Phase functions

The note describes a method to compute the phase function from the differential equations for some simple cases.

Let \(\phi(x(t))\) be the phase function. Then,

\[\frac{d\phi}{dt} = \frac{d\phi}{dx}.\frac{dx}{dt}=1\]

However, we know, from the definition of the differential equations,

\[\frac{dx}{dt} = f(x)\]

Therefore,

\[\frac{d\phi}{dx}.\frac{dx}{dt} = \frac{d\phi}{dx}.f(x) = 1\]

This helps us get the following equation to compute phase function.

\[\frac{d\phi}{dx}.f(x) = 1\]

Where \(\frac{d\phi}{dx}\) is the PRC.

Approximately,

\[PRC = \frac{1}{f(x(t))} = \frac{1}{\dot{\zeta}(t)}\]

Where \(\zeta (t)\) is the limit cycle solution of the differential equation!

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