3,446 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
Bi-continuous semigroups for flows in infinite networks
We study transport processes on infinite metric graphs with non-constant
velocities and matrix boundary conditions in the -setting.
We apply the theory of bi-continuous operator semigroups to obtain
well-posedness of the problem under different assumptions on the velocities and
for general stochastic matrices appearing in the boundary conditions.Comment: 12 page
The Stokes boundary layer for a thixotropic or antithixotropic fluid
We present a mathematical investigation of the oscillatory boundary layer (‘Stokes layer’) in a semi-infinite fluid bounded by an oscillating wall (the socalled ‘Stokes problem’), when the fluid has a thixotropic or antithixotropic rheology. We obtain asymptotic solutions in the limit of small-amplitude oscillations, and we use numerical integration to validate the asymptotic solutions and to explore the behaviour of the system for larger-amplitude oscillations. The solutions that we obtain differ significantly from the classical solution for a Newtonian fluid. In particular, for antithixotropic fluids the velocity reaches zero at a finite distance from the wall, in contrast to the exponential decay for a thixotropic or a Newtonian fluid. For small amplitudes of oscillation, three regimes of behaviour are possible: the structure parameter may take values defined instantaneously by the shear rate, or by a long-term average; or it may behave hysteretically. The regime boundaries depend on the precise specification of structure build-up and breakdown rates in the rheological model, illustrating the subtleties of complex fluid models in non-rheometric settings. For larger amplitudes of oscillation the dominant behaviour is hysteretic. We discuss in particular the relationship between the shear stress and the shear rate at the oscillating wall
Nonlinear dynamics of the viscoelastic Kolmogorov flow
The weakly nonlinear regime of a viscoelastic Navier--Stokes fluid is
investigated. For the purely hydrodynamic case, it is known that large-scale
perturbations tend to the minima of a Ginzburg-Landau free-energy functional
with a double-well (fourth-order) potential. The dynamics of the relaxation
process is ruled by a one-dimensional Cahn--Hilliard equation that dictates the
hyperbolic tangent profiles of kink-antikink structures and their mutual
interactions. For the viscoelastic case, we found that the dynamics still
admits a formulation in terms of a Ginzburg--Landau free-energy functional. For
sufficiently small elasticities, the phenomenology is very similar to the
purely hydrodynamic case: the free-energy functional is still a fourth-order
potential and slightly perturbed kink-antikink structures hold. For
sufficiently large elasticities, a critical point sets in: the fourth-order
term changes sign and the next-order nonlinearity must be taken into account.
Despite the double-well structure of the potential, the one-dimensional nature
of the problem makes the dynamics sensitive to the details of the potential. We
analysed the interactions among these generalized kink-antikink structures,
demonstrating their role in a new, elastic instability. Finally, consequences
for the problem of polymer drag reduction are presented.Comment: 26 pages, 17 figures, submitted to The Journal of Fluid Mechanic
Temporal stability of network partitions
We present a method to find the best temporal partition at any time-scale and
rank the relevance of partitions found at different time-scales. This method is
based on random walkers coevolving with the network and as such constitutes a
generalization of partition stability to the case of temporal networks. We show
that, when applied to a toy model and real datasets, temporal stability
uncovers structures that are persistent over meaningful time-scales as well as
important isolated events, making it an effective tool to study both abrupt
changes and gradual evolution of a network mesoscopic structures.Comment: 15 pages, 12 figure
A singular limit for an age structured mutation problem
The spread of a particular trait in a cell population often is modelled by an
appropriate system of ordinary differential equations describing how the sizes
of subpopulations of the cells with the same genome change in time. On the
other hand, it is recognized that cells have their own vital dynamics and
mutations, leading to changes in their genome, mostly occurring during the cell
division at the end of its life cycle. In this context, the process is
described by a system of McKendrick type equations which resembles a network
transport problem. In this paper we show that, under an appropriate scaling of
the latter, these two descriptions are asymptotically equivalent
Distributed MPC with time-varying communication network: A density-dependent population games approach
© 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.This work addresses distributed control design by using density-dependent population dynamics. Furthermore, stability of the equilibrium point under this proposed class of population dynamics is studied, and the relationship between the equilibrium point of density-dependent population games (DDPG) and the solution of constrained optimization problems is shown. Finally, a distributed predictive control is designed with the proposed density-dependent dynamics, and contemplating a time-varying communication network.Peer ReviewedPostprint (author's final draft
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