DS Arnold Tongue

A one-dimensional mapping of the circle to itself defined by

f(x)= x + w + epsilon/(2 Pi) sin( 2 Pi x)

The set of points for which f(x)-x=0 are the fixed points for this mapping. The set of points for which f(x)-x = f'(x)-1 = 0 are called points of saddle-node bifurcation. The set of parameter values leading to fixed points is called the resonance region for the map. See Chapter 6 of J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983.

Since this model has a one-dimensional domain, one or more of the parameters must be permuted into the domain before attempting to trace level sets.

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