132 research outputs found
Invariant submanifold for series arrays of Josephson junctions
We study the nonlinear dynamics of series arrays of Josephson junctions in
the large-N limit, where N is the number of junctions in the array. The
junctions are assumed to be identical, overdamped, driven by a constant bias
current and globally coupled through a common load. Previous simulations of
such arrays revealed that their dynamics are remarkably simple, hinting at the
presence of some hidden symmetry or other structure. These observations were
later explained by the discovery of (N - 3) constants of motion, each choice of
which confines the resulting flow in phase space to a low-dimensional invariant
manifold. Here we show that the dimensionality can be reduced further by
restricting attention to a special family of states recently identified by Ott
and Antonsen. In geometric terms, the Ott-Antonsen ansatz corresponds to an
invariant submanifold of dimension one less than that found earlier. We derive
and analyze the flow on this submanifold for two special cases: an array with
purely resistive loading and another with resistive-inductive-capacitive
loading. Our results recover (and in some instances improve) earlier findings
based on linearization arguments.Comment: 10 pages, 6 figure
Dynamics of fully coupled rotators with unimodal and bimodal frequency distribution
We analyze the synchronization transition of a globally coupled network of N
phase oscillators with inertia (rotators) whose natural frequencies are
unimodally or bimodally distributed. In the unimodal case, the system exhibits
a discontinuous hysteretic transition from an incoherent to a partially
synchronized (PS) state. For sufficiently large inertia, the system reveals the
coexistence of a PS state and of a standing wave (SW) solution. In the bimodal
case, the hysteretic synchronization transition involves several states.
Namely, the system becomes coherent passing through traveling waves (TWs), SWs
and finally arriving to a PS regime. The transition to the PS state from the SW
occurs always at the same coupling, independently of the system size, while its
value increases linearly with the inertia. On the other hand the critical
coupling required to observe TWs and SWs increases with N suggesting that in
the thermodynamic limit the transition from incoherence to PS will occur
without any intermediate states. Finally a linear stability analysis reveals
that the system is hysteretic not only at the level of macroscopic indicators,
but also microscopically as verified by measuring the maximal Lyapunov
exponent.Comment: 22 pages, 11 figures, contribution for the book: Control of
Self-Organizing Nonlinear Systems, Springer Series in Energetics, eds E.
Schoell, S.H.L. Klapp, P. Hoeve
Chapman-Enskog method and synchronization of globally coupled oscillators
The Chapman-Enskog method of kinetic theory is applied to two problems of
synchronization of globally coupled phase oscillators. First, a modified
Kuramoto model is obtained in the limit of small inertia from a more general
model which includes ``inertial'' effects. Second, a modified Chapman-Enskog
method is used to derive the amplitude equation for an O(2) Takens-Bogdanov
bifurcation corresponding to the tricritical point of the Kuramoto model with a
bimodal distribution of oscillator natural frequencies. This latter calculation
shows that the Chapman-Enskog method is a convenient alternative to normal form
calculations.Comment: 7 pages, 2-column Revtex, no figures, minor change
Role of Network Topology in the Synchronization of Power Systems
We study synchronization dynamics in networks of coupled oscillators with
bimodal distribution of natural frequencies. This setup can be interpreted as a
simple model of frequency synchronization dynamics among generators and loads
working in a power network. We derive the minimum coupling strength required to
ensure global frequency synchronization. This threshold value can be
efficiently found by solving a binary optimization problem, even for large
networks. In order to validate our procedure, we compare its results with
numerical simulations on a realistic network describing the European
interconnected high-voltage electricity system, finding a very good agreement.
