Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection

We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this ‘cycling chaos’ manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets. Such behaviour is robust to perturbations that preserve the symmetry of the system; we examine bifurcations of this state. We discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace. This differs from the standard scenario for the blowout bifurcation in that in our case, the blowout is neither subcritical nor supercritical. The robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors. The model we consider is a ninth-order truncated ordinary differential equation (ODE) model of three-dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient. Symmetries of the model lead to the existence of invariant subspaces for the dynamics; in particular there are invariant subspaces that correspond to regimes of two-dimensional flows, with variation in the vertical but only one of the two horizontal directions. Stable two-dimensional chaotic flow can go unstable to three-dimensional flow via the cross-roll instability. We show how the bifurcations mentioned above can be located by examination of various transverse Liapunov exponents. We also consider a reduction of the ODE to a map and demonstrate that the same behaviour can be found in the corresponding map. This allows us to describe and predict a number of observed transitions in these models. The dynamics we describe is new but nonetheless robust, and so should occur in other applications

Edge-transitivity of Cayley graphs generated by transpositions

Let $S$ be a set of transpositions generating the symmetric group $S_n$. The transposition graph of $S$ is defined to be the graph with vertex set $\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in S$. In the present note, it is proved that two transposition graphs are isomorphic if and only if the corresponding two Cayley graphs are isomorphic. It is also proved that the transposition graph $T(S)$ is edge-transitive if and only if the Cayley graph $Cay(S_n,S)$ is edge-transitive

Diameter of Cayley graphs of permutation groups generated by transposition trees

Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $\Gamma$ is bounded as \diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}, where the maximization is over all permutations $\pi$, $c(\pi)$ denotes the number of cycles in $\pi$, and \dist_T is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upper bound is sharp for all trees of maximum diameter and also for all trees of minimum diameter, and we exhibit some families of trees for which the bound is strict. We then show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Observe that evaluating this upper bound requires on the order of $n!$ (times a polynomial) computations. We provide an algorithm that obtains an estimate of the diameter, but which requires only on the order of (polynomial in) $n$ computations; furthermore, the value obtained by our algorithm is less than or equal to the previously known diameter upper bound. This result is possible because our algorithm works directly with the transposition tree on $n$ vertices and does not require examining any of the permutations (only the proof requires examining the permutations). For all families of trees examined so far, the value $\beta$ computed by our algorithm happens to also be an upper bound on the diameter, i.e. \diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}.Comment: This is an extension of arXiv:1106.535