Simulations for Cucker-Smale-Model and Its Extensions

Contents

• Introduction

• Simple Instruction for Models with Numerical Results

• Bibliography

• License

Introduction

In this repository, I did simulations about the Cucker-Smale model [1], second-order-nonlinear ODEs describing the flocking behaviors, and its extensions. Mostly, the works have been done for writing my master's degree thesis; [2].

In the directory named 'model', there are two directories, 'ODE' and 'SDE'. In the directory 'ODE', there are simulation codes for solving ODE systems, Cucker-Smale model, Cucker-Smale model with decentralized formation control term in [3], and an extension of the model in [3] on more general graphs. In the directory 'SDE', there are simulation codes for solving the SDE system introduced in [2], which has a formation controller and multiplicative noises.

The main goals of codes in directories 'ODE' and 'SDE' are simulating ODE and SDE systems mentioned in a preceding paragraph and showing that solutions converge when initial conditions satisfy specific conditions. The main features implemented in the codes are visualizing movements of agents as video and showing numerical results support the theoretical results in [1], [3] and [2] with plots.

Also, I made a code in the directory 'etc' for visualizing networks and it is used for writing [2]. In the directory 'figure', there are video files and plots made by the codes. Aimed patterns through simulations were shaped pattern and Einstein's face image pattern. I get the curve equations for the shapes from https://wolframalpha.com.

On my git-hub page, instructions are focused on numerical and coding parts, so if you want to get mathematical information, I recommend you reading my thesis paper; [2]. More specific instructions for how I made plots will be written in the directory 'figure' and instruction of numerical methods for solving ODE and SDE will be written in the directory 'model'.

Simple Instruction for Models with Numerical Results

Fig. 1 Simulation for Cucker-Smale model

Fig. 2 The average of 100 realizations for the energies, which shows movements of the SDE model converge.

Fig. 3 A simulation for an Einstien's face image pattern model.

Bibliography

1. CS07 : Felipe Cucker and Steve Smale. Emergent behavior in flocks. IEEE Transactions on Automatic Control, 52(5):852–862, 2007.

2. O21 : Tackgeun Oh, Flocking Behavior in Stochastic Cucker-Smale Model with Formation Control on Symmetric Digraphs, Yonsei Univ., 2021. / ( I changed my name from 'Tackgeun Oh' to 'Doeun Oh'. )

3. CKPP19 : Young-Pil Choi, Dante Kalise, Jan Peszek, and Andrés A. Peters. A collisionless singular Cucker-Smale model with decentralized formation control. SIAM J. Appl. Dyn. Syst., 18(4):1954–1981, 2019.

License

MIT