The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

Abstract

This paper considers the computation of the simplest parameterized normal forms (SPNF) of Hopf and generalized Hopf bifurcations. Although the notion of the simplest normal form has been studied for more than two decades, most of the efforts have been spent on the systems that do not involve perturbation parameters due to the restriction of the computational complexity. Very recently, two singularities -single zero and Hopf bifurcation -have been investigated, and the SPNFs for these two cases have been obtained. This paper extends a recently developed method for Hopf bifurcation to compute the SPNF of generalized Hopf bifurcations. The attention is focused on a codimension-2 generalized Hopf bifurcation. It is shown that the SPNF cannot be obtained by using only a near-identity transformation. Additional transformations such as time and parameter rescaling are further introduced. Moreover, an efficient recursive formula is presented for computing the SPNF. Examples are given to demonstrate the applicability of the new method.

Spectral sequences and parametric normal forms

“…The following corollary provides a different parametric normal form for generalized Hopf singularity from the ones in [32, Theorem 3] and [14,33] but identical with the parametric normal form presented in [12,13]. This, of course, is a consistent alternative form with those of [14,32,33]. where µ ∈ R N 0 , have a parametric dimension of N 0 .…”

Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

“…Zhang and Leung [19] considered a general four-dimensional normal form of a double Hopf bifurcation. Yu and his associates [20][21][22] developed efficient computing methods for parametric normal forms. They also applied the new method to consider controlling bifurcations of the nonlinear dynamical systems.…”

Infinite Order Parametric Normal Form of Hopf Singularity

“…However, in reality all systems contain some parameters, and thus parametric normal forms are the only useful tool in the direct analysis of engineering and practical problems. Recently, several researchers [Yu, 2002;Yu & Leung, 2003;Liao et al, 2007;Yu & Chen, 2007] have paid particular attention to this problem. They have extended the efficient computing normal form theory with their novel formulas in which the computation of the simplest normal form (SNF) with unfolding is only involved with some successive algebraic equations at each degree.…”

“…The implementation of the formulas and results obtained here generates simpler and more systematic Maple programs. It has been noticed that the parametric simplest normal form may not be obtained without time rescaling and reparametrization [Yu, 2002;Yu & Leung, 2003;Liao et al, 2007;Yu & Chen, 2007]. Thus, our algebraic structure develops the necessary tools for time rescaling and reparametrization.…”

“…Thus, our algebraic structure develops the necessary tools for time rescaling and reparametrization. For a good background and some original ideas used in this paper, we refer the reader to [Jacobson, 1966;Baider & Churchill, 1988;Chua & Kokubu, 1988;Baider, 1989;Baider & Sanders, 1991, 1992Wang, 1993;Kokubu et al, 1996;Chen & Dora, 2000;Wang et al, 2000;Yu, 2002;Algaba et al, 2003;Yu & Leung, 2003;Peng & Wang, 2004;Liao et al, 2007;Yu & Chen, 2007].…”