Network of mobile systems: mutual influence of oscillators and agents

Abstract

This work presents a network of mobile systems whose nodes are constituted by a moving agent with an internal state (an oscillator), which influences each other. The coupling topology of the agents and internal oscillators changes over time according to the interaction range (also called vision range or vision sizes (Majhi et al. Phys Rev E 99: 012308, 2019)) of their corresponding counterparts. The goal is to investigate the dynamics of the oscillators and the agents in the considered systems. Our results show that the synchronization between agents and that between oscillators depends on the coupling parameter of the oscillators, the velocity of the agents and the interaction range of both agents and oscillators. We have found that the vision range of the oscillators has a great influence on the dynamics of the agents. Among this dynamics, we can mention phase synchronization and clusters formation in the mobile system and complete synchronization as well as clusters formation on the oscillators. The stability of the synchronization in the oscillators is investigated using the Master Stability Function (MSF) developed by Pecora and Carroll (Phys Rev Lett 80: 2109, 1998).

Appendices

Appendix A: Evaluation of Order parameter

The Order parameter, defined by Kuramoto and Battogtokh [27], is a good tool used to analyse phase synchronization in coupled systems as well as in networks. The computation of the Order parameter needs the phase of each system. To calculate this phase, we consider an arbitrary time signal $s\left( \tau \right)$ and ${\tilde{s}}\left( \tau \right)$ being its Hilbert transform. We have:

$$\begin{aligned} \psi \left( \tau \right) = s\left( \tau \right) + j{\tilde{s}}\left( \tau \right) = R\left( \tau \right) {e^{j\varphi \left( \tau \right) }} \end{aligned}$$

(A.1)

where $R\left( \tau \right)$ is the amplitude and $\varphi \left( \tau \right)$ the phase of the variable $s\left( \tau \right)$. If we denote by ${\varphi _i} \left( \tau \right)$ the instantaneous phase, then it can be determined by:

$$\begin{aligned} {\varphi _i}\left( \tau \right) = {\tan ^{ - 1}}\left( {\frac{{{{\tilde{s}}_i}\left( \tau \right) }}{{{s_i}\left( \tau \right) }}} \right) \end{aligned}$$

(A.2)

The Order parameter for a system with N oscillators is expressed as:

$$\begin{aligned} r = \frac{1}{N}\sum \limits _{i = 1}^N {{e^{j{\varphi _i}}}} \end{aligned}$$

(A.3)

Where ${j^2} = - 1$. Phase synchronization is effective when the value of $r=1$ and when $r=0$, the network is completely desynchronized.

Appendix B: Evaluation of Master Stability Function

Developed by Pecora and Carroll in 1998, the Master Stability Function (MSF) is the best tool used in a network of coupled identical systems to demonstrate the stability of the complete synchronization of the network [24, 26]. To facilitate the comprehension of some results presented in this paper, we develop here some important points of this method. Let us consider an isolated system (oscillator) defined by Eq. B.1:

$$\begin{aligned} {\dot{x}_i} = F\left( {x_i} \right) \end{aligned}$$

(B.1)

Where ${x_i}$ is a vector of m-components used to describe the state of the ith oscillator; F is a function defined from ${R^{m} \longrightarrow R^{m} }$ used to define the local synchrony of the oscillators. Taking into account the interaction or the connection between the N oscillators of the network, Eq. B.1 is not sufficient to describe the dynamics of the network. Taking into account the interactions the governing law of the $i^\mathrm{th}$ oscillator is given by:

$$\begin{aligned} {\dot{x}_i} = F\left( {{x_i}} \right) + k\sum \limits _{j = 1}^N {{G_{ij}}H\left( {{x_j}} \right) } \end{aligned}$$

(B.2)

In this equation (Eq. B.2), k represents the coupling strength; $H: {R^{m} \longrightarrow R^{m} }$ is an arbitrary coupling function and G is a Laplacian matrix. The oscillators of the network are synchronized if all oscillators converge toward the same state s such as ${x_1} = {x_2} =... = {x_N} = s$. It should be noted that, for the N nodes we can design N state variables, N coupling functions and N local functions F contained into the matrix described respectively by Eqs. B.3, B.4 and B.5.

