Limit cycles, part 1: introduction and examples

Limit cycles are isolated periodic orbits which are inherently nonlinear, and form the main mechanism of oscillations in many systems. We give several physical examples of stable limit cycles, such as aeroelastic limit cycle oscillations, flapping of window blinds in a breeze, shaking bridges, heart beats, predator-prey cycles, and even walking. Analytical examples are also given, including the Van der Pol oscillator.

► Next, we look at techniques for proving the existence (or nonexistence) of 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).
Playlist https://is.gd/NonlinearDynamics

► For background on 2D dynamical systems, see
Phase plane introduction https://youtu.be/U4IM7HFzcuY
Classifying 2D fixed points https://youtu.be/7Ewe_tVa5Fs
Systems with special structure https://youtu.be/uGUzPZzvPWQ
Index theory https://youtu.be/bO8FxxpocNQ

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► Make your own phase portrait
https://is.gd/phaseplane

► Course lecture notes (PDF)
https://is.gd/NonlinearDynamicsNotes

References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 7: Limit Cycles
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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Видео Limit cycles, part 1: introduction and examples канала Ross Dynamics Lab