The problem of determining the bounds of applicability of perturbation expansions in terms both of the system parameters and the state-space variable amplitude is a key point in the perturbation analysis of nonlinear systems. In the present paper an analysis in a finite neighborhood of a Hopf bifurcation is presented in order to analyze the conditions under which a Normal Form zero-divisors-based approach fails to describe the local dynamics and, therefore, a small divisor approach is required. The condition of “smallness” referred to the divisors is analyzed from both a qualitative and a quantitative point of view. Finally, a simple but effective analytical and numerical example is introduced to illustrate the theoretical issues along with an interpretation within a codimension-two framework.

A normal form analysis in a finite neighborhood of a hopf bifurcation: on the center manifold dimension / Eugeni, Marco; Dessi, Daniele; Mastroddi, Franco. - In: NONLINEAR DYNAMICS. - ISSN 0924-090X. - ELETTRONICO. - 91:3(2018), pp. 1461-1472. [10.1007/s11071-017-3958-3]

A normal form analysis in a finite neighborhood of a hopf bifurcation: on the center manifold dimension

Marco Eugeni
Writing – Original Draft Preparation
;
Franco Mastroddi
Writing – Original Draft Preparation
2018

Abstract

The problem of determining the bounds of applicability of perturbation expansions in terms both of the system parameters and the state-space variable amplitude is a key point in the perturbation analysis of nonlinear systems. In the present paper an analysis in a finite neighborhood of a Hopf bifurcation is presented in order to analyze the conditions under which a Normal Form zero-divisors-based approach fails to describe the local dynamics and, therefore, a small divisor approach is required. The condition of “smallness” referred to the divisors is analyzed from both a qualitative and a quantitative point of view. Finally, a simple but effective analytical and numerical example is introduced to illustrate the theoretical issues along with an interpretation within a codimension-two framework.
2018
normal form; small divisors; hopf bifurcation; nonlinear systems
01 Pubblicazione su rivista::01a Articolo in rivista
A normal form analysis in a finite neighborhood of a hopf bifurcation: on the center manifold dimension / Eugeni, Marco; Dessi, Daniele; Mastroddi, Franco. - In: NONLINEAR DYNAMICS. - ISSN 0924-090X. - ELETTRONICO. - 91:3(2018), pp. 1461-1472. [10.1007/s11071-017-3958-3]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1028702
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