Normal form with bifurcation theory

Abstract EN

This paper deals with the effective computation of normal forms, centre manifolds and first integrals in Hamiltonian mechanics.

These kind of calculations are very useful since they allow, for instance, to give explicit estimates on the diffusion time or to compute invariant tori.

The approach presented here is based on using algebraic manipulation for the formal series but taking numerical coefficients for them.

This, jointly with a very efficient implementation of the software, allows big savings in both memory and execution time of the algorithms if we compare with the use of commercial algebraic manipulators.

and they are applied to some concrete examples coming from celestial mechanics .The dynamics of a diffusive Nicholson's blowflies equation with a finite delay and Dirichlet boundary condition have been investigated.

The occurrence of steady state bifurcation with the changes of parameter is proved by applying phase plane ideas.

The existence of Hopf bifurcation at the positive equilibrium with the changes of specify parameters is obtained, and the phenomenon that the unstable positive equilibrium state without dispersion may become stable with dispersion under certain conditions is found by analyzing the distribution of the eigenvalues.

By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived.