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Nonlinear dynamics of networks: the groupoid formalism
 Bull. Amer. Math. Soc
, 2006
"... Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which ..."
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Cited by 35 (6 self)
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Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the grouptheoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend grouptheoretic methods to more general networks, and in particular it leads to a complete classification of ‘robust ’ patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a highdimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood. Contents 1.
Some curious phenomena in coupled cell networks
 J. Nonlinear Sci
, 2004
"... We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of ..."
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Cited by 15 (13 self)
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We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a threecell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic 2n×2n arrays of cells that generate 2 n different patterns of synchrony from one symmetry generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form. 1
Averaged equations for Josephson junction series arrays
 Phys. Rev. E
, 1995
"... We derive the averaged equations describing a series array of Josephson junctions shunted by a parallel inductorresistorcapacitor load. We assume that the junctions have negligable capacitance (β = 0), and derive averaged equations which turn out to be completely tractable: in particular the stabi ..."
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Cited by 11 (1 self)
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We derive the averaged equations describing a series array of Josephson junctions shunted by a parallel inductorresistorcapacitor load. We assume that the junctions have negligable capacitance (β = 0), and derive averaged equations which turn out to be completely tractable: in particular the stability of both inphase and splay states depends on a single parameter, δ. We find an explicit expression for δ in terms of the load parameters and the bias current. We recover (and refine) a common claim found in the technical literature, that the inphase state is stable for inductive loads and unstable for capacitive loads.
A dynamical theory of spike train transitions in networks of integrateandfire oscillators
 SIAM J. Appl. Math
"... Abstract. A dynamical theory of spike train transitions in networks of pulsecoupled integrateandfire (IF) neural oscillators is presented. We begin by deriving conditions for 1:1 frequencylocking in a network with noninstantaneous synaptic interactions. This leads to a set of phase equations deter ..."
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Cited by 9 (1 self)
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Abstract. A dynamical theory of spike train transitions in networks of pulsecoupled integrateandfire (IF) neural oscillators is presented. We begin by deriving conditions for 1:1 frequencylocking in a network with noninstantaneous synaptic interactions. This leads to a set of phase equations determining the relative firing times of the oscillators and the selfconsistent collective period. We then investigate the stability of phaselocked solutions by constructing a linearized map of the firing times and analyzing its spectrum. We establish that previous results concerning the stability properties of IF oscillator networks are incomplete since they only take into account the effects of weak coupling instabilities. We show how strong coupling instabilities can induce transitions to nonphase locked states characterized by periodic or quasiperiodic variations of the interspike intervals on attracting invariant circles. The resulting spatiotemporal pattern of network activity is compatible with the behavior of a corresponding firing rate (analog) model in the limit of slow synaptic interactions.
Unfolding Isochronicity in Weakly Dissipative Coupled Oscillators
, 1999
"... We consider the dynamics of networks of oscillators that are weakly dissipative perturbations of identical Hamiltonian oscillators with weak coupling. Suppose the Hamiltonian oscillators have angular frequency !(ff) when their energy is ff. We address the problem of what happens in a neighbourhood o ..."
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We consider the dynamics of networks of oscillators that are weakly dissipative perturbations of identical Hamiltonian oscillators with weak coupling. Suppose the Hamiltonian oscillators have angular frequency !(ff) when their energy is ff. We address the problem of what happens in a neighbourhood of where d!=dff = 0; we refer to this as a point of isochronicity for the oscillators. If the coupling is much weaker than the dissipation we can use averaging to reduce the system to equations on a torus. We consider example applications to two and three weakly diffusively coupled oscillators with points of isochronicity and reduce to approximating flows on tori. We use this to identify changes of stability of various periodic solutions caused by perturbing away from a point of isochronicity. 1 Introduction Coupled oscillator systems have long been used as models for a variety of physical applications in fields as diverse as biology and solid mechanics. In addition to applications they have...
