Poincaré-Bendixson Theorem: Worked Examples (Glycolysis) | Limit Cycles Part 3

The Poincaré-Bendixson theorem — the powerful tool for proving the existence of a limit cycle in 2D dynamical systems. The idea: find a region in phase space that trajectories enter but cannot leave, AND that contains no fixed points. By topological necessity, those trapped trajectories must accumulate on a closed orbit — a limit cycle. The challenge in practice is constructing the right trapping region. We work through two concrete examples: 🎯 An analytical example in polar coordinates (where the radial dynamics are explicit and the trapping region is an annulus between two circles) 🎯 A biological example: the Sel'kov model of glycolysis, a biochemical oscillator. Glycolysis — the metabolic pathway breaking down sugar into ATP — exhibits sustained oscillations in concentrations of ADP and fructose-6-phosphate. Using nullclines, we carefully construct the trapping region, and identify the range of parameter values where a stable limit cycle exists. A deep consequence of Poincaré-Bendixson: in 2D continuous dynamical systems, *chaos is impossible*. The theorem says trajectories trapped in a bounded region without fixed points must approach a periodic orbit — there's no room for chaotic behavior in two dimensions. Chaos requires three or more dimensions, which is why the Lorenz system (3D) can be chaotic but pendulums and Van der Pol oscillators (2D) cannot. From the Nonlinear Dynamics and Chaos online course (based on Strogatz Chapter 7: Limit Cycles). ▶️ Chapters 0:00 Review of the Poincaré-Bendixson theorem 1:27 Analytical example in polar coordinates 7:33 Biological example: glycolysis (the Sel'kov model) 9:44 Nullclines and the geometry of the trapping region 14:50 Constructing the trapping region for the biochemical oscillator 24:00 Region in parameter space where a stable limit cycle exists 📘 What you'll learn What the Poincaré-Bendixson theorem says (and why it requires no fixed points in the trapping region) How to construct a trapping region in practice using nullclines How to apply the theorem to a polar-coordinate example How to model glycolysis as a 2D dynamical system (the Sel'kov model) Why biochemical oscillators are limit cycles (sustained oscillations, not damped) Why 2D continuous systems cannot exhibit chaos (a major consequence of the theorem) 🎓 Course Nonlinear Dynamics and Chaos (AOE 4514, cross-listed as ESM 4114, Virginia Tech) Full playlist:    • Nonlinear Dynamics & Chaos — Full Course F...   🔗 Limit Cycles series (this mini-series within Nonlinear Dynamics and Chaos) Part 1: Introduction to limit cycles —    • What Is a Limit Cycle? Introduction with E...   Part 2: Testing for limit cycles (Lyapunov, Dulac, Poincaré-Bendixson) —    • Lyapunov Functions and Dulac's Criterion (...   Part 3: Poincaré-Bendixson worked examples (this video) Part 4: Van der Pol — strongly nonlinear —    • Limit Cycles Part 4: Van der Pol — Strongl...   Part 5: Van der Pol — weakly nonlinear —    • Van der Pol Oscillator: Weakly Nonlinear L...   ▶️ Next — Van der Pol Equation in the Strongly Nonlinear Limit    • Limit Cycles Part 4: Van der Pol — Strongl...   🔗 Related background videos in this course Phase plane introduction —    • Phase Portrait Explained: The Pendulum Exa...   Classifying 2D fixed points —    • Fixed Points Explained: Nodes, Saddles, Sp...   Gradient systems —    • Gradient Systems: Nonlinear ODEs with Spec...   Index theory —    • Index Theory Explained: Fixed Points, Peri...   🛠️ Make your own phase portrait https://aeb019.hosted.uark.edu/pplane... 📄 Course lecture notes (PDF) https://drive.google.com/drive/folder... 📖 References Steven Strogatz — Nonlinear Dynamics and Chaos, Chapter 7: Limit Cycles The glycolysis example appears in Strogatz §7.3, based on: Sel'kov, E.E. (1968). "Self-oscillations in glycolysis." European Journal of Biochemistry, 4, 79–86. https://doi.org/10.1111/j.1432-1033.1... ⸻ 👨‍🏫 Instructor Dr. Shane Ross Professor of Aerospace Engineering, Virginia Tech (Caltech PhD, former NASA/JPL and Boeing) Research: https://ross.aoe.vt.edu Follow: https://x.com/RossDynamicsLab Subscribe: https://www.youtube.com/user/RossDyna... ⸻ 🔗 Related courses Hamiltonian Dynamics —    • Hamiltonian Mechanics: Full Graduate Cours...   Lagrangian & Rigid Body Dynamics —    • Lagrangian Mechanics & 3D Rigid Body Dynam...   Center Manifolds, Normal Forms & Bifurcations —    • Local Bifurcation Theory: Full Graduate Co...   3-Body Problem Orbital Dynamics —    • Three-Body Problem: Trajectory Design & Lo...   #PoincareBendixson #PoincareBendixsonTheorem #LimitCycles #Glycolysis #BiochemicalOscillator #SelkovModel #TrappingRegion #Nullclines #NonlinearDynamics #DynamicalSystems #Strogatz #VirginiaTech