# 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!