Bifurcations in 2D Part 4 Global Bifurcations Limit Cycle Creation Homoclinic Bifurcation

>> YOUR LINK HERE: ___ http://youtube.com/watch?v=iphb2GlNtCk

In two-dimensional systems, there are four common ways in which limit cycles are created or destroyed. The Hopf bifurcation is the best known and occurs 'locally' in phase space, in the small neighborhood of a fixed point. The other three involve large regions of the phase plane and are therefore called 'global' bifurcations. • Chapters • 0:00 Introduction • 0:51 saddle-node bifurcation of cycles, or fold bifurcation • 9:11 saddle-node infinite-period bifurcation on a cycle, or SNIPER • 18:07 connection with heartbeat or nerve firing time-series • 22:00 homoclinic bifurcation, or saddle-loop • 26:11 universal behavior of bifurcations of cycles, a summary of the amplitude and period of the resulting limit cycles as a function of the distance of the parameter from the critical value • ► Next, quasi-periodicity, phase-locking, and dynamics on the torus • • Action-Angle Variables in Hamiltonian... • ► From 'Nonlinear Dynamics and Chaos' (online course). • Playlist https://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 • 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 https://is.gd/RossLabSubscribe • ► Follow me on Twitter • / rossdynamicslab • ► Make your own phase portrait • https://is.gd/phaseplane • ► Course lecture notes (PDF) • https://is.gd/NonlinearDynamicsNotes • ► 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 • References: • Steven Strogatz, Nonlinear Dynamics and Chaos , Chapter 8: Bifurcations Revisited • Arnaldo Rodriguez-Gonzalez, Strogatz's example of an infinite-period bifurcation, • • Strogatz's example of an infinite-per... • Arnaldo Rodriguez-Gonzalez, Strogatz's example of a homoclinic bifurcation, • • Strogatz's example of a homoclinic bi... • • cardiac 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 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

#############################

New on site