107 research outputs found
A Lower Bound for Chaos on the Elliptical Stadium
The elliptical stadium is a plane region bounded by a curve constructed by
joining two half-ellipses by two parallel segments of equal length. The
billiard inside it, as a map, generates a two parameters family of dynamical
systems. It is known that the system is ergodic for a certain region of the
parameter space. In this work we study the stability of a particular family of
periodic orbits obtaining good bounds for the chaotic zone.Comment: 13 pages, LaTeX. 7 postscript low resolution figures included. High
resolution figures avaiable under request to [email protected]
Robustness of Cucker-Smale flocking model
Consider a system of autonomous interacting agents moving in space, adjusting
each own velocity as a weighted mean of the relative velocities of the other
agents. In order to test the robustness of the model, we assume that each pair
of agents, at each time step, can fail to connect with certain probability, the
failure rate. This is a modification of the (deterministic) Flocking model
introduced by Cucker and Smale in Emergent behavior in flocks, IEEE Trans. on
Autom. Control, 2007, 52 (May) pp. 852-862. We prove that, if this random
failures are independent in time and space, and have linear or sub-linear
distance dependent rate of decay, the characteristic behavior of flocking
exhibited by the original deterministic model, also holds true under random
failures, for all failure rates.Comment: 9 pages, 3 figure
From weighted to unweighted graphs in Synchronizing Graph Theory
A way to associate unweighted graphs from weighted ones is presented, such
that linear stable equilibria of the Kuramoto homogeneous model associated to
both graphs coincide, i.e., equilibria of the system , where means vertices and
are adjacent in the corresponding graph. As a consequence, the existence of
linearly stable equilibrium is proved to be NP-Hard as conjectured by R. Taylor
in 2015 and a new lower bound for the minimum degree that ensures
synchronization is found
Network reliability analysis and intractability of counting diameter crystal graphs
Consider a stochastic network, where nodes are perfect but links fail independently, ruled by failure probabilities. Additionally, we are given distinguished nodes, called terminals, and a positive integer, called diameter. The event under study is to connect terminals by paths not longer than the given diameter. The probability of this event is called diameter-constrained reliability (DCR, for short). Since the DCR subsumes connectedness probability of random graphs, its computation belongs to the class of NP-Hard problems. The computational complexity for DCR is known for fixed values of the number of terminals k n and diameter d, being n the number of nodes in the network. The contributions of this article are two-fold. First, we extend the computational complexity of the DCR when the terminal size is a function of the number of nodes, this is, when k = k(n). Second, we state counting diameter-critical graphs belongs to the class of NP-Hard problems
Recursive variance reduction in reliability analysis
Network reliability deals with reliability metrics of large classes of mul- ticomponent systems. Recursive Variance Reduction (RVR) is a powerful pointwise estimation method, widely applied in network reliability anal- ysis. In this paper, RVR is extended to arbitrary Stochastic Binary Sys- tems, with minor requirements. Additionally, its variance is again lower than Crude Monte Carlo (CMC), in this general context
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