Hierarchical cluster structures in a one-dimensional swarm oscillator model

Abstract

Swarm oscillator model derived by one of the authors (Tanaka), where interacting motile elements form various kinds of patterns, is investigated. We particularly focus on the cluster patterns in one-dimensional space. We mathematically derive all static and stable configurations in final states for a particular but a large set of parameters. In the derivation, we introduce renormalized expression of this model. We find that the static final states are hierarchical cluster structures in which a cluster consists of smaller clusters in a nesting manner.

Oscillators that sync and swarm

“…Tanaka and colleagues also made an early contribu-tion to the modeling of swarmalators [48,49]. They analyzed a broad class of models in the hope of finding phenomena which were not system-specific.…”

“…The oscillators' consumption of this chemical depends on their internal states, thereby completing the bidirectional space-phase coupling. Tanaka et al [48,49] began with a general model with these ingredients, from which they derived a simpler model by means of center manifold and phase-reduction methods.…”

Cohesive motion in one-dimensional flocking

“…Regarding the dimensionality of the models developed, most of them have been defined in dimensions higher than one [10,11,12,13,14,15,20,21,22,23,24,25,27,28,29,30], since the velocity of the self-propelled particles (SPP) in these models can have continuous values. In contrast, only a few one-dimensional (1D) models have been studied [5,16,17,18,19,26,31]. For these, the direction of the particles velocity can only take discrete values, in fact, only two.…”

“…The interest in the study of this kind of systems comes from the fact that they represent prototypical out-of-equilibrium systems. The formulations to study the flocking phenomenon can be succinctly classified in rule-based [10,11,12,13,14,15,16,17,18], Lagrangian (trajectory-based) [19,20,21,22,23,24,25,26] and Eulerian (continuum) models [27,28,29]. Regarding the dimensionality of the models developed, most of them have been defined in dimensions higher than one [10,11,12,13,14,15,20,21,22,23,24,25,27,28,29,30], since the velocity of the self-propelled particles (SPP) in these models can have continuous values.…”

Juggling motion in a system of motile coupled oscillators

“…The indirect interaction is effectively transformed into the direct interaction between elements in the derivation. With numerical simulations of the swarm oscillator model, it has been confirmed that this model exhibits a rich variety of collective behavior such as a group of moving clusters, rotating circles with phase waves, aggregation into a single point, and construction of a lattice structure, depending on the four real parameters included in the model, the number of elements, and the system size [12,13]. This variety indicates that the investigation of this model in detail would play a significant role not only in understanding chemotaxis but also in revealing the basic mathematical structure that underlies various kinds of collective behavior.…”