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.
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► 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)
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► From 'Nonlinear Dynamics and Chaos' (online course).
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► 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
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► 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
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