Emergent dynamics of the time-discrete infinite Cucker–Smale model

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We study the emergent dynamics of the (time-) discrete infinite Cucker–Smale model (in short DICS model), such as the existence of quasi-steady velocity configuration, mono-cluster flocking behaviors, uniform-in-time stability, and uniform-in-time continuous transition from discrete dynamics to continuous dynamics. First, we provide sufficient frameworks for quasi-steady velocity configuration in both continuous and discrete infinite Cucker–Smale models, showing that weak network connectivity can prevent mono-cluster flocking by keeping the velocity diameter unchanged. We also present a framework for flocking dynamics in the DICS model, which can be applied to the discrete finite Cucker–Smale model. Without requiring average velocity conservation, we construct a system of dissipative differential inequalities (SDDI) to derive an exponential decay of velocity diameter. Additionally, we explore two uniform-in-time stability and continuous transition under the sender network. By using the flocking estimate and conservation of the average velocity, we construct corresponding SDDI to show the uniform-in-time stability. We also derive the uniform-in-time continuous transition using the flocking estimate and classical finite-in-time transition result.