Claire M. Postlethwaite

On designing heteroclinic networks from graphs

Abstract Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a complicated structure determined to a large extent by the constraints and dimension of the system. As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand how particular directed graphs can be realised as attracting robust heteroclinic networks between states in phase space. This paper presents two methods of realising arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs—we say the ODEs realise the graphs as heteroclinic networks between equilibria that represent the vertices. Suppose we have a directed graph on n v vertices with n e edges. The “simplex realisation” embeds the graph as an invariant set of a flow on an ( n v − 1 ) -simplex. This method realises the graph as long as it is one- and two-cycle free. The “cylinder realisation” embeds a graph as an invariant set of a flow on a ( n e + 1 ) -dimensional space. This method realises the graph as long as it is one-cycle free. In both cases we realise the graph as an invariant set within an attractor, and discuss some illustrative examples, including the influence of noise and parameters on the dynamics. In particular we show that the resulting heteroclinic network may or may not display “memory” of the vertices visited.

Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

We show that the Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [B. Fiedler, V. Flunkert, M. Georgi, P. Hovel, E. Scholl, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett. 98(11) (2007) 114101], who demonstrated that such a feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. The Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two-parameter plane. In each case, the Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the feedback phase angle satisfies a certain restriction.

Effects of Temporal Resolution on an Inferential Model of Animal Movement

Recently, there has been much interest in describing the behaviour of animals by fitting various movement models to tracking data. Despite this interest, little is known about how the temporal ‘grain’ of movement trajectories affects the outputs of such models, and how behaviours classified at one timescale may differ from those classified at other scales. Here, we present a study in which random-walk state-space models were fit both to nightly geospatial lifelines of common brushtail possums (Trichosurus vulpecula) and synthetic trajectories parameterised from empirical observations. Observed trajectories recorded by GPS collars at 5-min intervals were sub-sampled at periods varying between 10 and 60 min, to approximate the effect of collecting data at lower sampling frequencies. Markov-Chain Monte-Carlo fitting techniques, using information about movement rates and turning angles between sequential fixes, were employed using a Bayesian framework to assign distinct behavioural states to individual location estimates. We found that in trajectories with higher temporal granularities behaviours could be clearly differentiated into ‘slow-area-restricted’ and ‘fast-transiting’ states, but for trajectories with longer inter-fix intervals this distinction was markedly less obvious. Specifically, turning-angle distributions varied from being highly peaked around either or at fine temporal scales, to being uniform across all angles at low sampling intervals. Our results highlight the difficulty of comparing model results amongst tracking-data sets that vary substantially in temporal grain, and demonstrate the importance of matching the observed temporal resolution of tracking devices to the timescales of behaviours of interest, otherwise inter-individual comparisons of inferred behaviours may be invalid, or important biological information may be obscured.

A mechanism for switching near a heteroclinic network

We describe an example of a robust heteroclinic network for which nearby orbits exhibit irregular but sustained switching between the various sub-cycles in the network. The mechanism for switching is the presence of spiralling due to complex eigenvalues in the flow linearized about one of the equilibria common to all cycles in the network. We construct and use return maps to investigate the asymptotic stability of the network, and show that switching is ubiquitous near the network. Some of the unstable manifolds involved in the network are two-dimensional; we develop a technique to account for all trajectories on those manifolds. A simple numerical example illustrates the rich dynamics that can result from the interplay between the various cycles in the network.

Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems

Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.

The Feed-Forward Chain as a Filter-Amplifier Motif

Hudspeth, Magnasco, and collaborators have suggested that the auditory system works by tuning a collection of hair cells near Hopf bifurcation, but each with a different frequency. An incoming sound signal to the cochlea then resonates most strongly with one of these hair cells, which then informs the auditory neuronal system of the frequency of the incoming signal. In this chapter, we discuss two mathematical issues. First, we describe how periodic forcing of systems near a point of Hopf bifurcation is generally more complicated than the description given in these auditory system models. Second, we discuss how the periodic forcing of coupling identical systems whose internal dynamics is each tuned near a point of Hopf bifurcation leads naturally to successive amplification of the incoming signal. We call this coupled system a feed-forward chain and suggest that it is a mathematical candidate for a motif.

