Bounds on the hausdorff dimension of a renormalisation map arising from an excitable reaction-diffusion system on a fractal lattice

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Abstract

A renormalisation approach to investigate travelling wave solutions of an excitable reaction- diusion system on a deterministic fractal structure has recently been derived. The dynamics of a particular class of solutions which are governed by a two dimensional subspace of these renormalisation recursion relationships are discussed in this paper. The bifurcations of this mapping are discussed with reference to the discontinuities which arise at the singularities. The map is chaotic for a bounded region in parameter space and bounds on the Hausdor dimension of the associated invariant hyperbolic set are calculated.
Original languageEnglish
Pages (from-to)274-284
Number of pages11
JournalChaos, Solitons and Fractals
Volume35
Issue number2
DOIs
Publication statusPublished - Jan 2008

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Hyperbolic Set
Excitable Systems
Fractal Structure
Invariant Set
Traveling Wave Solutions
Reaction-diffusion System
Hausdorff Dimension
Recursion
traveling waves
Renormalization
Fractals
Parameter Space
Fractal
Discontinuity
fractals
discontinuity
Bifurcation
Subspace
Singularity
Relationships

Keywords

  • Hausdorff dimension
  • renormalisation map
  • reaction-diffusion system
  • fractal lattice

Cite this

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abstract = "A renormalisation approach to investigate travelling wave solutions of an excitable reaction- diusion system on a deterministic fractal structure has recently been derived. The dynamics of a particular class of solutions which are governed by a two dimensional subspace of these renormalisation recursion relationships are discussed in this paper. The bifurcations of this mapping are discussed with reference to the discontinuities which arise at the singularities. The map is chaotic for a bounded region in parameter space and bounds on the Hausdor dimension of the associated invariant hyperbolic set are calculated.",
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AB - A renormalisation approach to investigate travelling wave solutions of an excitable reaction- diusion system on a deterministic fractal structure has recently been derived. The dynamics of a particular class of solutions which are governed by a two dimensional subspace of these renormalisation recursion relationships are discussed in this paper. The bifurcations of this mapping are discussed with reference to the discontinuities which arise at the singularities. The map is chaotic for a bounded region in parameter space and bounds on the Hausdor dimension of the associated invariant hyperbolic set are calculated.

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