The Simplest Normal Form of Generalized Hopf Bifurcation with Perturbation Parameters
نویسندگان
چکیده
This paper extends a recently developed method for Hopf bifurcation to compute the SNF of generalized Hopf bifurcations. The particular attention is focused on a codimensiontwo generalized Hopf bifurcation. It is shown that the SNF with perturbation parameters cannot be obtained using only near-identity transformation. Additional transformations such as time and parameter rescaling need be introduced. Moreover, an efficient, recursive formula is presented for computing the SNF. An electrical circuit is given to demonstrate the applicability of the new method.
منابع مشابه
The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations
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 singu...
متن کاملNormal forms of Hopf Singularities: Focus Values Along with some Applications in Physics
This paper aims to introduce the original ideas of normal form theory and bifurcation analysis and control of small amplitude limit cycles in a non-technical terms so that it would be comprehensible to wide ranges of Persian speaking engineers and physicists. The history of normal form goes back to more than one hundreds ago, that is to the original ideas coming from Henry Poincare. This tool p...
متن کاملSimplest Normal Forms of Hopf and Generalized Hopf Bifurcations
The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form. In this paper, three theorem...
متن کاملHOPF BIFURCATION CONTROL WITH PD CONTROLLER
In this paper, we investigate the problem of bifurcation control for a delayed logistic growth model. By choosing the timedelay as the bifurcation parameter, we present a Proportional - Derivative (PD) Controller to control Hopf bifurcation. We show that the onset of Hopf bifurcation can be delayed or advanced via a PD Controller by setting proper controlling parameter. Under consideration mode...
متن کاملThe simplest normal form and its application to bifurcation control
This paper is concerned with the computation of the simplest normal forms with perturbation parameters, associated with codimension-one singularities, and applications to control systems. First, an efficient method is presented to compute the normal forms for general semi-simple cases, which combines center manifold theory and normal form theory in one unified procedure. The efficient approach ...
متن کامل