11.2 Nonlinear oscillators and pendulums

Nonlinear oscillators and pendulums are fascinating systems that show complex behaviors. They go beyond simple back-and-forth motion, displaying wild swings, multiple equilibrium points, and even chaos. These systems pop up everywhere, from electrical circuits to biological processes.

In this part, we'll check out some famous nonlinear oscillators like the Duffing and Van der Pol. We'll also look at how external forces and changing parameters affect these systems. Finally, we'll dive into pendulums and see how they behave in phase space.

Nonlinear Oscillators

Duffing Oscillator and Van der Pol Oscillator

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• Duffing oscillator exhibits nonlinear behavior due to a cubic term in its restoring force
  • Described by the equation: $\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos(\omega t)$
  • Displays phenomena such as multiple stable equilibrium points (bistability) and chaotic motion depending on parameter values
• Van der Pol oscillator is a nonlinear oscillator with a nonlinear damping term
  • Governed by the equation: $\ddot{x} - \mu (1 - x^2) \dot{x} + x = 0$
  • Exhibits self-sustained oscillations and limit cycle behavior
  • Used to model various systems such as electrical circuits and biological processes (heart rhythms)

Forced and Parametric Oscillations

• Forced oscillations occur when an external periodic force is applied to a nonlinear oscillator
  • The external force can lead to various phenomena such as frequency entrainment and chaotic motion
  • The response of the oscillator depends on the frequency and amplitude of the external force
• Parametric oscillations arise when a parameter of the oscillator varies periodically with time
  • The equation of motion contains a time-varying parameter, e.g., $\ddot{x} + \omega^2(t) x = 0$
  • Parametric resonance can occur when the frequency of the parameter variation matches specific ratios of the natural frequency of the oscillator
  • Leads to exponential growth of oscillation amplitude under certain conditions (parametric instability)

Resonance in Nonlinear Oscillators

• Resonance is a phenomenon where the oscillator's amplitude reaches a maximum at specific frequencies of the external force
  • In linear oscillators, resonance occurs when the external force frequency matches the natural frequency of the oscillator
• Nonlinear oscillators can exhibit more complex resonance behavior
  • Multiple resonance peaks can occur at different frequencies due to the nonlinear terms in the equation of motion
  • The shape and location of the resonance peaks depend on the specific nonlinearity and parameter values
• Practical applications of resonance include energy harvesting, vibration control, and signal processing (filters)

Pendulums and Phase Space

Simple Harmonic Oscillator and Nonlinear Pendulum

• Simple harmonic oscillator is a linear oscillator described by the equation: $\ddot{x} + \omega^2 x = 0$
  • Exhibits sinusoidal motion with a constant amplitude and frequency
  • Serves as a fundamental model for many physical systems (mass-spring system)
• Nonlinear pendulum is a pendulum with a nonlinear restoring force due to the sine term in its equation of motion
  • Described by the equation: $\ddot{\theta} + \frac{g}{L} \sin(\theta) = 0$, where $\theta$ is the angular displacement, $g$ is the acceleration due to gravity, and $L$ is the length of the pendulum
  • Exhibits more complex behavior compared to the simple harmonic oscillator, such as amplitude-dependent frequency and chaotic motion for large amplitudes

Phase Space and Limit Cycles

• Phase space is a mathematical space in which all possible states of a system are represented
  • Each point in the phase space corresponds to a unique state of the system
  • The evolution of the system over time is represented by a trajectory in the phase space
• Limit cycles are isolated closed trajectories in the phase space
  • Represent self-sustained oscillations in nonlinear systems
  • Nearby trajectories are attracted to or repelled from the limit cycle
  • Examples include the Van der Pol oscillator and the Belousov-Zhabotinsky reaction (chemical oscillator)
• Phase space analysis provides insights into the long-term behavior and stability of nonlinear systems
  • Fixed points, limit cycles, and chaotic attractors can be identified and characterized in the phase space
  • Bifurcations, which are qualitative changes in the system's behavior, can be studied using phase space techniques (bifurcation diagrams)