42,584 research outputs found
Boolean Delay Equations: A simple way of looking at complex systems
Boolean Delay Equations (BDEs) are semi-discrete dynamical models with
Boolean-valued variables that evolve in continuous time. Systems of BDEs can be
classified into conservative or dissipative, in a manner that parallels the
classification of ordinary or partial differential equations. Solutions to
certain conservative BDEs exhibit growth of complexity in time. They represent
therewith metaphors for biological evolution or human history. Dissipative BDEs
are structurally stable and exhibit multiple equilibria and limit cycles, as
well as more complex, fractal solution sets, such as Devil's staircases and
``fractal sunbursts``. All known solutions of dissipative BDEs have stationary
variance. BDE systems of this type, both free and forced, have been used as
highly idealized models of climate change on interannual, interdecadal and
paleoclimatic time scales. BDEs are also being used as flexible, highly
efficient models of colliding cascades in earthquake modeling and prediction,
as well as in genetics. In this paper we review the theory of systems of BDEs
and illustrate their applications to climatic and solid earth problems. The
former have used small systems of BDEs, while the latter have used large
networks of BDEs. We moreover introduce BDEs with an infinite number of
variables distributed in space (``partial BDEs``) and discuss connections with
other types of dynamical systems, including cellular automata and Boolean
networks. This research-and-review paper concludes with a set of open
questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular
the discussion on partial BDEs is updated and enlarge
Dissipative Stabilization of Linear Systems with Time-Varying General Distributed Delays (Complete Version)
New methods are developed for the stabilization of a linear system with
general time-varying distributed delays existing at the system's states, inputs
and outputs. In contrast to most existing literature where the function of
time-varying delay is continuous and bounded, we assume it to be bounded and
measurable. Furthermore, the distributed delay kernels can be any
square-integrable function over a bounded interval, where the kernels are
handled directly by using a decomposition scenario without using
approximations. By constructing a Krasovski\u{i} functional via the application
of a novel integral inequality, sufficient conditions for the existence of a
dissipative state feedback controller are derived in terms of matrix
inequalities without utilizing the existing reciprocally convex combination
lemmas. The proposed synthesis (stability) conditions, which take dissipativity
into account, can be either solved directly by a standard numerical solver of
semidefinite programming if they are convex, or reshaped into linear matrix
inequalities, or solved via a proposed iterative algorithm. To the best of our
knowledge, no existing methods can handle the synthesis problem investigated in
this paper. Finally, numerical examples are presented to demonstrate the
effectiveness of the proposed methodologies.Comment: Accepted by Automatic
Distributed finite-time stabilization of entangled quantum states on tree-like hypergraphs
Preparation of pure states on networks of quantum systems by controlled
dissipative dynamics offers important advantages with respect to circuit-based
schemes. Unlike in continuous-time scenarios, when discrete-time dynamics are
considered, dead-beat stabilization becomes possible in principle. Here, we
focus on pure states that can be stabilized by distributed, unsupervised
dynamics in finite time on a network of quantum systems subject to realistic
quasi-locality constraints. In particular, we define a class of quasi-locality
notions, that we name "tree-like hypergraphs," and show that the states that
are robustly stabilizable in finite time are then unique ground states of a
frustration-free, commuting quasi-local Hamiltonian. A structural
characterization of such states is also provided, building on a simple yet
relevant example.Comment: 6 pages, 3 figure
A step towards holistic discretisation of stochastic partial differential equations
The long term aim is to use modern dynamical systems theory to derive
discretisations of noisy, dissipative partial differential equations. As a
first step we here consider a small domain and apply stochastic centre manifold
techniques to derive a model. The approach automatically parametrises subgrid
scale processes induced by spatially distributed stochastic noise. It is
important to discretise stochastic partial differential equations carefully, as
we do here, because of the sometimes subtle effects of noise processes. In
particular we see how stochastic resonance effectively extracts new noise
processes for the model which in this example helps stabilise the zero
solution.Comment: presented at the 5th ICIAM conferenc
An application of maximal dissipative sets in control theory
AbstractA model of a distributed-boundary control system is considered. Assume the uncontrolled system possesses an exponential asymptotically stable zero solution. We then construct suboptimal feedback controls for the distributed and boundary control problems via the direct method of Liapunov. Furthermore, existence-uniqueness of the synthesized control systems is proven by applying the theory of nonlinear semigroups and maximal dissipative sets. Applications to diffusion equations are given
Temporal dissipative solitons in time-delay feedback systems
Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, auto-solitons, spot or pulse solitons, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studied in spatially extended systems, temporally localized states are gaining attention only recently, driven primarily by applications from fiber or semiconductor lasers. Here we present a theory for temporal dissipative solitons (TDS) in systems with time-delayed feedback. In particular, we derive a system with an advanced argument, which determines the profile of the TDS. We also provide a complete classification of the spectrum of TDS into interface and pseudo-continuous spectrum. We illustrate our theory with two examples: a generic delayed phase oscillator, which is a reduced model for an injected laser with feedback, and the FitzHugh--Nagumo neuron with delayed feedback. Finally, we discuss possible destabilization mechanisms of TDS and show an example where the TDS delocalizes and its pseudo-continuous spectrum develops a modulational instability
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