Chaotic Dynamics in Nonlinear Theory (eBook)

Using phase-plane analysis, findings from the theory of topological horseshoes and linked-twist maps, this book presents a novel method to prove the existence of chaotic dynamics. In dynamical systems, complex behavior in a map can be indicated by showing the existence of a Smale-horseshoe-like structure, either for the map itself or its iterates. This usually requires some assumptions about the map, such as a diffeomorphism and some hyperbolicity conditions. In this text, less stringent definitions of a horseshoe have been suggested so as to reproduce some geometrical features typical of the Smale horseshoe, while leaving out the hyperbolicity conditions associated with it. This leads to the study of the so-called topological horseshoes. The presence of chaos-like dynamics in a vertically driven planar pendulum, a pendulum of variable length, and in other more general related equations is also proved.

LAKSHMI BURRA is associate professor of mathematics at Jawaharlal Nehru Technological University (JNTU), Hyderabad. Professor Burra has bagged two PhDs in mathematics: first from Osmania University, Hyderabad, and then from the University of Udine, Italy. She is also a postgraduate in philosophy. With a large number of research publications in leading international journals, her research topics include mathematical modeling of real-life problems such as epidemics, population dynamics, mathematical physics and climate modeling to chaotic dynamics. Some of her important results have been on oscillating population models and long-term models in climate including the link with sunspots. As well as her teaching responsibilities, Professor Burra continues to carry out research both in India and in Europe, where she has an active and ongoing collaboration. She has presented her work in invited talks in several international conferences in India, Europe and the USA. Professor Burra is married and has two sons.

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Using phase-plane analysis, findings from the theory of topological horseshoes and linked-twist maps, this book presents a novel method to prove the existence of chaotic dynamics. In dynamical systems, complex behavior in a map can be indicated by showing the existence of a Smale-horseshoe-like structure, either for the map itself or its iterates. This usually requires some assumptions about the map, such as a diffeomorphism and some hyperbolicity conditions. In this text, less stringent definitions of a horseshoe have been suggested so as to reproduce some geometrical features typical of the Smale horseshoe, while leaving out the hyperbolicity conditions associated with it. This leads to the study of the so-called topological horseshoes. The presence of chaos-like dynamics in a vertically driven planar pendulum, a pendulum of variable length, and in other more general related equations is also proved.