Our synchronization threshold can be used to test the stability of frequency
synchronization to link removals. As the threshold value changes only in very
few cases when aplied to the European realistic network, we conclude that
network is resilient in this regard. Since the threshold calculation depends on
the local connectivity, it can also be used to identify critical network
partitions acting as synchronization bottlenecks. In our stability experiments
we observe that when a link removal triggers a change in the critical
partition, its limits tend to converge to national borders. This phenomenon,
which can have important consequences to synchronization dynamics in case of
cascading failure, signals the influence of the uncomplete topological
integration of national power grids at the European scale.Comment: The final publication is available at http://www.epj.org (see
http://www.springerlink.com/content/l22k574x25u6q61m/
Solvable model of a phase oscillator network on a circle with infinite-range Mexican-hat-type interaction
We describe a solvable model of a phase oscillator network on a circle with
infinite-range Mexican-hat-type interaction. We derive self-consistent
equations of the order parameters and obtain three non-trivial solutions
characterized by the rotation number. We also derive relevant characteristics
such as the location-dependent distributions of the resultant frequencies of
desynchronized oscillators. Simulation results closely agree with the
theoretical ones
Interacting universes and the cosmological constant
We study some collective phenomena that may happen in a multiverse scenario.
First, it is posed an interaction scheme between universes whose evolution is
dominated by a cosmological constant. As a result of the interaction, the value
of the cosmological constant of one of the universes becomes very close to zero
at the expense of an increasing value of the cosmological constant of the
partner universe. Second, we found normal modes for a 'chain' of interacting
universes. The energy spectrum of the multiverse, being this taken as a
collective system, splits into a large number of levels, some of which
correspond to a value of the cosmological constant very close to zero. We
finally point out that the multiverse may be much more than the mere sum of its
parts.Comment: 7 page
How to suppress undesired synchronization
It is delightful to observe the emergence of synchronization in the blinking
of fireflies to attract partners and preys. Other charming examples of
synchronization can also be found in a wide range of phenomena such as, e.g.,
neurons firing, lasers cascades, chemical reactions, and opinion formation.
However, in many situations the formation of a coherent state is not pleasant
and should be mitigated. For example, the onset of synchronization can be the
root of epileptic seizures, traffic congestion in communication networks, and
the collapse of constructions. Here we propose the use of contrarians to
suppress undesired synchronization. We perform a comparative study of different
strategies, either requiring local or total knowledge of the system, and show
that the most efficient one solely requires local information. Our results also
reveal that, even when the distribution of neighboring interactions is narrow,
significant improvement in mitigation is observed when contrarians sit at the
highly connected elements. The same qualitative results are obtained for
artificially generated networks as well as two real ones, namely, the Routers
of the Internet and a neuronal network
Multiple dynamical time-scales in networks with hierarchically nested modular organization
Many natural and engineered complex networks have intricate mesoscopic
organization, e.g., the clustering of the constituent nodes into several
communities or modules. Often, such modularity is manifested at several
different hierarchical levels, where the clusters defined at one level appear
as elementary entities at the next higher level. Using a simple model of a
hierarchical modular network, we show that such a topological structure gives
rise to characteristic time-scale separation between dynamics occurring at
different levels of the hierarchy. This generalizes our earlier result for
simple modular networks, where fast intra-modular and slow inter-modular
processes were clearly distinguished. Investigating the process of
synchronization of oscillators in a hierarchical modular network, we show the
existence of as many distinct time-scales as there are hierarchical levels in
the system. This suggests a possible functional role of such mesoscopic
organization principle in natural systems, viz., in the dynamical separation of
events occurring at different spatial scales.Comment: 10 pages, 4 figure
A measure of individual role in collective dynamics
Identifying key players in collective dynamics remains a challenge in several
research fields, from the efficient dissemination of ideas to drug target
discovery in biomedical problems. The difficulty lies at several levels: how to
single out the role of individual elements in such intermingled systems, or
which is the best way to quantify their importance. Centrality measures
describe a node's importance by its position in a network. The key issue
obviated is that the contribution of a node to the collective behavior is not
uniquely determined by the structure of the system but it is a result of the
interplay between dynamics and network structure. We show that dynamical
influence measures explicitly how strongly a node's dynamical state affects
collective behavior. For critical spreading, dynamical influence targets nodes
according to their spreading capabilities. For diffusive processes it
quantifies how efficiently real systems may be controlled by manipulating a
single node.Comment: accepted for publication in Scientific Report
Evolution of an intra-plate rift basin: the Latest Jurassic-Early Cretaceous Cameros Basin (Northwest Iberian Ranges, North Spain)
Depto. de Geodinámica, Estratigrafía y PaleontologíaFac. de Ciencias GeológicasTRUEpu
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