Differential Equations › Section E · Systems of differential equations › Topic E.12
Limit cycles, energy, and Lyapunov functions
Nonlinear systems can sustain rhythms that no linear system can, and their stability can often be proved without solving anything. This topic covers limit cycles, the theorem that guarantees them in the plane, and the energy and Lyapunov arguments that settle the cases linearisation leaves open.
In this topic
- Rhythms with a fixed size
- Worked example: a cycle in polar form
- The van der Pol oscillator
- The Poincaré-Bendixson theorem
- Conservative systems and energy
- Lyapunov's direct method
- Worked example: where linearisation fails
- Worked example: energy with friction
- Hopf bifurcations and chaos
- Common mistakes to avoid
- Key takeaways
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