In contrast, in section 12 we discuss the limited applicability of the method in higher dimensions. One notes that although MNF are quite effective for planar problems, it has become clear primarily through the work of Dr. Etienne Forest that they are not suitable for the study of higher-dimensional systems (Forest and Murray).