Bifurcation Dynamics of a Damped Parametric Pendulum

by Yu Guo (Author), Albert C. J. Luo (Author)

The inherent complex dynamics of a parametrically excited pendulum is of great interest in nonlinear dynamics, which can help one better understand the complex world.

Even though the parametrically excited pendulum is one of the simplest nonlinear systems, until now, complex motions in such a parametric pendulum cannot be achieved. In this book, the bifurcation dynamics of periodic motions to chaos in a damped, parametrically excited pendulum is discussed. Complete bifurcation trees of periodic motions to chaos in the parametrically excited pendulum include:

period-1 motion (static equilibriums) to chaos, and
period- motions to chaos ( = 1, 2, ···, 6, 8, ···, 12).
The aforesaid bifurcation trees of periodic motions to chaos coexist in the same parameter ranges, which are very difficult to determine through traditional analysis. Harmonic frequency-amplitude characteristics of such bifurcation trees are also presented to show motion complexity and nonlinearity in such a parametrically excited pendulum system. The non-travelable and travelable periodic motions on the bifurcation trees are discovered. Through the bifurcation trees of travelable and non-travelable periodic motions, the travelable and non-travelable chaos in the parametrically excited pendulum can be achieved. Based on the traditional analysis, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the parametrically excited pendulum. The results in this book may cause one rethinking how to determine motion complexity in nonlinear dynamical systems

About the Author
Dr. Yu Guo was born on January 14, 1984. He is currently working at Midwestern State University Texas as an Associate Professor. He previously worked at Caterpillar Inc. as an engine structural and dynamics engineer. Dr. Guo was originally from China. He has his B.S. in Computer Engineering from South China University of Technology. He received his M.S. and Ph.D. in Mechanical Engineering at Southern Illinois University Edwardsville. Dr. Guo conducts research focusing on nonlinear vibration and impact dynamics. He has published 14 peer-reviewed journal papers, more than 15 conference articles, 4 book chapters, and 1 monograph. He has also conducted many professional presentations or invited lectures all over the world.

Professor Albert C.J. Luo has worked at Southern Illinois University Edwardsville. For over 30 years, Dr. Luo's contributions on nonlinear dynamical systems and mechanics lie in: (i) the local singularity theory for discontinuous dynamical systems; (ii) dynamical systems synchronization; (iii) analytical solutions of periodic and chaotic motions in nonlinear dynamical systems; (iv) the theory for stochastic and resonant layer in nonlinear Hamiltonian systems; and (v) the full nonlinear theory for a deformable body. Such contributions have been scattered into 20 monographs and over 300 peer-reviewed journal and conference papers. Dr. Luo has served as an editor for the journal Communications in Nonlinear Science and Numerical Simulation, and book series on Nonlinear Physical Science (HEP) and Nonlinear Systems and Complexity (Springer). Dr. Luo was an editorial member for IMeCh E Part K Journal of Multibody Dynamics and Journal of Vibration and Control; and has also organized over 30 international symposiums and conferences on dynamics and control.