Bifurcations in 2D, Part 2: Hopf Bifurcation- Birth of a Limit Cycle from a Fixed Point

A new bifurcation that can occur in two or more dimensions is the Hopf bifurcation, where a limit cycle is created from a fixed point, due purely to nonlinear terms. This can occur when the Jacobian of the fixed point has a complex conjugate pair of eigenvalues that cross the imaginary axis together, going from a stable spiral to an unstable spiral or vice-versa. ► Next, examples and phenomena of Hopf bifurcations    • Bifurcations in 2D, Part 3: Hopf Bifu...   ► From 'Nonlinear Dynamics and Chaos' (online course). Playlist is.gd/NonlinearDynamics ► Bifurcations in 2D Zero eigenvalue bifurcations    • Bifurcations in 2D, Part 1: Introduct...   Hopf bifurcation theory    • Bifurcations in 2D, Part 2: Hopf Bifu...   Hopf physical examples    • Bifurcations in 2D, Part 3: Hopf Bifu...   Bifurcations of limit cycles    • Bifurcations in 2D, Part 4: Global Bi...   ► Bifurcations in 1D (the zero eigenvalue bifurcations) Saddle-node    • Bifurcations Part 1, Saddle-Node Bifu...   Trans-critical    • Bifurcations Part 2- Transcritical Bi...   Pitchfork    • Bifurcations Part 3- Pitchfork Bifurc...   Robustness under perturbation    • Bifurcations Part 4- Robustness of Bi...   ► Additional background on 2D dynamical systems Phase plane introduction    • Phase Portrait Introduction- Pendulum...   Classifying 2D fixed points    • Classifying Fixed Points of 2D Systems   Gradient systems    • Gradient Systems - Nonlinear Differen...   Index theory    • Index Theory for Dynamical Systems, P...   Limit cycles    • Limit Cycles, Part 1: Introduction & ...   Averaging theory    • Averaging Theory for Weakly Nonlinear...   ► Advanced lecture on Hopf bifurcations    • Hopf Bifurcation Example- Normal Form...   ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) Subscribe is.gd/RossLabSubscribe​ ► Follow me on Twitter twitter.com/RossDynamicsLab ► Make your own phase portrait is.gd/phaseplane ► Course lecture notes (PDF) is.gd/NonlinearDynamicsNotes References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 8: Bifurcations Revisited Stephen Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos" (2003) ► Courses and Playlists by Dr. Ross 📚Attitude Dynamics and Control is.gd/SpaceVehicleDynamics 📚Nonlinear Dynamics and Chaos is.gd/NonlinearDynamics 📚Hamiltonian Dynamics is.gd/AdvancedDynamics 📚Three-Body Problem Orbital Mechanics is.gd/SpaceManifolds 📚Lagrangian and 3D Rigid Body Dynamics is.gd/AnalyticalDynamics 📚Center Manifolds, Normal Forms, and Bifurcations is.gd/CenterManifolds stable focus unstable focus supercritical subcritical topological equivalence genetic switch structural stability Andronov-Hopf Andronov-Poincare-Hopf small epsilon method of multiple scales two-timing Van der Pol Oscillator Duffing oscillator nonlinear oscillators nonlinear oscillation nerve cells driven current nonlinear circuit glycolysis biological chemical oscillation adenosine diphosphate ADP fructose 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 Verhulst #NonlinearDynamics #DynamicalSystems #Bifurcation #Hopf #HopfBifurcation #NonlinearOscillators #AveragingTheory #LimitCycle #Oscillations #nullclines #RelaxationOscillations #VanDerPol #VanDerPolOscillator #LimitCycles #VectorFields #topology #IndexTheory #EnergyConservation #Hamiltonian #Streamfunction #Streamlines #Vortex #SkewGradient #Gradient #PopulationBiology #FixedPoint #DifferentialEquations #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