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Limit Cycles, Part 1: Introduction & Examples

Limit cycles are isolated periodic orbits which are inherently nonlinear, and form the main mechanism of oscillations in many systems. We give several physical examples of stable limit cycles, such as aeroelastic limit cycle oscillations, flapping of window blinds in a breeze, shaking bridges, heart beats, predator-prey cycles, and even walking. Analytical examples are also given, including the Van der Pol oscillator. ► Next, we look at techniques for proving the existence (or nonexistence) of limit cycles    • Limit Cycles, Part 2: Analytical Test...   ► Previous: Index theory for dynamical systems, part 2: Poincaré-Hopf index theorem or Why you can't comb a coconut    • Index Theory for Dynamical Systems, P...   ► 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 ► For background on 2D dynamical systems, see Phase plane introduction    • Phase Portrait Introduction- Pendulum...   Classifying 2D fixed points    • Classifying Fixed Points of 2D Systems   Systems with special structure    • Gradient Systems - Nonlinear Differen...   Index theory    • Index Theory for Dynamical Systems, P...   ► Follow me on Twitter   / 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 ► Courses and Playlists by Dr. Ross 📚Attitude Dynamics and Control https://is.gd/SpaceVehicleDynamics 📚Nonlinear Dynamics and Chaos https://is.gd/NonlinearDynamics 📚Hamiltonian Dynamics https://is.gd/AdvancedDynamics 📚Three-Body Problem Orbital Mechanics https://is.gd/SpaceManifolds 📚Lagrangian and 3D Rigid Body Dynamics https://is.gd/AnalyticalDynamics 📚Center Manifolds, Normal Forms, and Bifurcations https://is.gd/CenterManifolds ► Chapters 0:00 What is a limit cycle? 3:00 A continuum of periodic orbits are NOT limit cycles 4:58 Physical examples of limit cycles 12:18 Analytical examples 17:59 Van der Pol equation 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 #LimitCycles #DynamicalSystems #Oscillations #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