$$\begin{aligned} {x={[{x_1};\,\, {x_2};\,\,... \,; \,\,{x_N}]}} \end{aligned}$$

(B.3)

$$\begin{aligned} {H\left( {x} \right) ={[H\left( {{x_1}} \right) ;\,\, H\left( {{x_2}} \right) ;\,\,... ;\,\, H\left( {{x_N}} \right) ]}} \end{aligned}$$

(B.4)

$$\begin{aligned} {F\left( {x} \right) ={[F\left( {{x_1}} \right) ;\,\, F\left( {{x_2}} \right) ;\,\,...;\,\, F\left( {{x_N}} \right) ]}} \end{aligned}$$

(B.5)

Based on Eqs. B.3, B.4 and B.5, B.2 can be expressed in compact form as:

$$\begin{aligned} \dot{x} = F\left( x \right) + kG \otimes H\left( x \right) \end{aligned}$$

(B.6)

where $\otimes$ is the Kronecker product. For the case of our network, we have:

$$\begin{aligned} F\left( {{x_i}} \right) = \left\{ \begin{array}{l} - {y_i} - {z_i} \\ {x_i} + a{y_i} \\ b + \left( {{x_i} - c} \right) {z_i} \\ \end{array} \right. \,\,\mathrm{{and}}\,\,\,\,H = \left( {\begin{array}{*{20}{c}} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array}} \right) \qquad \end{aligned}$$

(B.7)

Initially, we suppose that $\delta {x_i}$ is a small perturbation of the $i^\mathrm{th}$ oscillators of the network. After perturbation, the state variable of the $i^\mathrm{th}$ oscillators becomes ${x_i} = s + \delta {x_i}$. For the whole network, the collection of the variation of the N oscillators is expressed by $\delta x={[\delta {x_i}\,;\,\, \delta {x_i}\,;\,\,... \,\,; \,\,\delta {x_i}]}$. Replacing the perturbation in Eq. B.2 and using the Taylor expansion theorem for $F\left( {s + \delta {x_i}} \right)$ and $H\left( {s + \delta {x_i}} \right)$ to first order, the following variational equation is obtained:

$$\begin{aligned} \dot{\delta {x_i}}= DF\left( s \right) .\delta {x_i} + k \sum \limits _{j = 1}^N {{G_{ij}}DH\left( s \right) .\delta {x_j}} \end{aligned}$$

(B.8)

where $DF\left( s \right)$ and $DH\left( s \right)$ are the $N \times N$ Jacobian matrices of the corresponding vector functions evaluated at the synchronous state $s\left( t \right)$. Equation B.8 is used to explore if the synchronous state is stable or unstable. According to Peccora and Carroll the use of tensor notation leads to Eq. B.9:

$$\begin{aligned} \dot{\delta {x}}= [{1_N} \otimes DF\left( s \right) + k{G \otimes DH\left( s \right) }].\delta {x} \end{aligned}$$

(B.9)

Due to the degree of Eq. B.9, the solution can be in the form $\delta {x_i} \sim exp({\lambda _i}t)$ where exponent $\lambda$ helps to us to know if the perturbation grows $(\lambda > 0)$ or decays $(\lambda < 0)$. So, the digitalization of the second term of Eq. B.9 helps us to obtain the following variational equation expressed as:

$$\begin{aligned} \dot{\delta {x_k}} = \left[ {DF\left( s \right) + k {\alpha _k} DH\left( s \right) } \right] .\delta {x_k} \end{aligned}$$

(B.10)

${\alpha _k}$ is the eigenvalue of the matrix G, $k=1,\,\, 2,\,\,...,\,\, N$. Finally, these steps help us to design the following Master Stability Eq. B.11:

$$\begin{aligned} \dot{\delta {x}} = \left[ {DF\left( s \right) + {\alpha _k} DH\left( s \right) } \right] .\delta {x} \end{aligned}$$

(B.11)

ased on refs [24, 28] the synchronization is stable if the largest Lyapunov Exponent computed from Eq. B.11 which corresponds to the largest eigenvalue is negative and unstable otherwise.