Emergence of mono-cluster flocking in the thermomechanical Cucker–Smale model under switching topologies

Abstract

We study emergent dynamics of the discrete Cucker-Smale (in short, DCS) model with randomly switching network topologies. For this, we provide a sufficient framework leading to the stochastic flocking with probability one. Our sufficient framework is formulated in terms of an admissible set of network topologies realized by digraphs and probability density function for random switching times. As examples for the law of switching times, we use the Poisson process and the geometric process and show that these two processes satisfy the required conditions in a given framework so that we have a stochastic flocking with probability one. As a corollary of our flocking analysis, we improve the earlier result [15] on the continuous C-S model.

Emergent behaviors of Kuramoto model with frustration under switching topology

“…When the initial configuration is confined in half circle, under the structural assumption that the network topology in any mode contains a spanning tree, the method in [42] can be directly applied in a sufficient regime before the first network switching occurs, so that we can find a finite time at which the oscillators concentrate into a small arc less than a quarter circle. Then we apply the method based on matrix-graph theories in [9] for the second-order Kuramoto system and present sufficient conditions leading to the exponentially fast emergence of frequency synchronization. In our framework, the size of frustration is sufficiently small and the coupling strength is sufficiently large.…”

“…However, our analytical approach requires that the switching interaction topology contains a spanning tree in any mode. It is an interesting issue whether we can relax this structural constraint to the union digraph with a spanning tree in a sequence of time-blocks in [9] for the half circle case. This question will be investigated in the future work.…”

“…First, under the Assumption 1.2 of the switching topology with spanning tree structure in any mode of P, before the first network switching, we apply the idea of digraph decomposition in [21] and similar procedure in [42], to find some finite time at which the oscillators stay in a small arc less than a quarter circle (see Lemma 3.3). Second, based on the estimate in Lemma 3.3, under a priori condition, we apply the matrix and graph theories and the method in [5,9] for the second-order Kuramoto model (1.2) to derive the frequency diameter a priori tends to zero exponentially fast (see Lemma 4.3). Lastly, we show that a priori condition can be ensured by the sufficient frameworks, which ultimately yields the complete synchronization.…”

“…In this section, we provide a proof of Theorem 1.1 on the frequency synchronization of the Kuramoto model with uniform frustration and switching topology. In the work [5,9] on the flocking estimates for the Cucker-Smale model with switching topology, the linear structure in velocity coupling is the key ingredient for using induction argument based on matrix-graph theories. Therefore, although the sinusoidal coupling in the Kuramoto model is nonlinear, based on the preparatory estimate (3.12) at t * in Lemma 3.3, we consider system (1.1) starting from t * , and lift it to the second-order formulation (1.2), which enjoys the linearity in frequency coupling similar to the Cucker-Smale model.…”