Part I: Nonlinear Dynamical Systems.- 1. Mathematical analysis on chaos of a parametrically excited pendulum with damping force.- 2. Energy in hybrid system nonlinear dynamics.- 3.Nonlinear characteristics diagnosis of dynamical systems in engineering.- 4. Synchronization analysis for two coupled populations of phase oscillators.- 5. A Pendulum Synchronizing with a Duffing Oscillator under a Feedback Control.- 6. The Ring Problem of (N+1) Bodies:...