Phase functions

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

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

dϕdt=dϕdx.dxdt=1

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

dxdt=f(x)

Therefore,

dϕdx.dxdt=dϕdx.f(x)=1

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

dϕdx.f(x)=1

Where dϕdx is the PRC.

Approximately,

PRC=1f(x(t))=1˙ζ(t)

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

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