Converting system of ODE from polar to rectangular coordinates

Question

I am trying to convert the following system of equations from polar coordinates to rectangular coordinates. (A,phi)->(u,v)

 {Derivative[1][A][t] == -A β Sin[τ/2] + 
   1/2 A α Sin[2 ϕ], 
 Derivative[1][ϕ][t] == (3 A^2 γ)/
  4 + δ1 - β Cos[τ/2] + 1/2 α Cos[2 ϕ]}

I am trying to reproduce the attached:

Polar form:

Rectangular form:

Answer 1

First of all, you've mixed up dependent variables and function relationship, so we need to modify the equation to

eq = With[{A = A[t], ϕ = ϕ[t]}, 
   {D[A, t] == -A β Sin[τ/2] + 1/2 A α Sin[2 ϕ], 
    D[ϕ, t] == (3 A^2 γ)/4 + δ1 - β Cos[τ/2] + 1/2 α Cos[2 ϕ]}];

Then we make use of DChange:

neweq = DChange[eq, {A[t] == Sqrt[u[t]^2 + v[t]^2], ϕ[t] == ArcTan[u@t, v@t]}]

Solve[TrigExpand@neweq, {u'[t], v'[t]}][[1]]

So the result in the given screenshot seems to be wrong, unless it has made use of certain special convention for the coordinates.