The Chaotic Pendulum

Pendulum is the simplest nonlinear system, which, however, provides the means for the description of different phenomena in Nature that occur in physics, chemistry, biology, medicine, communications, economics and sociology. The chaotic behavior of pendulum is usually associated with the random force acting on a pendulum (Brownian motion). Another type of chaotic motion (deterministic chaos) occurs in nonlinear systems with only few degrees of freedom. This book presents a comprehensive description of these phenomena going on in underdamped and overdamped pendula subject to additive and multiplicative periodic and random forces. No preliminary knowledge, such as complex mathematical or numerical methods, is required from a reader other than undergraduate courses in mathematical physics. A wide group of researchers, along with students and teachers will, thus, benefit from this definitive book on nonlinear dynamics. 1. Pendulum equations. 1.1. Mathematical pendulum. 1.2. Period of oscillations. 1.3. Underdamped pendulum. 1.4. Nonlinear vs linear equation. 1.5. Isomorphic models. 1.6. General concepts -- 2. Deterministic chaos. 2.1. Damped, periodically driven pendulum. 2.2. Analytic methods. 2.3. Parametric periodic force. 2.4. Parametrically driven pendulum. 2.5. Periodic and constant forces. 2.6. Parametric and constant forces. 2.7. External and parametric periodic forces -- 3. Pendulum subject to a random force. 3.1. Noise. 3.2. External random force. 3.3. Constant and random forces. 3.4. External periodic and random forces. 3.5. Pendulum with multiplicative noise. 3.6. Parametric periodic and random forces. 3.7. Damped pendulum subject to a constant torque, periodic force and noise. 3.8. Overdamped pendulum -- 4. Systems with two degrees of freedom. 4.1. Spring pendulum. 4.2. Double pendulum. 4.3. Spherical pendulum -- 5. Conclusions