Luciano Buono

Université du Québec à Rimouski UQAR | uqar

We present the first circuit realization of an animal (quadruped) robot controlled by a Central Pattern Generator (CPG) network of neurons, whose model and design are biologically-inspired [Golubitsky et al., 1998, 1999]. We demonstrate, through hardware simulations of the CPG network and through video snapshots, that our electronic CPG and our ani...

Computational and experimental works reveal that the coupling of similar crystal oscillators leads to a variety of collective patterns, mainly various forms of discrete rotating waves and synchronization patterns, which have the potential for developing precision timing devices through phase drift reduction. Among all observed patterns, the standar...

The ability for a Spin Torque Nano Oscillator (STNO) to perform as a nano-scaled microwave voltage oscillator continues to be the focus of extensive research. Due to their small size (on the order of 10nm), low power consumption, and ultrawide frequency range STNOs demonstrate significant potential for practical applications in microwave generation...

At the National Observatory in Washington D.C., time is measured by averaging the times of an uncoupled ensemble. The measurements show a scaling law for phase-error reduction as, where is the number of crystals in the ensemble. Analytical and computational works show that certain patterns of collective behavior produced by a network of nonlinear o...

In this study we start by reviewing a class of 1D hyperbolic/kinetic models (with two velocities) used to investigate the collective behaviour of cells, bacteria or animals. We then focus on a restricted class of nonlocal models that incorporate various inter-individual communication mechanisms, and discuss how the symmetries of these models impact...

Precise time dissemination and synchronization have been some of the most important technological tasks for several centuries. No later than Harrison's time, it was realized that precise time-keeping devices having the same stable frequency and precisely synchronized can have important applications in navigation. In modern times, satellite-based gl...

Symmetry is used to investigate the existence and stability of collective patterns of oscillations in rings of coupled crystal oscillators. We assume N identical crystal oscillators, where each oscillator is described by a two-mode nonlinear oscillatory circuit. We also assume the coupling to be identical and consider two different topologies, unid...

Symmetry is used to investigate the existence and stability of collective patterns of oscillations in rings of coupled crystal oscillators. We assume $N$ identical crystal oscillators, where each oscillator is described by a two-mode nonlinear oscillatory circuit. We also assume the coupling to be identical and consider two different topologies, un...

Precise time dissemination and synchronization have been some of the most important technological tasks for several centuries. It was realized that precise time-keeping devices having the same stable frequency and precisely synchronized can have important applications in navigation. Satellite-based global positioning and navigation systems such as...

Synchronization of spin torque nano-oscillators (STNOs) has been a subject of extensive research as various groups try to harness the collective power of STNOs to produce a strong enough microwave signal at the nanoscale. Achieving synchronization has proven to be, however, rather difficult for even small arrays while in larger ones the task of syn...

We investigate the bifurcation structure of the Kuramoto–Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and O(2) symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection...

The growth and invasion of cancer cells are very complex processes, which can be regulated by the cross-talk between various signalling pathways, or by single signalling pathways that can control multiple aspects of cell behaviour. TGF-β is one of the most investigated signalling pathways in oncology, since it can regulate multiple aspects of cell...

The coupled cell formalism is a systematic way to represent and study coupled nonlinear differential equations using directed graphs. In this work, we focus on coupled cell systems in which individual cells are also Hamiltonian. We show that some coupled cell systems do not admit Hamiltonian vector fields because the associated directed graphs are...

The goal of this paper is to establish the applicability of the Lyapunov–Schmidt reduction and the Centre Manifold Theorem (CMT) for a class of hyperbolic partial differential equation models with nonlocal interaction terms describing the aggregation dynamics of animals/cells in a one-dimensional domain with periodic boundary conditions. We show th...

We study a network-based model of a high-precision, inexpensive, Coupled Crystal Oscillator System and Timing (CCOST) device. The model consist of N crystals coupled together, unidirectionally, in a ring configuration. Other coupling topologies might be feasible but they are deferred for future work. Preliminary results from computer simulations se...

We consider symmetric rings of delay-coupled lasers modeled using the Lang-Kobayashi (LK) rate equations with unidirectional and bidirectional coupling. Because of phase symmetry the networks have symmetry groups ℤn×S1 (unidirectional) and Dn×S1 (bidirectional). Our first main result is a characterization of isotropy subgroups of those actions from...

Over the past twelve years, ideas and methods from nonlinear dynamics system theory, in particular, group theoretical methods in bifurcation theory, have been used to study, design, and fabricate novel engineering technologies. For instance, the existence and stability of heteroclinic cycles in coupled bistable systems has been exploited to develop...

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 pre...

Modeling and bifurcation analysis of an energy harvesting system composed of coupled resonators using the Galfenol-based magnetostrictive material are presented. The analysis in this work should be broad enough to be applicable to a large class of vibratory-based energy harvesting systems since various types of vibratory harvesters share the same n...

The study of self-organised collective animal behaviour, such as swarms of insects or schools of fish, has become over the last decade a very active research area in mathematical biology. Parabolic and hyperbolic models have been used intensively to describe the formation and movement of various aggregative behaviours. While both types of models ca...

Background/Question/Methods The behavioral game taking place between predators and prey largely determines their spatial distributions. One possible, and surprising, outcome of this behavioral response race is the leapfrog effect, whereby predators match the distribution of their prey’s resources, while prey undermatch their own resources to redu...

