3,318 research outputs found
Birth of a Learning Law
Defense Advanced Research Projects Agency; Office of Naval Research (N00014-95-1-0409, N00014-95-1-0657, N00014-92-J-1309
Fractional dynamics of systems with long-range interaction
We consider one-dimensional chain of coupled linear and nonlinear oscillators
with long-range power wise interaction defined by a term proportional to
1/|n-m|^{\alpha+1}. Continuous medium equation for this system can be obtained
in the so-called infrared limit when the wave number tends to zero. We
construct a transform operator that maps the system of large number of ordinary
differential equations of motion of the particles into a partial differential
equation with the Riesz fractional derivative of order \alpha, when 0<\alpha<2.
Few models of coupled oscillators are considered and their synchronized states
and localized structures are discussed in details. Particularly, we discuss
some solutions of time-dependent fractional Ginzburg-Landau (or nonlinear
Schrodinger) equation.Comment: arXiv admin note: substantial overlap with arXiv:nlin/051201
Selection theorem for systems with inheritance
The problem of finite-dimensional asymptotics of infinite-dimensional dynamic
systems is studied. A non-linear kinetic system with conservation of supports
for distributions has generically finite-dimensional asymptotics. Such systems
are apparent in many areas of biology, physics (the theory of parametric wave
interaction), chemistry and economics. This conservation of support has a
biological interpretation: inheritance. The finite-dimensional asymptotics
demonstrates effects of "natural" selection. Estimations of the asymptotic
dimension are presented. After some initial time, solution of a kinetic
equation with conservation of support becomes a finite set of narrow peaks that
become increasingly narrow over time and move increasingly slowly. It is
possible that these peaks do not tend to fixed positions, and the path covered
tends to infinity as t goes to infinity. The drift equations for peak motion
are obtained. Various types of distribution stability are studied: internal
stability (stability with respect to perturbations that do not extend the
support), external stability or uninvadability (stability with respect to
strongly small perturbations that extend the support), and stable realizability
(stability with respect to small shifts and extensions of the density peaks).
Models of self-synchronization of cell division are studied, as an example of
selection in systems with additional symmetry. Appropriate construction of the
notion of typicalness in infinite-dimensional space is discussed, and the
notion of "completely thin" sets is introduced.
Key words: Dynamics; Attractor; Evolution; Entropy; Natural selectionComment: 46 pages, the final journal versio
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