Pietro-Luciano Buono

A modular network for legged locomotion

Abstract In this paper we use symmetry methods to study networks of coupled cells, which are models for central pattern generators (CPGs). In these models the cells obey identical systems of differential equations and the network specifies how cells are coupled. Previously, Collins and Stewart showed that the phase relations of many of the standard gaits of quadrupeds and hexapods can be obtained naturally via Hopf bifurcation in small networks. For example, the networks they used to study quadrupeds all had four cells, with the understanding that each cell determined the phase of the motion of one leg. However, in their work it seemed necessary to employ several different four-oscillator networks to obtain all of the standard quadrupedal gaits. We show that this difficulty with four-oscillator networks is unavoidable, but that the problems can be overcome by using a larger network. Specifically, we show that the standard gaits of a quadruped, including walk, trot and pace, cannot all be realized by a single four-cell network without introducing unwanted conjugacies between trot and pace — conjugacies that imply a dynamic equivalence between these gaits that seems inconsistent with observations. In this sense a single network with four cells cannot model the CPG of a quadruped. We also introduce a single eight-cell network that can model all of the primary gaits of quadrupeds without these unwanted conjugacies. Moreover, this network is modular in that it naturally generalizes to provide models of gaits in hexapods, centipedes, and millipedes. The analysis of models for many-legged animals shows that wave-like motions, similar to those obtained by Kopell and Ermentrout, can be expected. However, our network leads to a prediction that the wavelength of the wave motion will divide twice the length of the animal. Indeed, we reproduce illustrations of wave-like motions in centipedes where the animal is approximately one-and-a-half wavelength long — motions that are consistent with this prediction. We discuss the implications of these results for the development of modular control networks for adaptive legged robots.

Restrictions and unfolding of double Hopf bifurcation in functional differential equations

Abstract The normal form of a vector field generated by scalar delay-differential equations at nonresonant double Hopf bifurcation points is investigated. Using the methods developed by Faria and Magalhaes (J. Differential Equations 122 (1995) 181) we show that (1) there exists linearly independent unfolding parameters of classes of delay-differential equations for a double Hopf point which generically map to linearly independent unfolding parameters of the normal form equations (ordinary differential equations), (2) there are generically no restrictions on the possible flows near a double Hopf point for both general and Z 2 -symmetric first-order scalar equations with two delays in the nonlinearity, and (3) there always are restrictions on the possible flows near a double Hopf point for first-order scalar delay-differential equations with one delay in the nonlinearity, and in nth-order scalar delay-differential equations (n⩾2) with one delay feedback.

Heteroclinic cycles in rings of coupled cells

Abstract Symmetry is used to investigate the existence and stability of heteroclinic cycles involving steady-state and periodic solutions in coupled cell systems with D n -symmetry. Using the lattice of isotropy subgroups, we study the normal form equations restricted to invariant fixed-point subspaces and prove that it is possible for the normal form equations to have robust, asymptotically stable, heteroclinic cycles connecting periodic solutions with steady states and periodic solutions with periodic solutions. A center manifold reduction from the ring of cells to the normal form equations is then performed. Using this reduction we find parameter values of the cell system where asymptotically stable cycles exist. Simulations of the cycles show trajectories visiting steady states and periodic solutions and reveal interesting spatio-temporal patterns in the dynamics of individual cells. We discuss how these patterns are forced by normal form symmetries.

Movement Responses of Caribou to Human-Induced Habitat Edges Lead to Their Aggregation near Anthropogenic Features

The assessment of disturbance effects on wildlife and resulting mitigation efforts are founded on edge-effect theory. According to the classical view, the abundance of animals affected by human disturbance should increase monotonically with distance from disturbed areas to reach a maximum at remote locations. Here we show that distance-dependent movement taxis can skew abundance distributions toward disturbed areas. We develop an advection-diffusion model based on basic movement behavior commonly observed in animal populations and parameterize the model from observations on radio-collared caribou in a boreal ecosystem. The model predicts maximum abundance at 3.7 km from cutovers and roads. Consistently, aerial surveys conducted over 161,920 km2 showed that the relative probability of caribou occurrence displays nonmonotonic changes with the distance to anthropogenic features, with a peak occurring at 4.5 km away from these features. This aggregation near disturbed areas thus provides the predators of this top-down-controlled, threatened herbivore species with specific locations to concentrate their search. The edge-effect theory developed here thus predicts that human activities should alter animal distribution and food web properties differently than anticipated from the current paradigm. Consideration of such nonmonotonic response to habitat edges may become essential to successful wildlife conservation.

