Coupled Oscillators, Quasiperiodicity, Synchronization, Flows on the Torus

An important two-dimensional phase space is the torus, the natural setting for studying two coupled oscillators, that is, two flows on the circle which interact. Coupled oscillators have been used to model the interaction between human circadian rhythms and the sleep-wake cycle. New phenomena can occur, such as quasi-periodicity, synchronization, phase-locking. The Poincare-Hopf index theorem provides an important constraint on the dynamics.

Chapters
0:00 Introduction, other phase spaces
0:55 Flows on the torus
3:52 Coupled oscillators
6:26 Ratio of frequencies is rational ► periodic orbit
10:14 Ratio of frequencies is irrational ► quasiperiodic orbit
12:31 Coupled oscillators and the synchronized state
16:48 Other flows on the torus with fixed points

A kids toy, the EzyRoller (or TwistCar or Roller Racer) shows interesting dynamics on a torus phase space, also applicable to tractor trailers.

► Next, chaos in 3D systems and the Lorenz equations
3D Systems, Lorenz Equations Derived,...

► From 'Nonlinear Dynamics and Chaos'
Online course playlist https://is.gd/NonlinearDynamics

► Poincare-Hopf index theory
Index Theory for Dynamical Systems, P...

► Flows on the circle
Flows on the Circle | Ghosts and Bott...

► KAM tori in Hamiltonian systems
Action-Angle Variables in Hamiltonian...

► 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...

► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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Twitter: RossDynamicsLab

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


References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 8: Bifurcations Revisited






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