119 research outputs found
Floquet Stability Analysis of Ott-Grebogi-Yorke and Difference Control
Stabilization of instable periodic orbits of nonlinear dynamical systems has
been a widely explored field theoretically and in applications. The techniques
can be grouped in time-continuous control schemes based on Pyragas, and the two
Poincar\'e-based chaos control schemes, Ott-Gebogi-Yorke (OGY) and difference
control. Here a new stability analysis of these two Poincar\'e-based chaos
control schemes is given by means of Floquet theory. This approach allows to
calculate exactly the stability restrictions occuring for small measurement
delays and for an impulse length shorter than the length of the orbit. This is
of practical experimental relevance; to avoid a selection of the relative
impulse length by trial and error, it is advised to investigate whether the
used control scheme itself shows systematic limitations on the choice of the
impulse length. To investigate this point, a Floquet analysis is performed. For
OGY control the influence of the impulse length is marginal. As an unexpected
result, difference control fails when the impulse length is taken longer than a
maximal value that is approximately one half of the orbit length for small
Ljapunov numbers and decreases with the Ljapunov number.Comment: 13 pages. To appear in New Journal of Physic
Winner-Relaxing Self-Organizing Maps
A new family of self-organizing maps, the Winner-Relaxing Kohonen Algorithm,
is introduced as a generalization of a variant given by Kohonen in 1991. The
magnification behaviour is calculated analytically. For the original variant a
magnification exponent of 4/7 is derived; the generalized version allows to
steer the magnification in the wide range from exponent 1/2 to 1 in the
one-dimensional case, thus provides optimal mapping in the sense of information
theory. The Winner Relaxing Algorithm requires minimal extra computations per
learning step and is conveniently easy to implement.Comment: 14 pages (6 figs included). To appear in Neural Computatio
Discrete stochastic processes, replicator and Fokker-Planck equations of coevolutionary dynamics in finite and infinite populations
Finite-size fluctuations in coevolutionary dynamics arise in models of
biological as well as of social and economic systems. This brief tutorial
review surveys a systematic approach starting from a stochastic process
discrete both in time and state. The limit of an infinite
population can be considered explicitly, generally leading to a replicator-type
equation in zero order, and to a Fokker-Planck-type equation in first order in
. Consequences and relations to some previous approaches are
outlined.Comment: Banach Center publications, in pres
Cyclic dominance and biodiversity in well-mixed populations
Coevolutionary dynamics is investigated in chemical catalysis, biological
evolution, social and economic systems. The dynamics of these systems can be
analyzed within the unifying framework of evolutionary game theory. In this
Letter, we show that even in well-mixed finite populations, where the dynamics
is inherently stochastic, biodiversity is possible with three cyclic dominant
strategies. We show how the interplay of evolutionary dynamics, discreteness of
the population, and the nature of the interactions influences the coexistence
of strategies. We calculate a critical population size above which coexistence
is likely.Comment: Physical Review Letters, in print (2008
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