Dynamical equivalence between a network of heterogeneous oscillators and a translocating polymer.

What is it about?

Networks of heterogeneous oscillators represent a relevant class of dynamical systems that have a remarkable property: while a similar network composed of identical oscillator may not oscillate - and analogously single oscillators in isolation would not oscillate, coupling a set of oscillators diversified in some of their properties can start collective oscillations. In other words, the simple diversification of the oscillators can induce collective synchronized oscillations. This phenomenon resembles Coherence Resonance, where oscillations can be induced by adding disorder in the form of noise, while here the same effect is obtained by adding quenched disorder - fixed heterogeneity in a parameter of the oscillators - without the need to add any noise. The amplitude and coherence of the oscillations depends strongly on the level of heterogeneity: oscillations will not appear if the diversity level is too small or too large, but will be strongest at an optimal level of diversity. That is, the response of the system presents a resonant behavior as the heterogeneity level is varied. Despite many works have addressed this counterintuitive effect, its understanding is not yet complete. To fill this gap, we provide here a simple mechanical analog of a heterogeneous oscillator network: a polymer diffusing on a substrate potential. We diversify the external forcing acting on (perceived by) the single oscillators and show that the equations describing the system can be recast in a form equivalent to that of a polymer moving in an external potential. For example, a network of N heterogeneous quartic oscillators (particles in a double-well potential) subject to N different external forces is equivalent to to a polymer composed of N monomers. The monomers spontaneously order themselves in order of strength of the external force, trying to maintain their mutual distances constant as they move in the same double-well potential. See the enclosed links to the videos for a visual illustration.

Featured Image

Photo by Conny Schneider on Unsplash

Photo by Conny Schneider on Unsplash

Why is it important?

Networks of heterogeneous oscillators are ubiquitous in biology and technology, but the understanding of their collective response is far from being complete. The simple mechanical model introduced here lends itself to be an alternative viewpoint, from which to interpret and predict collective oscillations in different types of heterogeneous networks, in terms of general collective features that do not depend on the finer details of the system under study.

Perspectives