Preprint No. A-01-07
In the present paper, we do not require such a foliation by constant parameters. Instead, we only require the existence of a trivial equilibrium manifold which is parametrized over p. No further time-constant "parameters" are involved.
We discuss such bifurcations without parameters along manifolds of equilibria. In particular, we investigate the four-dimensional analogue of the Takens-Bogdanov bifurcation from two-dimensional equilibrium manifolds.
Following the foot steps of Floris Takens, our analysis includes normal form reductions to any finite order, blow-up and reduction to Hamiltonian form, direct calculation of the perturbations via Weierstrass elliptic integrals, and a discussion of higher order terms.
As a result we obtain three different phase portraits. In all cases, bounded non-trivial orbits are heteroclinic to the equilibrium manifold, only. In spite of this lack of recurrent dynamics, exponentially small as well as finite order splittings may give rise to complicated geometric patterns of heteroclinicity.
We conclude with an application to hyperbolic balance laws with stiff source terms. Only oscillatory shock profiles violating the Lax-condition are observed.
Keywords: Takens-Bogdanov bifurcation, normal form, blow up, averaging, hyperbolic balance laws, oscillatory profiles
Mathematics Subject Classification (MSC2000): Primary 34C23, Secondary 34C29, 34C37, 35L67
Language: ENGAvailable: Pr-A-01-07.ps Pr-A-01-07.ps.gz
Contact: Bernold Fiedler, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (fiedler@math.fu-berlin.de)
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