Optimal movement in the prey strikes of weakly electric fish: a case study of the interplay of body plan and movement capability

Animal behaviour arises through a complex mixture of biomechanical, neuronal, sensory and control constraints. By focusing on a simple, stereotyped movement, the prey capture strike of a weakly electric fish, we show that the trajectory of a strike is one which minimizes effort. Specifically, we model the fish as a rigid ellipsoid moving through a fluid with no viscosity, governed by Kirchhoffs equations. This formulation allows us to exploit methods of discrete mechanics and optimal control to compute idealized fish trajectories that minimize a cost function. We compare these with the measured prey capture strikes of weakly electric fish from a previous study. The fish has certain movement limitations that are not incorporated in the mathematical model, such as not being able to move sideways. Nonetheless, we show quantitatively that the computed least-cost trajectories are remarkably similar to the measured trajectories. Since, in this simplified model, the basic geometry of the idealized fish determines the favourable modes of movement, this suggests a high degree of influence between body shape and movement capability. Simplified minimal models and optimization methods can give significant insight into how body morphology and movement capability are closely attuned in fish locomotion.

A Global Bifurcation Analysis of the Subcritical Hopf Normal Form Subject to Pyragas Time-Delayed Feedback Control ∗

Unstable periodic orbits occur naturally in many nonlinear dynamical systems. They can generally not be observed directly, but a number of control schemes have been suggested to stabilize them. One such scheme is that by Pyragas [Phys. Lett. A, 170 (1992), pp. 421--428], which uses time-delayed feedback to target a specific unstable periodic orbit of a given period and stabilize it. This paper considers the global effect of applying Pyragas control to a nonlinear dynamical system. Specifically, we consider the standard example of the subcritical Hopf normal form subject to Pyragas control, which is a delay differential equation (DDE) that models how a generic unstable periodic orbit is stabilized. Our aim is to study how this DDE model depends on its different parameters, including the phase of the feedback and the imaginary part of the cubic coefficient, over their entire ranges. We show that the delayed feedback control induces infinitely many curves of Hopf bifurcations, from which emanate infinitely m...

Delayed Feedback Versus Seasonal Forcing: Resonance Phenomena in an El Nin͂o Southern Oscillation Model

Climate models can take many different forms, from very detailed highly computational models with hundreds of thousands of variables, to more phenomenological models of only a few variables that are designed to investigate fundamental relationships in the climate system. Important ingredients in these models are the periodic forcing by the seasons, as well as global transport phenomena of quantities such as air or ocean temperature and salinity. We consider a phenomenological model for the El Nino Southern Oscillation system, where the delayed effects of oceanic waves are incorporated explicitly into the model. This gives a description by a delay differential equation, which models underlying fundamental processes of the interaction between internal delay-induced oscillations and the external forcing. The combination of delay and forcing in differential equations has also found application in other fields, such as ecology and gene networks. Specifically, we present exemplary stable solutions of the model...

A new mechanism for stability loss from a heteroclinic cycle

Asymptotically stable robust heteroclinic cycles can lose stability through resonance or transverse bifurcations. In a transverse bifurcation, an equilibrium in the cycle undergoes a local bifurcation, causing a change in stability. A resonance bifurcation is a global phenomenon, determined by an algebraic condition on the eigenvalues, and is generically accompanied by the birth or death of a long-period periodic orbit. In this article we demonstrate a new mechanism causing loss of stability, which is neither resonant nor transverse in the usual sense. The location of the instability is determined by an algebraic condition on the eigenvalues, but the instability occurs in a transverse direction. Furthermore, after the bifurcation, when the cycle is unstable, open sets of trajectories are seen to initially approach the network for an extended period, before moving away in the unstable direction. This should serve as a warning to all those doing numerics near heteroclinic cycles who deduce that the cycle is stable merely because trajectories are observed to initially approach the cycle.