## Bifurcation and Chaos in EngineeringFor the many different deterministic non-linear dynamic systems (physical, mechanical, technical, chemical, ecological, economic, and civil and structural engineering), the discovery of irregular vibrations in addition to periodic and almost periodic vibrations is one of the most significant achievements of modern science. An in-depth study of the theory and application of non-linear science will certainly change one's perception of numerous non-linear phenomena and laws considerably, together with its great effects on many areas of application. As the important subject matter of non-linear science, bifurcation theory, singularity theory and chaos theory have developed rapidly in the past two or three decades. They are now advancing vigorously in their applications to mathematics, physics, mechanics and many technical areas worldwide, and they will be the main subjects of our concern. This book is concerned with applications of the methods of dynamic systems and subharmonic bifurcation theory in the study of non-linear dynamics in engineering. It has grown out of the class notes for graduate courses on bifurcation theory, chaos and application theory of non-linear dynamic systems, supplemented with our latest results of scientific research and materials from literature in this field. The bifurcation and chaotic vibration of deterministic non-linear dynamic systems are studied from the viewpoint of non-linear vibration. |

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### Contents

Dynamical Systems Ordinary Differential Equations | 1 |

Calculation of Flows | 35 |

Discrete Dynamical Systems | 66 |

Copyright | |

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### Common terms and phrases

amplitude autonomous system averaging method behaviour bifurcation diagram bifurcation equation bifurcation point bifurcation solutions bifurcation theory calculate called chaos characteristic closed orbit co(x codimension coefficients Consider corresponding curve damping defined Definition dimensional Duffing equation dynamical systems eigenvalues eigenvectors equilibrium point equivalent example exists finite fixed point Floquet flow frequency Hamiltonian system homoclinic homoclinic orbit Hopf bifurcation hyperbolic implicit function theorem initial conditions initial value problem integration invariant torus lemma limit cycle matrix negative neighbourhood normal form obtain ordinary differential equation oscillators parameter periodic orbit periodic solution perturbation phase portrait phase space plane polynomial Proof region resonant rotating RT(g saddle point satisfies shown in Fig solution of eq stable and unstable structural stability subharmonic subharmonic solutions subspace Substituting eq Suppose tangent topological trajectories transformation transition set transversal trivial solution two-dimensional unique universal unfolding unstable manifolds variable vector field zero eigenvalue