arXiv subjects

Alastair M Rucklidge

Publications and source records attributed to Alastair M Rucklidge.

4 recordsLinked to original sources

arXiv2026

Continuity of projected maps of heteroclinic networks in $\mathbb{R}^{4}$

Stability of robust heteroclinic cycles and networks is typically studied by constructing return maps and analysing their associated transition matrices. This analysis can be simplified with the network's projected map, derived by projecting the linear action of the transition matrix onto a simplex. This projection produces a piecewise-smooth map, which is one-dimensional for heteroclinic networks in $\mathbb{R}^{4}$. We consider two such networks, the Kirk--Silber network and the $Δ$-clique network. For the Kirk--Silber network, the projected map is discontinuous on its switching manifold, while it is continuous for the $Δ$-clique network. In this paper, we address the dynamical phenomena that produce a discontinuity in the case of the Kirk--Silber network, and explain the value of the projected map at the switching manifold. We construct a completed return map near both networks, which captures the behaviour of all trajectories that begin near the network but may move away from it temporarily. We show that the discontinuity in the projected map of the Kirk--Silber network emerges due to two phenomena: first, there exists a discontinuity in a component of the completed return map in the limit as trajectories approach the Kirk--Silber network, as a result of the presence of a separatrix near the network, and, second, the procedure that defines the projected map. For the $Δ$-clique network, there is no such separatrix, and so no such discontinuity emerges. This analysis is a necessary step towards understanding the more complicated dynamics observed near heteroclinic networks of five or more equilibria.

math.DS↗

arXiv2022

Patterns and quasipatterns from the superposition of two hexagonal lattices

When two-dimensional pattern-forming problems are posed on a periodic domain, classical techniques (Lyapunov-Schmidt, equivariant bifurcation theory) give considerable information about what periodic patterns are formed in the transition where the featureless state loses stability. When the problem is posed on the whole plane, these periodic patterns are still present. Recent work on the Swift-Hohenberg equation (an archetypal pattern-forming partial differential equation) has proved the existence of quasipatterns, which are not spatially periodic and yet still have long-range order. Quasipatterns may have 8-fold, 10-fold, 12-fold and higher rotational symmetry, which preclude periodicity. There are also quasipatterns with 6-fold rotational symmetry made up from the superposition of two equal-amplitude hexagonal patterns rotated by almost any angle $α$ with respect to each other. Here, we revisit the Swift-Hohenberg equation (with quadratic as well as cubic nonlinearities) and prove existence of several new quasipatterns. The most surprising are hexa-rolls: periodic and quasiperiodic patterns made from the superposition of hexagons and rolls (stripes) oriented in almost any direction with respect to each other and with any relative translation; these bifurcate directly from the featureless solution. In addition, we find quasipatterns made from the superposition of hexagons with unequal amplitude (provided the coefficient of the quadratic nonlinearity is small). We consider the periodic case as well, and extend the class of known solutions, including the superposition of hexagons and rolls. While we have focused on the Swift-Hohenberg equation, our work contributes to the general question of what periodic or quasiperiodic patterns should be found generically in pattern-forming problems on the plane.

nlin.PS↗

arXiv2021

Stability of cycling behaviour near a heteroclinic network model of Rock-Paper-Scissors-Lizard-Spock

The well-known game of Rock--Paper--Scissors can be used as a simple model of competition between three species. When modelled in continuous time using differential equations, the resulting system contains a heteroclinic cycle between the three equilibrium solutions representing the existence of only a single species. The game can be extended in a symmetric fashion by the addition of two further strategies (`Lizard' and `Spock'): now each strategy is dominant over two of the remaining four strategies, and is dominated by the remaining two. The differential equation model contains a set of coupled heteroclinic cycles forming a heteroclinic network. In this paper we carefully consider the dynamics near this heteroclinic network. We are able to identify regions of parameter space in which arbitrarily long periodic sequences of visits are made to the neighbourhoods of the equilibria, which form a complicated pattern in parameter space.

math.DS↗

arXiv2019

A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock-Paper-Scissors

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models.

nlin.PS↗