Symbolic computation of normal forms for semi-simple cases
A nonlinear system of ordinary differential equations is considered \[ \dot x=Ax+F(x),\quad x\in \mathbb{R}^{n},\quad F(x):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}. \] Here \(A\) is an \(n\times n\) matrix with semi-simple eigenvalues, the nonlinear function \(F\) is analytic, and \(F_{i}=0,\frac{\partial F_{i}}{\partial x_{j}}=0\) at \(x=0,\forall i,j=1,\cdots,n.\) Efficient symbolic computation programs are developed for computing explicit normal forms and associated nonlinear transformations based on the coefficients of the original differential system. This approach combines the center manifold theory and normal form theory into one step to simultaneously obtain closed form expressions with the aid of the suggested iterative procedure. It does not require the computation of large-dimensional inverse matrices and therefore reduces computation time. The computer language Maple is used to facilitate the application of the method. A number of illustrative examples is presented.
scientific article; zbMATH DE number 1331060
scientific article