

(Martin, Sascha and Mario, and collaboration of Eiswirth's, Baer's and Kevrekidis groups). A 1d partial differential equation (PDE) model of a catalysis on a surface has been extensively studied for large domains. The main results arise from the study of the travellingwave ODE (ordinary differential equation), where a heteroclinic cycle bifurcation (known as Tpoint) between two fixed points is found numerically. This heteroclinic cycle separates the region in parameter space where there exists 1humped pulse solutions to either one or the other stationary state. Also it delimits the region where spatiotemporal chaotic behaviour of the PDE is reported. We computed the stability of both pulses and one is found stable while the other unstable. (Martin and Mario, and collaboration of Solari's group). A 3d ODE model to a laser of injected signal was numerically investigated. It has been found that the chaotic dynamics is organized by a Shilnikovsaddlenode global bifurcation, close to the hopfsaddlenode (type III) local bifurcation. The main effect of such a degenerate bifurcation is that the homoclinic orbit is destroyed. The periodic orbit organization is studied via a return map close to the saddlenode bifurcation. It is shown that chaos dissapears as one approaches the degenerate point. We are extending our studies in the laser model to a region in parameter space where this is no longer the main scenario for the appearece/disapearence of chaotic dynamics. In this case the main bifurcation is a homoclinic bifurcation to a periodic orbit. It is found that in this case, chaos appears before the appearence of the locked states. In the catalysis equations above, it is found that on top of the 1loop heteroclinic cycle, there are also Nloop heteroclinic cycles (N=220 reported). We study a return map model to this manifold organization, taking into account that there is a local transcritical bifurcation coexisting. It is found that under suitable reinjections, there are countable many such heteroclinic cycles. With respect to the PDE these will separate regions in parameter space where Nhumped pulses to one homogeneous state is replaced by a Nhumped pulse to the other homogeneous state, the latter being unstable in the PDE. Thus the transition to spatiotemporal chaos reported in these equations can be seen to arise from the succesive replacement of stable Nhumped pulses into unstable Nhumped pulses, with decreasing N, until the last 1loop heteroclinic cycles occurs and the turbulent regime sets in. Also a Smale's horseshoe is found in some regions of parameter space. The laser equations showed that the hopfsaddlenode (HSN) local bifurcation was acting as an organizing center of all the main bifurcations. The Shilnikovsaddlenode appears for type III flows, while for the type I the global reinjection provided by the laser equations, gives the possibility of homoclinic connections to the small loop (periodic orbit) involved in the HSN bifurcation. How the periodic orbits are organized are studied again with a return map model. The complex GinzburgaLandau equation has been studied for a long time now. But a clear picture of what are the role of the unstable pulses (PRL of MVH) is still missing. We are trying to make sence out of the main phase transitions present in the CGL equation, studying the coherent structures and their stability. We have developed a stochastic extended system which develops spatiotemporal intermittency. It contains a multiplicative noise term which develops an absorving phase. Our model is known not to be in the Directed Percolation universality class. However, the spatiotemporal intermittency which develops looks similar to others which develop in reactiondiffusion PDE's. We show that the noise term influences the dynamics of a front, and changes its direction of propagation. At this point we find a phase transition which can be determined from a meanfield approach. This is an alternative account of spatiotemporal intermittency where the chaotic dynamics in other such models (maps and reactiondiffusion systems) is replaced by noise. We study the Prisoners Dilemma on a network. Our main motivation is to model how colaboration works in a scientific environment. This kind of spatial games has been studied previously by May and coworkers with a static network. Basically each player plays with each of its links, and for the next step, "imitates" the strategy of its best player among its neigbors. They showed that there is a parameter regime where interesting dynamical patterns appear. We show how in an evolving network we always arrive to a frozen state. 
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