Periodic Systems and Periodic Motion - Parametric Resonance Tongues of Instability, Mathieu Equation

Explore periodic systems and motion in this comprehensive lecture on Hamiltonian and nonlinear dynamics. Delve into the analysis of time-dependent systems, focusing on periodic time-dependence. Discover parametric resonance through the motion of a pendulum with a vibrating pivot, progressing from simple "square-wave" forcing to more realistic sinusoidal forcing, leading to the Mathieu equation. Investigate the fascinating Kapitza pendulum and learn about vibration-induced stability of the inverted pendulum. Gain insights into resonance tongues of instability, forcing response diagrams, and the geometry of stroboscopic Poincaré maps for forced systems. This in-depth exploration covers essential concepts in dynamical systems, nonlinear dynamics, and mechanics, providing a solid foundation for understanding complex periodic phenomena.

Time-periodic system introduction.
Square wave forcing of simple harmonic oscillator.
Forcing response diagram.
eigenvalues of the mapping matrix M.
Resonance tongues for square wave forcing.
Stable and unstable examples of resonant motion.
Going to sinusoidal forcing .
Mathieu equation.
Resonance tongues of instability.
Kapitza pendulum - vibration-induced stability of inverted pendulum.
Geometry of stroboscopic Poincare map for forced system.