3,126 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
From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation
A rigorous proof is given for the convergence of the solutions of a viscous
Cahn-Hilliard system to the solution of the regularized version of the
forward-backward parabolic equation, as the coefficient of the diffusive term
goes to 0. Non-homogenous Neumann boundary condition are handled for the
chemical potential and the subdifferential of a possible non-smooth double-well
functional is considered in the equation. An error estimate for the difference
of solutions is also proved in a suitable norm and with a specified rate of
convergence.Comment: Key words and phrases: Cahn-Hilliard system, forward-backward
parabolic equation, viscosity, initial-boundary value problem, asymptotic
analysis, well-posednes
Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition
In this paper, we derive optimal upper and lower bounds on the dimension of
the attractor AW for scalar reaction-diffusion equations with a Wentzell
(dynamic) boundary condition. We are also interested in obtaining explicit
bounds about the constants involved in our asymptotic estimates, and to compare
these bounds to previously known estimates for the dimension of the global
attractor AK; K \in {D;N; P}, of reactiondiffusion equations subject to
Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we
obtain show that the dimension of the global attractor AW is of different order
than the dimension of AK; for each K \in {D;N; P} ; in all space dimensions
that are greater or equal than three.Comment: to appear in J. Nonlinear Scienc
Minimality properties of set-valued processes and their pullback attractors
We discuss the existence of pullback attractors for multivalued dynamical
systems on metric spaces. Such attractors are shown to exist without any
assumptions in terms of continuity of the solution maps, based only on
minimality properties with respect to the notion of pullback attraction. When
invariance is required, a very weak closed graph condition on the solving
operators is assumed. The presentation is complemented with examples and
counterexamples to test the sharpness of the hypotheses involved, including a
reaction-diffusion equation, a discontinuous ordinary differential equation and
an irregular form of the heat equation.Comment: 33 pages. A few typos correcte
Polynomial cubic differentials and convex polygons in the projective plane
We construct and study a natural homeomorphism between the moduli space of
polynomial cubic differentials of degree d on the complex plane and the space
of projective equivalence classes of oriented convex polygons with d+3
vertices. This map arises from the construction of a complete hyperbolic affine
sphere with prescribed Pick differential, and can be seen as an analogue of the
Labourie-Loftin parameterization of convex RP^2 structures on a compact surface
by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report.
v2: Corrections in section 5 and related new material in appendix
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