Abstracts of Dynamical Systems Research Papers

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See also

• Research papers (full list)
• Hamiltonian abstracts
• Geometry and Singularity Theory abstracts

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Golden gaskets: variations on the Sierpinski sieve

(with Dave Broomhead and Nikita Sidorov) Nonlinearity 14 (2004), 1455-1480.

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor λ in (0,1).

As is well known, for λ=1/2 the invariant set, S_λ, is a fractal called the Sierpinski sieve, and for λ<1/2 it is also a fractal. Our goal is to study S_λ for this IFS for 1/2<λ<2/3, i.e., when there are "overlaps" in S_λ as well as "holes". In this introductory paper we show that despite the overlaps (i.e., the Open Set Condition breaking down completely), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic λ's — the so-called multinacci numbers. We evaluate dim_H(S_λ) for these special values by showing that S_λ is essentially the attractor for an infinite IFS which does satisfy the Open Set Condition. We also show that the set of points in the attractor with a unique "address" is self-similar, and compute its dimension.

For "non-multinacci" values of λ we show that if λ is close to 2/3, then S_λ has a nonempty interior and that if λ<1/sqrt(3) then S_λ has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of the model in question.

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Group theoretic conditions for existence of robust relative homoclinic trajectories

(with Peter Ashwin) Math. Proc. Camb. Phil. Soc. 133 (2002), 125-141.

We consider robust relative homoclinic trajectories (RHTs) for G-equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these result we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHTs, we construct some new examples of robust relative homoclinic trajectories. We also classify RHTs of the dihedral and low order symmetric groups by means of their symmetries.
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Real continuation from the complex quadratic family: Fixed point bifurcation sets

(with Bruce Peckham). Int. J. Bifurcation and Chaos 10 (2000), 391-414

We study the family of real maps of the plane given by f(z) = z² + z + C + Az_bar . For A=0 this is the usual complex squaring family (up to coordinate change). As A is varied, the fixed points of the complex squaring map undergo certain bifurcations. In particular, at (C,A) = (0,0) the map has a very degenerate fixed point at z=0, with a semisimple eigenvalue 1 of multiplicity 2. This is then unfolded by varying the parameters A and C.
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Bifurcation générique d'ondes rotatives d'isotropie maximale.

(with Pascal Chossat and Muriel Koenig), C.R.Acad. Sci. Paris 320 Série I (1995), 25-30.

We consider bifurcations of relative equilibria (or rotating waves), using orbit space methods.
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James Montaldi