Bifurcations in 2D Part 1 Introduction SaddleNode Pitchfork Examples

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Bifurcations in one dimension are extended to two-dimensional systems. We still find that fixed points can be created or destroyed or destabilized as parameters are varied—but now the same is true of closed orbits as well. Prototypical examples are considered and a genetic regulatory system. • ► Next, birth of periodic orbits via Hopf bifurcation • • Bifurcations in 2D, Part 2: Hopf Bifu... • ► 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 • 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 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... • ► Ghosts and bottlenecks in 1D • • Flows on the Circle | Ghosts and Bott... • ► Advanced lecture on center manifold theory • • Center Manifold Theory- Computing Cen... • ► Advanced lecture on bifurcation theory • • Bifurcation Theory: Saddle-Node, Hopf... • ► 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 • ► Chapters • 0:00 Introduction to bifurcations of fixed points in 2D • 3:39 Saddle-node bifurcation in 2D intro • 8:18 Ghosts of fixed points and bottlenecks in phase space • 11:12 Saddle-node bifurcation based on nullcline intersections • 14:32 Saddle-node example, genetic regulatory network • 35:03 Bifurcations occur along 1-dimensional center manifold • 38:39 Pitchfork and transcritical bifurcations in 2D • ► References: • Steven Strogatz, Nonlinear Dynamics and Chaos , Chapter 8: Bifurcations Revisited • ► 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 • • 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 Strogatz' bulldog • #NonlinearDynamics #DynamicalSystems #Bifurcation #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 #StrogatzBulldog

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