Toward Analytical Chaos in Nonlinear Systems

Preface

There has been interest in periodic solutions of dynamical systems for a few centuries. The periodic motion as a steady-state motion has attracted many scientists' attention. Until now, it still cannot be achieved analytically and accurately. In the second half of the twentieth century, using computer software, another steady-state motion (i.e., chaos) has been observed to be caused by bifurcations of periodic motions. Such a steady-state motion can be numerically simulated with computational errors. Other than that, it was not known how to obtain the bifurcation trees of periodic motions to chaos analytically in nonlinear dynamical systems, and the mathematical structures of solutions for chaos are still unknown. This is because it is not yet known how to obtain the exact solutions of periodic motions analytically. This book will try to solve this problem. Further, the title of this book is Toward Analytical Chaos in Nonlinear Systems. The author hopes this book can attract more attention to finding accurate analytical solutions of periodic motions to chaos, and the mathematical structures of chaos solutions can be achieved.

Since 1788, Lagrange has used the fundamental matrix of the linearized system as a moving coordinate transformation to obtain the Lagrange stand form, which is based on the variation of parameter procedure. From the Lagrange standard form, the method of averaging was used to investigate the gravitational three-body problem through a two-body problem ...