## Basics of Nonlinearities in Mathematical SciencesThis book is primarily an attempt to familiarize the reader with nonlinear systems, particularly qualitative characteristics, in a variety of systems amenable to mathematization. Differential equations form the bulk of the book, while the basics of nonlinearities are presented through theorems and problems, aiming to bring out the essence of some aspects of nonlinearities in the emerging discipline of mathematical science. Qualitative studies that reflect the evolution of nonlinearities have not thus far been approached in this way. The uniqueness of the book lies in coupling historical perspectives with the latest trends in nonlinearities. Appendices are intended for inquisitive users of the book for further developments. This book will be of interest to students of Mathematics at the postgraduate and undergraduate level, while those involved in the disciplines of Physics, Chemistry, Biology, Ecology, Technology and Economics should also find the work intriguing. Dilip Kumar Sinha, formerly Sir Rashbehary Ghose Professor of Applied Mathematics, University of Calcutta, Professor of Mathematics, Jadavpur University, Fellow, Institute of Mathematics and its Applications (UK), Fellow, International Academy of Mathematical Chemistry (USA Crotia), Fellow, National Academy of Sciences of India. He is considered among the world's top mathematicians having held the coveted position of the General President, Indian Science Congress Association. Professor Sinha has also been Vice-Chancellor, Visva Bharati, Shantiniketan and Pro- Vice-Chancellor, University of Calcutta, India. |

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### Contents

Orbits limit cycles Poincare map | 141 |

A prelude | 200 |

A prelude | 222 |

Copyright | |

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approach asymptotically stable attractor autonomous system axis Bendixson called catastrophe set catastrophe theory centre characteristic equation circle closed curve closed orbit closed path co-dimension constant continuously differentiable coordinates corresponding critical point 0,0 defined diffeomorphism differential equation domain dx dt dynamical system earlier eigen values equilibrium point example exists fixed point Hence homoeomorphism hyperbolic hyperbolic fixed point invariant set Investigate the behaviour Lemma Let us consider Liapunov function Liapunov's theorem limit cycle limit points limit set linear system mathematical matrix motion negative definite node nonlinear system origin parameter pendulum periodic orbit periodic solution perturbations phase plane phase portrait phase space Poincare map Poincare-Bendixson theorem positive definite proof Remark Rene Thom saddle point sequence shown in Fig spiral stability or otherwise stable equilibrium structurally stable system dx system given system x topologically trajectory transverse unstable manifold variables vector field zero solution