179 research outputs found
Alternative determinism principle for topological analysis of chaos
The topological analysis of chaos based on a knot-theoretic characterization
of unstable periodic orbits has proved a powerful method, however knot theory
can only be applied to three-dimensional systems. Still, the core principles
upon which this approach is built, determinism and continuity, apply in any
dimension. We propose an alternative framework in which these principles are
enforced on triangulated surfaces rather than curves and show that in dimension
three our approach numerically predicts the correct topological entropies for
periodic orbits of the horseshoe map.Comment: Accepted for publication as Rapid Communication in Physical Review
Stochastic Oscillations Induced by Intrinsic Fluctuations in a Self-Repressing Gene
AbstractBiochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that when the mRNA and protein half-lives are approximately equal to the gene response time, fluctuations can induce relatively regular oscillations in the protein concentration. To gain insight into this phenomenon at the crossroads of determinism and stochasticity, we use an intermediate theoretical approach, based on a moment-closure approximation of the master equation, which allows us to take into account the binary character of gene activity. We thereby obtain differential equations that describe how nonlinearity can feed-back fluctuations into the mean-field equations to trigger oscillations. Finally, our results suggest that the self-repressing Hes1 gene circuit exploits this phenomenon to generate robust oscillations, inasmuch as its time constants satisfy precisely the conditions we have identified
Topological signature of deterministic chaos in short nonstationary signals from an optical parametric oscillator
Although deterministic chaos has been predicted to occur in the triply
resonant optical parametric oscillator (TROPO) fifteen years ago, experimental
evidence of chaotic behavior in this system has been lacking so far, in marked
contrast with most nonlinear systems, where chaos has been actively tracked and
found. This situation is probably linked to the high sensitivity of the TROPO
to perturbations, which adversely affects stationary operation at high power.
We report the experimental observation in this system of a burst of irregular
behavior of duration 80 microseconds. Although the system is highly
nonstationary over this time interval, a topological analysis allows us to
extract a clearcut signature of deterministic chaos from a time series segment
of only 9 base cycles (3 microseconds). This result suggests that
nonstationarity is not necessarily an obstacle to the characterization of
chaos
On proving the absence of oscillations in models of genetic circuits
International audienceUsing computer algebra methods to prove that gene regulatory networks cannot oscillate appears to be easier than expected. We illustrate this claim on a family of models related to historical examples
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