Generalized Normal Forms in Dynamical Systems Near Equilibrium

Abstract

Chaotic behavior of continuous dynamical systems is usually investigated using numerical techniques where sensitivity requires correct choice of method. Calculations may lead to truncation errors so that the dynamical behavior of the system might be misunderstood. To get a better understanding of the system, a more comprehensive analytic approach should be used to complement numerical methods. The central manifold or normal form approach can enable one to understand the system if the eigenvalue spectrum of the linearized system has the required characteristics. An alternative approach is using non near identity transformations. In this study, we try to modify the eigenvalue spectrum by means of scaling or logarithmic transformations to selected nonlinear systems mostly with three degrees of freedom that can modify or push resonances to higher orders and remove or simplify continuous time dynamical systems, such as those studied by Sprott to get a better picture of its qualitative behavior.