Dynamics of large cooperative pulsed-coupled networks
Eleonora Catsigeras
Journal of Dynamics & Games 2014, 1(2): 255-281 doi: 10.3934/jdg.2014.1.255
We study the deterministic dynamics of networks ${\mathcal N}$ composed by $m$ non identical, mutually pulse-coupled cells. We assume weighted, asymmetric and positive (cooperative) interactions among the cells, and arbitrarily large values of $m$. We consider two cases of the network's graph: the complete graph, and the existence of a large core (i.e. a large complete subgraph). First, we prove that the system periodically eventually synchronizes with a natural "spiking period" $p \geq 1$, and that if the cells are mutually structurally identical or similar, then the synchronization is complete ($p= 1$) . Second, we prove that the amount of information $H$ that ${\mathcal N}$ generates or processes, equals $\log p$. Therefore, if ${\mathcal N}$ completely synchronizes, the information is null. Finally, we prove that ${\mathcal N}$ protects the cells from their risk of death.
keywords: synchronization impulsive ODE Pulse-coupled networks information. cooperative evolutive game
Dominated splitting, partial hyperbolicity and positive entropy
Eleonora Catsigeras Xueting Tian
Discrete & Continuous Dynamical Systems - A 2016, 36(9): 4739-4759 doi: 10.3934/dcds.2016006
Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting on a compact Riemanian manifold $M$ without boundary. We state and prove several sufficient conditions for the topological entropy of $f$ to be positive. The conditions deal with the dynamical behaviour of the (non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue measure is $\delta$-recurrent then the entropy of $f$ is positive. We give counterexamples showing that these sufficient conditions are not necessary. Finally, in the case of partially hyperbolic diffeomorphisms, we give a positive lower bound for the entropy relating it with the dimension of the unstable and stable sub-bundles.
keywords: SRB-like measures. SRB volume smooth positive topological entropy and positive metric entropy Dominated splitting and partial hyperbolicity
SRB measures of certain almost hyperbolic diffeomorphisms with a tangency
Eleonora Catsigeras Heber Enrich
Discrete & Continuous Dynamical Systems - A 2001, 7(1): 177-202 doi: 10.3934/dcds.2001.7.177
We study topological and ergodic properties of some almost hyperbolic diffeomorphisms on two dimensional manifolds. Under generic conditions, diffeomorphisms obtained from Anosov by an isotopy pushing together the stable and unstable manifolds to be tangent at a fixed point, are conjugate to Anosov. For a finite codimension subset at the boundary of Anosov there exist a SRB measure and an unique ergodic attractor.
keywords: Lyapounov functions expansive maps. boundary of Anosov SBR measures almost hyperbolic systems distortion bounds nonuniformly hyperbolic systems almost Anosov Measure preservation conjugation to Anosov ergodic attractors
Observable optimal state points of subadditive potentials
Eleonora Catsigeras Yun Zhao
Discrete & Continuous Dynamical Systems - A 2013, 33(4): 1375-1388 doi: 10.3934/dcds.2013.33.1375
For a sequence of subadditive potentials, a method of choosing state points with negative growth rates for an ergodic dynamical system was given in [5]. This paper first generalizes this result to the non-ergodic dynamics, and then proves that under some mild additional hypothesis, one can choose points with negative growth rates from a positive Lebesgue measure set, even if the system does not preserve any measure that is absolutely continuous with respect to Lebesgue measure.
keywords: subadditive potentials Optimal state points observable measures. milnor-like attractors
Simultaneous continuation of infinitely many sinks at homoclinic bifurcations
Eleonora Catsigeras Marcelo Cerminara Heber Enrich
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 693-736 doi: 10.3934/dcds.2011.29.693
We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinks near the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.
keywords: Homoclinic Bifurcations. Infinitely many Sinks

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