Periodic Solutions of Nonlinear Dynamical Systems: Numerical Computation, Stability, Bifurcation, and Transition to ChaosLimit cycles or, more general, periodic solutions of nonlinear dynamical systems occur in many different fields of application. Although, there is extensive literature on periodic solutions, in particular on existence theorems, the connection to physical and technical applications needs to be improved. The bifurcation behavior of periodic solutions by means of parameter variations plays an important role in transition to chaos, so numerical algorithms are necessary to compute periodic solutions and investigate their stability on a numerical basis. From the technical point of view, dynamical systems with discontinuities are of special interest. The discontinuities may occur with respect to the variables describing the configuration space manifold or/and with respect to the variables of the vector-field of the dynamical system. The multiple shooting method is employed in computing limit cycles numerically, and is modified for systems with discontinuities. The theory is supported by numerous examples, mainly from the field of nonlinear vibrations. The text addresses mathematicians interested in engineering problems as well as engineers working with nonlinear dynamics. |
Contents
Introduction | 3 |
Differentiable dynamical systems | 9 |
Representation in state space coordinates | 18 |
Copyright | |
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Other editions - View all
Periodic Solutions of Nonlinear Dynamical Systems: Numerical Computation ... Eduard Reithmeier No preview available - 2014 |
Periodic Solutions of Nonlinear Dynamical Systems: Numerical Computation ... Eduard Reithmeier No preview available - 1991 |
Common terms and phrases
according to equation algebraic constraint algorithm analytical boundary value problem BULIRSCH codimension computation of periodic connected component coordinates defined definition degrees of freedom DF(zo diffeomorph double pendulum dry friction dynamical behavior dynamical system eigenvalues energy value equation of motion Example excitation exists field f frame HAMILTONian function HAMILTONian systems Hence homoclinic orbit HOPF HOPF-bifurcation impact indicator function investigate k₂ limit cycle limit function linearized system manifold Math mathematical monodromy matrix neighborhood non-linear system normal forms numerical computation ODE's parameter periodic solutions PFEIFFER phase curve POINCARÉ polynomial RABINOWITZ railway vehicle rattling model real axis real physical space recursion resonant respect separatrix singular point space TM stability and bifurcation stability domain stable fixed points theorem trajectory transformation transversally unique unit circle unstable value h variables vector field velocity Verlag wheel set zeroset