Stability threshold approach for complex dynamical systems

Acknowledgments This paper was developed within the scope of the IRTG 1740/TRP 2011/50151-0, funded by the DFG/FAPESP, and supported by the Government of the Russian Federation (Agreement No. 14.Z50.31.0033 with the Institute of Applied Physics RAS). The first author thanks Dr Roman Ovsyannikov for valuable discussions regarding estimation of the mistake probability.Peer reviewedPreprintPublisher PD

Electrohydrodynamic linear stability analysis of dielectric liquids subjected to unipolar injection in a rectangular enclosure with rigid sidewalls

We investigate the linear stability threshold of a dielectric liquid subjected to unipolar injection in a 2D rectangular enclosure with rigid boundaries. A finite element formulation transforms the set of linear partial differential equations that governs the system into a set of algebraic equations. The resulting system poses an eigenvalue problem. We calculate the linear stability threshold, as well as the velocity field and charge density distribution, as a function of the aspect ratio of the domain. The stability parameter as a function of the aspect ratio describes paths of symmetry-breaking bifurcation. The symmetry properties of the different linear modes determine whether these paths cross each other or not. The resulting structure has important consequences in the non-linear behavior of the system after the bifurcation points.Ministerio de ciencia y tecnología FIS2011-25161Junta de Andalucía P10-FQM-5735Junta de Andalucía P09-FQM-458

Generalized Stability Condition for Generalized and Doubly-Generalized LDPC Codes

In this paper, the stability condition for low-density parity-check (LDPC) codes on the binary erasure channel (BEC) is extended to generalized LDPC (GLDPC) codes and doublygeneralized LDPC (D-GLDPC) codes. It is proved that, in both cases, the stability condition only involves the component codes with minimum distance 2. The stability condition for GLDPC codes is always expressed as an upper bound to the decoding threshold. This is not possible for D-GLDPC codes, unless all the generalized variable nodes have minimum distance at least 3. Furthermore, a condition called derivative matching is defined in the paper. This condition is sufficient for a GLDPC or DGLDPC code to achieve the stability condition with equality. If this condition is satisfied, the threshold of D-GLDPC codes (whose generalized variable nodes have all minimum distance at least 3) and GLDPC codes can be expressed in closed form.Comment: 5 pages, 2 figures, to appear in Proc. of IEEE ISIT 200

Stability of BTZ black strings

We study the dynamical stability of the BTZ black string against fermonic and gravitational perturbations. The BTZ black string is not always stable against these perturbations. There exist threshold values for $m^2$ related to the compactification of the extra dimension for fermonic perturbation, scalar part of the gravitational perturbation and the tensor perturbation, respectively. Above the threshold values, perturbations are stable; while below these thresholds, perturbations can be unstable. We find that this non-trivial stability behavior qualitatively agrees with that predicted by a thermodynamical argument, showing that the BTZ black string phase is not the privileged stable phase.Comment: 9 pages, revised version to appear in Phys. Rev.

Synchronization Probability in Large Random Networks

In a generalized framework, where multi-state and inter-state linkages are allowed, we derive a sufficient condition for the stability of synchronization in a network of chaotic attractors. This condition explicitly relates the network structure and the local and coupling dynamics to synchronization stability. For large Erd\"{o}s-R\'{e}nyi networks, the obtained condition is translated into a lower bound on the probability of stability of synchrony. Our results show that the probability of stability quickly increases as the randomness crosses a threshold which for large networks is inversely proportional to the network size