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. https://youtu.be/SpltxX_LZWs ► Previous: An introduction to limit cycles https://youtu.be/9rVscJwDpBo ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) Subscribe https://is.gd/RossLabSubscribe​ ► From 'Nonlinear Dynamics and Chaos' (online course). Playlist https://is.gd/NonlinearDynamics ► Additional background on 2D dynamical systems Phase plane introduction https://youtu.be/U4IM7HFzcuY Classifying 2D fixed points https://youtu.be/7Ewe_tVa5Fs Gradient systems https://youtu.be/uGUzPZzvPWQ Index theory https://youtu.be/bO8FxxpocNQ ► Follow me on Twitter https://twitter.com/RossDynamicsLab ► Make your own phase portrait https://is.gd/phaseplane ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes 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 Liapunov gradient systems passive dynamic biped walker Tacoma Narrows bridge collapse Charles Conley index theory gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations 2d ODE vector field topology cylinder bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field Two-Dimensional 2-dimensional Functions Hamiltonian Hamilton streamlines weather vortex dynamics point vortices pendulum Newton's Second Law Conservation of Energy topology #NonlinearDynamics #DynamicalSystems #Oscillations #LimitCycles #VectorFields #topology #IndexTheory #EnergyConservation #Hamiltonian #Streamfunction #Streamlines #Vortex #SkewGradient #Gradient #PopulationBiology #FixedPoint #DifferentialEquations #Bifurcation #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CuspBifurcation #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Wiggins #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions #PopulationGrowth #DynamicalSystems #PopulationDynamics #Population #Logistic #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion