In this chapter, we present the Cucker–Smale-type flocking models and discuss their mathematical structures and flocking theorems in terms of coupling strength, interaction topologies, and initial data. In 2007, two mathematicians Felipe Cucker and Steve Smale introduced a second-order particle model which resembles Newton’s equations in N-body system and present how their simple model can exhibit emergent flocking behavior under sufficient conditions expressed only in terms of parameters and initial data. After Cucker–Smale’s seminal works in [31, 32], their model has received lots of attention from applied math and control engineering communities. We discuss the state of the art for the flocking theorems to Cucker–Smale-type flocking models.

Emergent dynamics of the Cucker-Smale flocking model and its variants

In the literature, the study of the Cucker-Smale (C-S) flocking model is all restricted to connected undirected graphs or digraphs of some special types, containing only a single root. In this technical note, we establish the flocking behavior of the C-S model on general digraphs, which contain all existing interaction topologies in relation to the C-S model as special cases. In particular, the topology with multiple leaders is included. The flocking results are guaranteed under some conditions on the initial state of the flock only. The critical exponent below which unconditional convergence holds is given, depending only on the interaction topology. It is consistent with the ones obtained under some known special topologies in the literature.

Flocking of the Cucker-Smale Model on General Digraphs

We present the mathematical analysis of bi-cluster flocking phenomenon for the short-ranged Cucker–Smale model with some well-prepared initial data. For this, we derive a system of differential inequalities for the functionals measuring the local spatial and velocity fluctuations and differences of local velocity averages, and then estimate the upper bound of spatial fluctuations and the lower bound of the difference between local velocity averages. We explicitly present an admissible class of initial configurations leading to the asymptotic emergence of bi-cluster flocking phenomenon. Unlike global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.

Emergence of bi-cluster flocking for the Cucker–Smale model

Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership

We present a thermodynamically consistent particle (TCP) model motivated by the theory of multi-temperature mixture of fluids in the case of spatially homogeneous processes. The proposed model incorporates the Cucker-Smale (C-S) type flocking model as its isothermal approximation. However, it is more complex than the C-S model, because the mutual interactions are not only “mechanical” but are also affected by the “temperature effect” as individual particles may exhibit distinct internal energies. We develop a framework for asymptotic weak and strong flocking in the context of the proposed model.

Emergent Dynamics of a Thermodynamically Consistent Particle Model

In recent years, a number of articles have focused on mathematical models for emergent phenomena, for instance, the flocking of birds or the schooling of fish. In 2007, Cucker and Smale proposed an ingenious model which captures many of the observed features of moving animals. Subsequently, Shen extended the result to hierarchically structured flocks. Motivated by these works, in this paper we study the discrete Cucker–Smale flocking under rooted leadership, which means that there exists an overall leader such that any other agent is led, directly or indirectly, by the leader. The feature of our proposal, departing from the existing models, is that both the assumption of symmetry and the partial ordering of a hierarchy are dropped. The rooted leadership topology is a necessary condition for the group to converge towards a single leader's fixed constant velocity. The rates of convergence are established for flocks with fixed and switching topologies. The results may reveal the applicability and advantage o...

Cucker–Smale Flocking under Rooted Leadership with Fixed and Switching Topologies

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Xiaoping Xue

Author Tools

Chat about Author

Papers

Cucker–Smale Flocking under Rooted Leadership with Fixed and Switching Topologies