Periodic solutions of nonlinear dynamical systems. Numerical computation, stability, bifurcation and transition to chaos
type: Item
instance of: scholarly article
title: Periodic solutions of nonlinear dynamical systems. Numerical computation, stability, bifurcation and transition to chaos
zbMATH Open document ID: 0728.34043
published in: Lecture Notes in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: In these lecture notes, the term, ``dynamical system'' is understood in the sense of mechanical engineering, i.e. as the mathematical model of a mechanical system with f degrees of freedom (f a finite integer), exhibiting a combination of (free or forced) translational and rotational motions. The mathematical description is assumed to be given by \(\dot x=f(x,u,p,t)\), f: TM\(\times U\times P\times I\to TM\) (where M is the configuration space, a differentiable manifold, locally isomorphic to \({\mathbb{R}}^ f\), with dim M\(=F\), \(TM=M\times {\mathbb{R}}^ f\) is the state space, \(U\subset {\mathbb{R}}^ m\) is some range of admissible controls, \(P\subset {\mathbb{R}}^ k\) is the parameter space, \(I\subset {\mathbb{R}}\) is the range of time of excitation), eventually complemented by certain nonlinear algebraic constraints \(g(x,u,p,t)=0\). By using control laws \(u=\psi (x,p,t)\) and Lagrange multiplier techniques and, if necessary, by introducing the time t as an additional state variable, the system description is reduced to the standard form \(\dot x=f(x,p).\) For systems of this type, the complete spectrum of methods for the investigation of stationary and periodic solutions is presented: transformation to normal forms, classification of singularities, numerical calculation of periodic solutions for Hamiltonian systems and of limit cycles for dissipative and externally excited systems, calculation of periodic solutions with additional properties (e.g. minimum period for a certain \(p=p^*\in P)\), investigation of the stability properties of the obtained periodic solutions and of the types of possible bifurcations for p varying in P. All steps of the presentation are accompanied and illustrated by two typical examples: the double pendulum, and a dynamical model of a wheel set of a railway vehicle system with four degrees of freedom. For these examples, fascinating plots with families of numerically computed periodic solutions are presented. - In the last chap…
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 70-02
Mathematics Subject Classification ID: 70K05
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 65J99
Mathematics Subject Classification ID: 70K20
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 70K30
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 65L99
Mathematics Subject Classification ID: 70K40
zbMATH DE Number: 51254
zbMATH Keywords: dynamical system
zbMATH Keywords: mechanical engineering
zbMATH Keywords: differentiable manifold
zbMATH Keywords: periodic solutions
zbMATH Keywords: transformation to normal forms
zbMATH Keywords: classification of singularities
zbMATH Keywords: numerical calculation
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: limit cycles
zbMATH Keywords: stability
zbMATH Keywords: bifurcations
zbMATH Keywords: double pendulum
zbMATH Keywords: railway vehicle system
zbMATH Keywords: discontinuities
zbMATH Keywords: excitation model with dry friction
author: Eduard Reithmeier
MaRDI profile type: Publication
DOI: 10.1007/BFB0094521
description: scientific article; zbMATH DE number 51254