ANALYSIS OF HOPF/HOPF BIFURCATIONS IN NONLOCAL HYPERBOLIC MODELS FOR SELF-ORGANISED AGGREGATIONS

The modelling and investigation of complex spatial and spatio-temporal patterns exhibited by a various self-organised biological aggregations has become one of the most rapidly-expanding research areas. Generally, the majority of the studies in this area either try to reproduce numerically the observed patterns, or use existence results to prove analytically that the models can exhibit certain types of patterns. Here, we focus on a class of nonlocal hyperbolic models for self-organised movement and aggregations, and investigate the bifurcation of some spatial and spatio-temporal patterns observed numerically near a codimension-2 Hopf/Hopf bifurcation point. Using weakly nonlinear analysis and the symmetry of the model, we identify analytically all types of solutions that can exist in the neighbourhood of this bifurcation point. We also discuss the stability of these solutions, and the implication of these stability results on the observed numerical patterns.

A spatial theory for characterizing predator–multiprey interactions in heterogeneous landscapes

Trophic interactions in multiprey systems can be largely determined by prey distributions. Yet, classic predator–prey models assume spatially homogeneous interactions between predators and prey. We developed a spatially informed theory that predicts how habitat heterogeneity alters the landscape-scale distribution of mortality risk of prey from predation, and hence the nature of predator interactions in multiprey systems. The theoretical model is a spatially explicit, multiprey functional response in which species-specific advection–diffusion models account for the response of individual prey to habitat edges. The model demonstrates that distinct responses of alternative prey species can alter the consequences of conspecific aggregation, from increasing safety to increasing predation risk. Observations of threatened boreal caribou, moose and grey wolf interacting over 378 181 km2 of human-managed boreal forest support this principle. This empirically supported theory demonstrates how distinct responses of apparent competitors to landscape heterogeneity, including to human disturbances, can reverse density dependence in fitness correlates.

Bifurcation and branching of equilibria in reversible equivariant vector fields

We study steady-state bifurcation in reversible equivariant vector fields. We assume an action on the phase space of a compact Lie group G with a normal subgroup H of index two, and study vector fields that are H-equivariant and have all elements of the complement G \ H as time-reversal symmetries. We focus on separable bifurcation problems that can be reduced to equivariant steady-state bifurcation problems, possibly with parameter symmetries. We describe both bifurcations of equilibria that arise when external parameters are varied, and branching of families of equilibria that may arise in the phase space when external parameters are fixed. We also show how our results apply to bifurcation problems for reversible relative equilibria and reversible (relative) periodic orbits.

Versal unfoldings for linear retarded functional differential equations

We consider parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds, and address the question of versality of the resulting parametrized family of linear ordinary differential equations. A sufficient criterion for versality is given in terms of readily computable quantities. In the case where the unfolding is not versal, we show how to construct a perturbation of the original linear RFDE (in terms of delay differential operators) whose finite-dimensional projection generates a versal unfolding. We illustrate the theory with several examples, and comment on the applicability of these results to bifurcation analyses of nonlinear RFDEs.

Codimension-Two Bifurcations in Animal Aggregation Models with Symmetry

Pattern formation in self-organized biological aggregation is a phenomenon that has been studied intensively over the past 20 years. In general, the studies on pattern formation focus mainly on identifying the biological mechanisms that generate these patterns. However, identifying the mathematical mechanisms behind these patterns is equally important, since it can offer information on the biological parameters that could contribute to the persistence of some patterns and the disappearance of other patterns. Also, it can offer information on the mechanisms that trigger transitions between different patterns (associated with different group behaviors). In this article, we focus on a class of nonlocal hyperbolic models for self-organized aggregations and show that these models are

Dynamics, Bifurcations and Normal Forms in Arrays of Magnetostrictive Energy Harvesters with All-to-All Coupling

{{\bf O(2)}}