Limit Cycles, Part 3: Poincare-Bendixson Worked Examples- Glycolysis

Poincaré-Bendixson theorem is used to prove the existence of a limit cycle in two examples, including a biochemical oscillator, glycolysis, for certain values of the parameter. We carefully construct the necessary trapping region, guided by nullclines. We comment on the implication of the Poincare-Bendixson theorem for the absence of chaos in 2-dimensional differential equation systems. ► Next, the Van der Pol equation in the strongly nonlinear limit. https://youtu.be/qaeKi47z4ik ► Limit cycles Introduction to limit cycles https://youtu.be/9rVscJwDpBo Testing for limit cycles https://youtu.be/0uSTZFM0GdE ► 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 ► 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 Review of Poincare-Bendixson theorem 1:27 Analytical example in polar coordinates 7:33 Biological example: glycolysis 9:44 Nullclines 14:50 Trapping region for biochemical oscillator model 24:00 Region in parameter space where stable limit cycle exists References: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 7: Limit Cycles 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 #NonlinearDynamics #DynamicalSystems #Oscillations #Nullclines #PoincareBendixson #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