.4.3 Belyakov's scaling
"... to a homoclinic orbit to a saddle. The system we study is the following x = y \Gamma z y = 2:657466x +2:328733y + x 2 + xy + x 2 +0:83893461z (8.14) z = e 2:657466x \Gamma 0:83893461y 3:361277 \Gamma 0:83893461z This system was derived by augmenting a truncated unfolding of the normal ..."
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to a homoclinic orbit to a saddle. The system we study is the following x = y \Gamma z y = 2:657466x +2:328733y + x 2 + xy + x 2 +0:83893461z (8.14) z = e 2:657466x \Gamma 0:83893461y 3:361277 \Gamma 0:83893461z This system was derived by augmenting a truncated unfolding of the normal form of the Takens Bogdanov bifurcation x = y y = 2cx + (1 + c)y + x 2 + xy (8.15) which is numerically observed to have a homoclinic connection to the origin at c 1:328733,
unknown title
, 1996
"... Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance ..."
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Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance
ADYNAMICAL THEORYOF SPIKE TRAIN TRANSITIONS IN NETWORKS OF INTEGRATEANDFIRE OSCILLATORS ∗
"... Abstract. Adynamical theory of spike train transitions in networks of pulsecoupled integrateandfire (IF) neural oscillators is presented. We begin by deriving conditions for 1:1 frequency locking in anetwork with noninstantaneous synaptic interactions. This leads to aset of phase equations determ ..."
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Abstract. Adynamical theory of spike train transitions in networks of pulsecoupled integrateandfire (IF) neural oscillators is presented. We begin by deriving conditions for 1:1 frequency locking in anetwork with noninstantaneous synaptic interactions. This leads to aset of phase equations determining the relative firing times of the oscillators and the selfconsistent collective period. We then investigate the stability of phaselocked solutions by constructing a linearized map of the firing times and analyzing its spectrum. Weestablish that previous results concerning the stability properties of IF oscillator networks are incomplete since they only take into account the effects of weak coupling instabilities. We show how strong coupling instabilities can induce transitions to nonphase locked states characterized by periodic or quasiperiodic variations of the interspike intervals on attracting invariant circles. The resulting spatiotemporal pattern of network activity is compatible with the behavior of acorresponding firing rate (analog) model in the limit of slow synaptic interactions. Key words. pulsecoupled oscillators, integrateandfire, quasiperiodicity,Hopf bifurcations AMS subject classifications. 34,92
c ○ 2012 Society for Industrial and Applied Mathematics Splay States in Finite PulseCoupled Networks of Excitable Neurons ∗
"... Abstract. The emergence and stability of splay states is studied in fully coupled finite networks of N excitable quadratic integrateandfire neurons, connected via synapses modeled as pulses of finite amplitude and duration. For such synapses, by introducing two distinct types of synaptic events (p ..."
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Abstract. The emergence and stability of splay states is studied in fully coupled finite networks of N excitable quadratic integrateandfire neurons, connected via synapses modeled as pulses of finite amplitude and duration. For such synapses, by introducing two distinct types of synaptic events (pulse emission and termination), we were able to write down an exact eventdriven map for the system and to evaluate the splay state solutions. For M overlapping postsynaptic potentials, the linear stability analysis of the splay state should also take in account, besides the actual values of the membrane potentials, the firing times associated with the M previous pulse emissions. As a matter of fact, it was possible, by introducing M complementary variables, to rephrase the evolution of the network as an eventdriven map and to derive an analytic expression for the Floquet spectrum. We find that, independently of M, the splay state is marginally stable with N −2 neutral directions. Furthermore, we have identified a family of periodic solutions surrounding the splay state and sharing the same neutral stability directions. In the limit of δpulses, it is still possible to derive an eventdriven formulation for the dynamics; however, the number of neutrally stable directions associated with the splay state becomes N. Finally, we prove a link between the results for our system and a previous theory [S. Watanabe and S. H. Strogatz, Phys. D, 74 (1994), pp. 197–253] developed for networks of phase oscillators with sinusoidal coupling.