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Periodic solutions of nonlinear dynamical systems. Numerical computation, stability, bifurcation and transition to chaos. (English) Zbl 0728.34043

Lecture Notes in Mathematics, 1483. Berlin etc.: Springer-Verlag. vi, 171 p. DM 36.00 (1991).
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 chapter (40 pp.) an analogous treatment is given for differentiable dynamical systems with discontinuities along certain submanifolds. As examples, a one-staged gear wheel set and an excitation model with dry friction are considered.
Reviewer: W.Müller (Berlin)

MSC:

34C25 Periodic solutions to ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C23 Bifurcation theory for ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
65J99 Numerical analysis in abstract spaces
70K20 Stability for nonlinear problems in mechanics
34D20 Stability of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
70K30 Nonlinear resonances for nonlinear problems in mechanics
37C75 Stability theory for smooth dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
65L99 Numerical methods for ordinary differential equations
70K40 Forced motions for nonlinear problems in mechanics
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