Periodic solutions of nonlinear dynamical systems. Numerical computation, stability, bifurcation and transition to chaos
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.
scientific article; zbMATH DE number 51254