In this paper, we consider a class of equivariant neutral functional differential equations (NFDEs) with stable D operator. We showthe existence of a centre manifold near periodic solutions with finite spatio-temporal symmetry group (a.k.a discrete rotating waves) invariant with respect to the spatio-temporal symmetry group. This is done by extendi...

We consider the question of linear stability of a periodic solution z(t) with finite spatio-temporal symmetry group of a reversible-equivariant Hamiltonian system obtained as a minimizer of the action functional. Our main theorem states that z(t) is unstable if the second variation at the minimizer has positive directions and a subspace W associate...

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 import...

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 an...

The evolution of a large class of biological, physical and engineering systems can be studied through both dynamical systems theory and Hamiltonian mechanics. The former theory, in particular its specialization to study systems with symmetry, is already well developed and has been used extensively on a wide variety of spatio-temporal systems. There...

Abstract 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-dep...

The hip-hop orbit is an interesting symmetric periodic family of orbits whereby the global existence methods of variational analysis applied to the N-body problem result in a collision free solution of (1). Perturbation techniques have been applied to study families of hip-hop like orbits bifurcating from a uniformly rotating planar 2N-gon [4] with...

In this paper, we study rings of symmetrically coupled fast∕slow systems with a coupling from the fast variables to the slow variable. We focus mainly on the case of coupled Hindmarsh‐Rose systems for which we investigate the symmetry‐breaking steady‐state bifurcations from the homogeneous equilibrium solution. We classify the primary bifurcations...

Let $G$ be a finite group acting on vector spaces $V$ and $W$ and consider a smooth $G$-equivariant mapping $f:V\to W$. This paper addresses the question of the zero set near a zero $x$ of $f$ with isotropy subgroup $G$. It is known from results of Bierstone and Field on $G$-transversality theory that the zero set in a neighborhood of $x$ is a stra...

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 GH as time-reversal symmetries. We focus on separable bifurcation problems that can b...

This paper studies the link between the number of critical eigenvalues and the number of delays in certain classes of delay-differential equations. There are two main results. The first states that for k purely imaginary numbers which are linearly independent over the rationals, there exists a scalar delay-differential equation depending on k fixed...

The lectures in this 2005 book are intended to bring young researchers to the current frontier of knowledge in geometrical mechanics and dynamical systems. They succinctly cover an unparalleled range of topics from the basic concepts of symplectic and Poisson geometry, through integrable systems, KAM theory, fluid dynamics, and symmetric bifurcatio...

The heartbeat of the medicinal leech is driven by direct contact between two arrays of motorneurons and two lateral blood vessels. At any given time, motorneurons exhibit one of two alternating states so that, on one side of the animal, the heart beats in a rear-to-front fashion (peristaltic), while on the other side the heart beats synchronously....

Drug therapies are often designed to reproduce physiological fluctuations in normal biological agents including hormones. These regimens entail the need for a periodic, yet sustained administration, and thus require the generation of oscillatory variations in concentrations. We extend (in part, by the explicit use of time-delayed arguments) a syste...

The heartbeat of the medicinal leech consists of two intricate patterns of oscillatory behavior, which are driven by two lateral arrays of motorneurons. On one side of the animal the motorneurons oscillate synchronously, while on the other side they produce a peristaltic wave of oscillations. Then every 20 heartbeats, approximately, the two sides a...

We continue our investigation of versality for parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds. In this paper, we consider RFDEs equivariant with respect to the action of a compact Lie group. In a previous paper (Buono and LeBlanc, to appear in J. Diff. Eqs.),...

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 Magalhães (J. Differential Equations 122 (1995) 181) we show that (1) there exists linearly independent unfolding parameters of classes of delay-differential equat...

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 quant...

We continue the analysis of the network of symmetrically coupled cells modeling central pattern generators (CPG) for quadruped locomotion proposed by Golubitsky, Stewart, Buono and Collins by studying secondary gaits. Secondary gaits are modeled by output signals from the CPG where each cell emits one of two different output signals along with exac...

In this paper we continue the analysis of a network of symmetrically coupled cells modeling central pattern generators for quadruped locomotion proposed by Golubitsky, Stewart, Buono, and Collins. By a cell we mean a system of ordinary differential equations and by a coupled cell system we mean a network of identical cells with coupling terms. We h...

Symmetry is used to investigate the existence and stability of heteroclinic cycles involving steady-state and periodic solutions in coupled cell systems with Dn-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 equatio...

this paper, we explore numerically the existence of heteroclinic cycles when O(2) symmetry is replaced by Dn symmetry. Here we show that cycles corresponding to those in Figures 2 and 3 for systems with O(2) symmetry also occur in systems with Dn symmetry. These cycles are richer in that they connect equilibria and time periodic solutions with more...

Animal locomotion is controlled, in part, by a central pattern generator (CPG), which is an intraspinal network of neurons capable of generating a rhythmic output. The spatio-temporal symmetries of the quadrupedal gaits walk, trot and pace lead to plausible assumptions about the symmetries of locomotor CPGs. These assumptions imply that the CPG of...

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...

The purpose of these notes is to give a brief survey of bifurcation theory of Hamiltonian systems with symmetry; they are a slightly extended version of the 5 lectures given by JM on Hamiltonian Systems with Symmetry at the Peyresq Summer School. Attention is focussed on bifurcations near equilibrium solutions and relative equilibria. [Taken from i...