34,740 research outputs found
Delayed Feedback Control near Hopf Bifurcation
The stability of functional differential equations under delayed feedback is
investigated near a Hopf bifurcation. Necessary and sufficient conditions are
derived for the stability of the equilibrium solution using averaging theory.
The results are used to compare delayed versus undelayed feedback, as well as
discrete versus distributed delays. Conditions are obtained for which delayed
feedback with partial state information can yield stability where undelayed
feedback is ineffective. Furthermore, it is shown that if the feedback is
stabilizing (respectively, destabilizing), then a discrete delay is locally the
most stabilizing (resp., destabilizing) one among delay distributions having
the same mean. The result also holds globally if one considers delays that are
symmetrically distributed about their mean
Delay-Controlled Reactions
When the entities undergoing a chemical reaction are not available
simultaneously, the classical rate equation of a reaction or, alternatively for
the evolution of a population, should be extended by including non-Markovian
memory effects. We consider the two cases of an external feedback, realized by
fixed functions and an internal feedback originated in a self-organized manner
by the relevant concentration itself. Whereas in the first case the fixed
points are not changed, although the dynamical process is altered, the second
case offers a complete new behaviour, characterized by the existence of a time
persistent solution. Due to the feedback the reaction may lead to a finite
concentration in the stationary limit even in case of a single-species pair
annihilation process. We argue that the different cases are
similar to a coupling of additive or multiplicative noises in stochastic
processes.Comment: 18 pages, 3 figure
On the stability of periodic orbits in delay equations with large delay
We prove a necessary and sufficient criterion for the exponential stability
of periodic solutions of delay differential equations with large delay. We show
that for sufficiently large delay the Floquet spectrum near criticality is
characterized by a set of curves, which we call asymptotic continuous spectrum,
that is independent on the delay.Comment: postprint versio
Distributed delays stabilize negative feedback loops
Linear scalar differential equations with distributed delays appear in the
study of the local stability of nonlinear differential equations with feedback,
which are common in biology and physics. Negative feedback loops tend to
promote oscillation around steady states, and their stability depends on the
particular shape of the delay distribution. Since in applications the mean
delay is often the only reliable information available about the distribution,
it is desirable to find conditions for stability that are independent from the
shape of the distribution. We show here that the linear equation with
distributed delays is asymptotically stable if the associated differential
equation with a discrete delay of the same mean is asymptotically stable.
Therefore, distributed delays stabilize negative feedback loops
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