807 research outputs found
Slim Fractals: The Geometry of Doubly Transient Chaos
Traditional studies of chaos in conservative and driven dissipative systems
have established a correspondence between sensitive dependence on initial
conditions and fractal basin boundaries, but much less is known about the
relation between geometry and dynamics in undriven dissipative systems. These
systems can exhibit a prevalent form of complex dynamics, dubbed doubly
transient chaos because not only typical trajectories but also the (otherwise
invariant) chaotic saddles are transient. This property, along with a manifest
lack of scale invariance, has hindered the study of the geometric properties of
basin boundaries in these systems--most remarkably, the very question of
whether they are fractal across all scales has yet to be answered. Here we
derive a general dynamical condition that answers this question, which we use
to demonstrate that the basin boundaries can indeed form a true fractal; in
fact, they do so generically in a broad class of transiently chaotic undriven
dissipative systems. Using physical examples, we demonstrate that the
boundaries typically form a slim fractal, which we define as a set whose
dimension at a given resolution decreases when the resolution is increased. To
properly characterize such sets, we introduce the notion of equivalent
dimension for quantifying their relation with sensitive dependence on initial
conditions at all scales. We show that slim fractal boundaries can exhibit
complex geometry even when they do not form a true fractal and fractal scaling
is observed only above a certain length scale at each boundary point. Thus, our
results reveal slim fractals as a geometrical hallmark of transient chaos in
undriven dissipative systems.Comment: 13 pages, 9 figures, proof corrections implemente
Why Optimal States Recruit Fewer Reactions in Metabolic Networks
The metabolic network of a living cell involves several hundreds or thousands
of interconnected biochemical reactions. Previous research has shown that under
realistic conditions only a fraction of these reactions is concurrently active
in any given cell. This is partially determined by nutrient availability, but
is also strongly dependent on the metabolic function and network structure.
Here, we establish rigorous bounds showing that the fraction of active
reactions is smaller (rather than larger) in metabolic networks evolved or
engineered to optimize a specific metabolic task, and we show that this is
largely determined by the presence of thermodynamically irreversible reactions
in the network. We also show that the inactivation of a certain number of
reactions determined by irreversibility can generate a cascade of secondary
reaction inactivations that propagates through the network. The mathematical
results are complemented with numerical simulations of the metabolic networks
of the bacterium Escherichia coli and of human cells, which show,
counterintuitively, that even the maximization of the total reaction flux in
the network leads to a reduced number of active reactions.Comment: Contribution to the special issue in honor of John Guckenheimer on
the occasion of his 65th birthda
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