25 research outputs found
Synchronization of chaotic networks with time-delayed couplings: An analytic study
Networks of nonlinear units with time-delayed couplings can synchronize to a
common chaotic trajectory. Although the delay time may be very large, the units
can synchronize completely without time shift. For networks of coupled
Bernoulli maps, analytic results are derived for the stability of the chaotic
synchronization manifold. For a single delay time, chaos synchronization is
related to the spectral gap of the coupling matrix. For networks with multiple
delay times, analytic results are obtained from the theory of polynomials.
Finally, the analytic results are compared with networks of iterated tent maps
and Lang-Kobayashi equations which imitate the behaviour of networks of
semiconductor lasers
Isospectral Compression and Other Useful Isospectral Transformations of Dynamical Networks
It is common knowledge that a key dynamical characteristic of a network is
its spectrum (the collection of all eigenvalues of the network's weighted
adjacency matrix). In \cite{BW10} we demonstrated that it is possible to reduce
a network, considered as a graph, to a smaller network with fewer vertices and
edges while preserving the spectrum (or spectral information) of the original
network. This procedure allows for the introduction of new equivalence
relations between networks, where two networks are spectrally equivalent if
they can be reduced to the same network. Additionally, using this theory it is
possible to establish whether a network, modeled as a dynamical system, has a
globally attracting fixed point (is strongly synchronizing). In this paper we
further develop this theory of isospectral network transformations and
demonstrate that our procedures are applicable to families of parameterized
networks and networks of arbitrary size.Comment: 26 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1010.327
Markovian evolution of quantum coherence under symmetric dynamics
Both conservation laws and practical restrictions impose symmetry constraints on the dynamics of open quantum systems. In the case of time-translation symmetry, which arises naturally in many physically relevant scenarios, the quantum coherence between energy eigenstates becomes a valuable resource for quantum information processing. In this work we identify the minimum amount of decoherence compatible with this symmetry for a given population dynamics. This yields a generalisation to higher-dimensional systems of the relation T2 2T1 for qubit decoherence and relaxation times. It also enables us to witness and assess the role of non-Markovianity as a resource for coherence preservation and transfer. Moreover, we discuss the relationship between ergodicity and the ability of Markovian dynamics to indenitely sustain a superposition of diferent energy states. Finally, we establish a formal connection between the resource-theoretic and the master equation approaches to thermodynamics, with the former being a non-Markovian generalisation of the latter. Our work thus brings the abstract study of quantum coherence as a resource towards the realm of actual physical applications
Multistability in lossy power grids and oscillator networks
Networks of phase oscillators are studied in various contexts, in particular, in the modeling of the electric power grid. A functional grid corresponds to a stable steady state such that any bifurcation can have catastrophic consequences up to a blackout. Also, the existence of multiple steady states is undesirable as it can lead to transitions or circulatory flows. Despite the high practical importance there is still no general theory of the existence and uniqueness of steady states in such systems. Analytic results are mostly limited to grids without Ohmic losses. In this article, we introduce a method to systematically construct the solutions of the real power load-flow equations in the presence of Ohmic losses and explicitly compute them for tree and ring networks.We investigate different mechanisms leading to multistability and discuss the impact of Ohmic losses on the existence of solutions. Published under license by AIP Publishing