Recent work in a number of areas has highlighted ways in which heteroclinic and homoclinic connections within dynamical attractors can be responsible for intermittent and bursting behaviour in nonlinear systems. Such dynamical behaviour appears in models from a wide range of applications including fluid dynamics, population dynamics and neuroscience, with related dynamics having been found in other contexts (e.g. unstable attractor networks, chaotic itinerancy and cycling chaos). Moreover, heteroclinic dynamics may serve as an appropriate mathematical framework for transient processes that can be treated as an itinerary past metastable states.
Analysis of the dynamics associated with attractors containing heteroclinic and homoclinic connections can present a number of formidable challenges, such as identifying and determining the structure of the attractors, understanding mechanisms for selection of connections and for switching between alternative itineraries for typical trajectories, design of efficient and accurate numerical methods that capture the intricate dynamics in such systems and consideration of robustness of the dynamics to perturbations. This special issue brings together papers on a number of important research topics related to these concerns, ranging from new applications in which robust heteroclinic connections can be found (Buono, Castro et al., Chawanya and Ashwin, Afraimovich et al.) to studies of mechanisms for switching (Kirk et al., Homburg and Knobloch) and stability (Postlethwaite) of networks and the response of networks to noise (Bakhtin).
The article of Buono develops ideas and new methods for the analysis of robust heteroclinic cycles (RHCs) in the context of delay differential equations, and shows that, as well as RHCs that can be understood by reduction to finite-dimensional centre manifolds, there can be inherently infinite-dimensional heteroclinic cycles in such systems.
Postlethwaite shows that, in addition to the well-studied instabilities of RHCs due to resonance between eigenvalues and transverse instabilities, there may be more subtle resonances between eigenvalues that lead to a periodic orbit being emitted in a transverse direction.
Kirk, Lane, Postlethwaite, Rucklidge and Silber examine a particular case of switching dynamics, for example an ordinary differential equation with symmetry, and show via a geometrical and numerical analysis that there can be sustained irregular switching of trajectories approaching the network. The switching is due to a combination of complex eigenvalues and two-dimensional connecting manifolds.
Homburg and Knobloch give results relating to switching dynamics in a class of robust homoclinic networks in R 5. They use rigorous geometric constructions to demonstrate that there can be dynamical switching near the network corresponding to the suspension of a Smale horseshoe.
Castro, Labouriau and Podvigina present a detailed study of examples of robust heteroclinic networks with symmetry that arise generically in a mode interaction on a plane layer. Their examples are motivated by a problem in Boussinesq convection in a plane layer where there is an interaction between modes with spatial wavelengths in the ratio 2:√3.
Chawanya and Ashwin consider a robust heteroclinic network on a manifold with boundary (on a cube in R 3) that has ‘depth two’ connections – namely that includes robust connections from equilibria to sub-cycles in the network. An approximate return map is used to derive criteria for stability and to investigate phenomena such as the existence of an infinite family of chaotic attractors that accumulate on the network.
Bakhtin examines the limiting behaviour of the dynamics near a heteroclinic network on addition of noise that is weak. By summarising a number of recent rigorous results on stochastic differential equations, the author shows that as the noise amplitude converges to zero, the trajectories under a suitable rescaling of time converge to a point process with instantaneous jumps between equilibria in the cycle.
Finally, Afraimovich, Rabinovich and Varona discuss a novel application of heteroclinic cycles in neuroscience – an application to the problem of binding, namely how sensory information from a number of sources is combined to give an understanding of a single concept. Using a ‘winnerless competition’ model the authors investigate how heteroclinic networks, and stable heteroclinic channels in particular, can be used to model transient binding.
We hope that this special issue will serve as a useful summary of the current state-of-the-art for robust heteroclinic and switching dynamics in the years to come.