%0 Journal Article
%T Time-Delayed Interactions in Networks of Self-Adapting Hopf Oscillators
%A Julio Rodriguez
%A Max-Olivier Hongler
%A Philippe Blanchard
%J ISRN Mathematical Analysis
%D 2013
%R 10.1155/2013/816353
%X A network of coupled limit cycle oscillators with delayed interactions is considered. The parameters characterizing the oscillator¡¯s frequency and limit cycle are allowed to self-adapt. Adaptation is due to time-delayed state variables that mutually interact via a network. The self-adaptive mechanisms ultimately drive all coupled oscillators to a consensual cyclostationary state, where the values of the parameters are identical for all local systems. They are analytically expressible. The interplay between the spectral properties of the coupling matrix and the time delays determines the conditions for which convergence towards a consensual state takes place. Once reached, this consensual state subsists even if interactions are removed. In our class of models, the consensual values of the parameters depend neither on the delays nor on the network¡¯s topologies. The farther back you can look, the farther forward you are likely to see Winston Churchill 1. Introduction The harmonic excitation of an elementary-damped harmonic oscillator with , , and produces the well-known asymptotic response (c.f. [1]): where and depend on the control parameters , , and . By construction, the oscillating environment here materialized by the input is totally insensitive to the oscillator , implying that in (1) is slaved by the external forcing. The next stage of complexity is to replace the harmonic oscillator in (1) by a Lienard system: where is a nonlinear controller. In absence of external excitation in (3) (i.e., when ), we assume to asymptotically drive the orbits towards a stable limit cycle which is independent of the initial conditions¡ªthe paradigmatic illustration being here the Van der Pol oscillator. When and for a suitably selected range of parameters, the time asymptotic response of (3) can be qualitatively written as (c.f. [2, 3]) with being a synchronized signal with the same periodicity as the environment (i.e., ). By construction, the external forcing in (3) is, as before, insensitive to the Lienard oscillator. In the resulting synchronized regime, the oscillator is caught by the external excitation¡ªin other words, the system adjusts itself to the environment but the environment remains insensible to the system. Observe that the dynamical response given by (4) only subsists as long as acts on the system. That is, as soon as the environment effect is removed (i.e., in (3)), the system (i.e., the limit cycle oscillator), after a transient time, recovers its original behavior¡ªconverges towards its limit cycle. In our present paper, we will extend the previous
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2013/816353/