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Nonlinear oscillators
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Duffing and van der Pol oscillators show chaos and limit cycles.
With nonlinear restoring forces or driving, systems show amplitude-dependent frequency, jumps (Duffing), limit cycles (van der Pol) and chaos.
Heartbeats modelled by limit cycles.
No superposition principle.
In more detail
Nonlinear oscillators have equations of motion with non-linear terms, which can produce behaviour not seen in simple harmonic motion, such as multiple stable states and self-sustained oscillations.
Real-world uses
- Heart rhythms and electronic oscillators are modelled with nonlinear equations.
- Clock escapements are nonlinear systems that sustain oscillation.
- Engineers studying the chaotic response of a driven pendulum use nonlinear models such as the Duffing oscillator.
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Frequently asked questions
What is a limit cycle?
A closed trajectory that a nonlinear system settles into, giving self-sustained oscillation.
Are nonlinear oscillators predictable?
Some are, but many show sensitive dependence on initial conditions.
What is a limit cycle?
A closed, stable orbit in phase space that a nonlinear oscillator settles into regardless of its starting point.
Related topics in Oscillations & SHM
Superposition of SHMs · Lissajous figures · Anharmonic oscillation · Chaos and the double pendulum · Parametric resonance · Oscillation of a floating body