Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling
A nonlinear ordinary differential equation \[ {dx\over dt}=f(x,\mu),\quad x\in {\mathbb{R}^n},\quad \mu\in{\mathbb{R}^m}, \] analytic with respect to \(x\) and a parameter \(\mu\), with the equilibrium point \((0,\mu)\) is considered. The aim of this work is to find a possibly simple form of any term homogeneous with respect to \(x\) and \(\mu\) of the Taylor expansion of \(f(x,\mu)\) at the equilibrium point. Such a procedure may be very useful in studying the singularities such as bifurcation, instability and so on of the equation discussed . Here the important tool is the Lie transformation of any homogeneous \(k\)-degree polynomial \(U_k\) \[ U_k\mapsto[U_k,v]=DvU_k-DU_kv, \] where \(v\) is of degree 1. The author develops basic ideas involved in the procedure of finding the so called conventional normal form. The number of variables in the system of linear algebraic equations of the above mentioned procedure is greater than the number of equations. Instead of setting to zero the variables which cannot be determined from the system, the author retains all the variables. It gives him the possibility to use them in the process of further reduction. Such a procedure leads to the new `form' of the normal form, the so called simplest normal form (SNF). Theorem 1 gives formulae connecting any homogeneous term of the SNF with corresponding terms of the nonlinear transformation. The formulae contain time rescaling factors, and are suitable for recursive computation, for example, with use of a kind of computer algebra system as Maple. The case of single zero singularity is discussed more in details.
nonlinear ordinary differential equation
scientific article; zbMATH DE number 1787669