When ignoring the parameter-dependency, the takens normal form can be derived formally by analyzing the resonant terms. If we want to preserve the parameter-dependency, how can the simplest form of the system be computed in a way that enables the investigation of bifurcations that occur when the fixed point is perturbed in the parameters?
Working with parameter-dependent coefficients of the k-jet of the system (in the variables only) does not work in my case, as this prevents the system from being simplified (and these simplifications are essential to make it solvable).
Use the same procedure as you do in the parameter independent case. The best and simple idea is : first use a transformation that will simplify the second order term,
y=x+h_2(x), h_2(x) the homogeneous polynomial of second degree with coefficients in R. Then compare and equate, only resonant terms will contribute in the normal form. Comparing co-efficients in h_2(x) with the parameters involved in the vector field, it may possible that you can solve all such equations implying that you have chosen h_2(x) whose coefficients relate with parameters and you kill the qudratic term (remember it may not happen, but at least you can simplify the vector field) the reduced system and the original systems are topologically equivalent. Proceed in the same way for higher order terms.
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