ClearAll["Global`*"]
polarRFun[r_,\[Theta]_]=\[Mu] *r - r^3;
polar\[Theta]Fun[r_,\[Theta]_]=\[Omega] + \[Nu]*r^2;
polarJ={{D[polarRFun[r,\[Theta]],r],D[polarRFun[r,\[Theta]],\[Theta]]},
	{D[polar\[Theta]Fun[r,\[Theta]],r],D[polar\[Theta]Fun[r,\[Theta]],\[Theta]]}};
polarM=MatrixExp[polarJ*T]/.r->Sqrt[\[Mu]]
Eigenvalues[polarM]
xyToR[x_,y_]=Sqrt[x^2 + y^2];
xyTo\[Theta][x_,y_]=ArcTan[y/x];
xyToPolarJ ={{D[xyToR[x,y],x],D[xyToR[x,y],y]},{D[xyTo\[Theta][x,y],x],D[xyTo\[Theta][x,y],y]}}/.
	{x->Sqrt[\[Mu]],y->0}
cM = Simplify[Inverse[xyToPolarJ].polarM.xyToPolarJ]
Eigenvalues[cM]
cM/.{\[Mu]->1/10,\[Nu]->1,T->2*\[Pi]/(1.1)}