Limit Cycles, Part 2: Analytical Tests for Limit Cycles- Lyapunov Functions, Dulac's Criterion

We describe techniques for proving the existence (or nonexistence) of limit cycles in 2-dimensional systems, that is, testing for isolated closed orbits. Methods for ruling out closed orbits include index theory, Lyapunov functions, and Dulac's criterion. A method for establishing the existence of a limit cycle is the Poincare-Bendixson theorem.

► Next, worked examples of using the Poincare-Bendixson theorem.
   • Limit Cycles, Part 3: Poincare-Bendixson W...  

► Previous: An introduction to limit cycles
   • Limit Cycles, Part 1: Introduction & Examples  

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► From 'Nonlinear Dynamics and Chaos' (online course).
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► Additional background on 2D dynamical systems
Phase plane introduction    • Phase Portrait Introduction- Pendulum Example  
Classifying 2D fixed points    • Classifying Fixed Points of 2D Systems - L...  
Gradient systems    • Gradient Systems - Nonlinear Differential ...  
Index theory    • Index Theory for Dynamical Systems, Part 1...  

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References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 7: Limit Cycles


Chapters
0:00 Testing for limit cycles
1:08 Rule out closed orbits with index theory
1:51 Rule out closed orbits if gradient system
2:33 Lyapunov function to rule out closed orbits
7:41 Dulac's criterion to rule out closed orbits
11:29 Establish existence of closed orbit: Poincaré-Bendixson theorem
13:32 Constructing a trapping region

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