Towards a computer-assisted existence proof for the C-type renormalisation 2-cycle #

Andrew Burbanks, Andrew Osbaldestin

11:50 Wednesday in 4Q04.

Part of the Nonlinear dynamics and applications session.

Abstract #

The existence of fixed points of renormalisation operators and the spectra of the tangent maps at the fixed point are important in explaining universality observed in the bifurcation structures of families of maps, for which the low dimension of unstable manifolds helps to explain quantitative universality of the dynamics.

Historically, existence proofs for these fixed points have been hard to come by. A number of questions were first settled via rigorous computer-assisted means, by bounding operations on spaces of functions in order to show that quasi-Newton operators for the associated fixed-point problems are contraction maps.

We present a proof of the existence of renormalisation fixed points for period doubling in pairs of maps with a particular type of unidirectional coupling, leading to behaviour in the so-called FS-type universality class, and work-in-progress on extending these results to the C-type criticality.

The C-type universality class is of interest in a variety of problems, including the driven Rössler oscillator at the period-doubling accumulation on the edge of synchronisation tongues, certain maps displaying the Neimark–Sacker bifurcation, and to biologically-relevant models of kidney blood flow regulation.