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15
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 76 (13 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.
Interior symmetry and local bifurcation in coupled cell networks
 Dynamical Systems
, 2004
"... Abstract. A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells an ..."
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Cited by 19 (2 self)
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Abstract. A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the ‘symmetry groupoid’, has shown that global grouptheoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structurepreserving bijections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: ‘interior symmetry’. This concept is closely related to the groupoid structure, but imposes stronger constraints of a grouptheoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for ‘synchronybreaking’ bifurcations of the Equivariant Branching Lemma for steadystate bifurcation, and the Equivariant Hopf Theorem for bifurcation to timeperiodic states.
An Evolutionary Game Perspective to ALOHA with power control
"... We study a large population of communicating terminals using an ALOHA protocol with two possible levels of transmission power. We pose the problem of how to choose between these power levels. We study two noncooperative optimization concepts: the Nash equilibrium and the Evolutionary Stable Strateg ..."
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Cited by 11 (3 self)
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We study a large population of communicating terminals using an ALOHA protocol with two possible levels of transmission power. We pose the problem of how to choose between these power levels. We study two noncooperative optimization concepts: the Nash equilibrium and the Evolutionary Stable Strategy. The latter were introduced in mathematical biology in the context of Evolutionary Games, which allow to describe and to predict properties of large populations whose evolution depends on many local interactions, each involving a finite number of individuals. We compare the performances of these noncooperative notions with the global cooperative solution. The payoffs that we consider are functions of the throughputs and of the cost for the power levels. We study in particular the impact of the pricing for the use of the power levels on the system performance.
A Stochastic Evolutionary Game Approach to Energy Management in a Distributed Aloha Network
 IEEE INFOCOM
, 1988
"... A major contribution of biology to competitive decision making is the area of evolutionary games. It describes the evolution of sizes of large populations as a result of many local interactions, each involving a small number of randomly selected individuals. An individual plays only once; it plays ..."
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Cited by 7 (3 self)
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A major contribution of biology to competitive decision making is the area of evolutionary games. It describes the evolution of sizes of large populations as a result of many local interactions, each involving a small number of randomly selected individuals. An individual plays only once; it plays in a one shot game against another randomly selected player with the goal of maximizing its utility (fitness) in that game. We introduce here a new more general type of games: a Stochastic Evolutionary Game where each player may be in different states; the player may be involved in several local interactions during his life time and his actions determine not only the utilities but also the transition probabilities and his life duration. This is used to study a large population of mobiles forming a sparse adhoc network, where mobiles compete with their neighbors on the access to a radio channel. We study the impact of the level of energy in the battery on the aggressiveness of the access policy of mobiles at equilibrium. We obtain properties of the ESS (Evolutionary Stable Strategy) equilibrium which, Unlike the Nash equilibrium concept, is robust against deviations of a whole positive fraction of the population. We further study dynamical properties of the system when it is not in equilibrium.
2003b, Speciation: a case study in symmetric bifurcation theory
 Universitatis Iagellonicae Acta Mathematica
"... Abstract. Symmetric bifurcation theory is the study of how the trajectories of symmetric vector fields behave as parameters are varied. We introduce some of the basic ideas of this theory in the context of dynamical system models of speciation in evolution. Abstractly, these models are dynamical s ..."
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Cited by 5 (1 self)
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Abstract. Symmetric bifurcation theory is the study of how the trajectories of symmetric vector fields behave as parameters are varied. We introduce some of the basic ideas of this theory in the context of dynamical system models of speciation in evolution. Abstractly, these models are dynamical systems that are equivariant under the natural permutation action of the symmetric group SN on R kN for some integers N, k ≥ 1. The general theory, which is grouptheoretic in nature, makes it possible to analyse such systems in a systematic manner. The results explain several phenomena that can be observed in simulations of specific equations. In particular, in steadystate bifurcation, primary branches involve bifurcation to twospecies states; such bifurcations are generically jumps; and the weighted mean phenotype of the organisms changes smoothly, whereas the standard deviation jumps. In particular, classical meanfield genetics, which focusses on allele proportions in the population, cannot detect this kind of speciation event. 1.
Design of Nonlinear Autopilots for High Angle of Attack Missiles
 AIAA Guidance, Navigation, and Control Conference
, 1996
"... Two nonlinear autopilot design approaches for a tailcontrolled high angle of attack airtoair missile are described. The research employs a highly nonlinear, time varying pitch plane rigidbody dynamical model of a short range missile. Feedback linearization technique together with linear control t ..."
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Cited by 5 (2 self)
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Two nonlinear autopilot design approaches for a tailcontrolled high angle of attack airtoair missile are described. The research employs a highly nonlinear, time varying pitch plane rigidbody dynamical model of a short range missile. Feedback linearization technique together with linear control theory are then used for autopilot design. In order to manage the difficulties associated with "zerodynamics" that arise in tail controlled missiles, two distinct approaches for approximate feedback linearization are advanced. The first approach imposes a timescale structure in the closedloop dynamics, while the second technique redefines the output. Performance of these autopilots are illustrated in a nonlinear simulation.
Multiple Access Game in Adhoc Network
"... We study a Modified Multiple Access Game (MMAG) in two approach: repeated game approach and evolutionary game approach. We compute Nash equilibria and Evolutionary Stable Strategy (ESS) in the two cases. We study the delay impact on the performance of the evolutionary MMAG describing competition bet ..."
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Cited by 3 (2 self)
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We study a Modified Multiple Access Game (MMAG) in two approach: repeated game approach and evolutionary game approach. We compute Nash equilibria and Evolutionary Stable Strategy (ESS) in the two cases. We study the delay impact on the performance of the evolutionary MMAG describing competition between mobile terminals over the access to a channel. We discuss about the convergence to the ESS in replicator dynamics and imitate better dynamics with delays.
An Evolutionary Game Perspective to ALOHA with power control
"... Abstract: We study a large population of communicating terminals using an ALOHA protocol with two possible levels of transmission power. We pose the problem of how to choose between these power levels. We study two noncooperative optimization concepts: the Nash equilibrium and the Evolutionary Stab ..."
Abstract
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Abstract: We study a large population of communicating terminals using an ALOHA protocol with two possible levels of transmission power. We pose the problem of how to choose between these power levels. We study two noncooperative optimization concepts: the Nash equilibrium and the Evolutionary Stable Strategy. The latter was introduced in mathematical biology in the context of Evolutionary Games, which allows to describe and to predict properties of large populations whose evolution depends on many local interactions, each involving a finite number of individuals. We compare the performances of these noncooperative notions with the global cooperative solution. The payoffs that we consider are functions of the throughputs and of the cost for the power levels. We study in particular the impact of the pricing for the use of the power levels on the system performance.