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We present emergent flocking dynamics of a thermodynamic Cucker-Smale (TCS) flock on a general digraph with spanning trees under the effect of communication time-delays. The TCS model describes a temporal evolution of mechanical and thermodynamic observables such as position, velocity and temperature of CS particles. In this paper, we study how variations in mechanical and thermodynamic variables can decay to zero along a time-independent network with position dependent weights from initial state configuration. For this, we provide a sufficient framework for a mechanical and thermodynamical flocking in terms of initial configuration, network topology, and system parameters. We also present several numerical examples and compare them with analytical results.

We study emergent behaviors of thermomechanical Cucker-Smale (TCS) ensemble confined in a harmonic potential field. In the absence of external force field, emergent dynamics of TCS particles has been extensively studied recently under various frameworks formulated in terms of initial configuration, system parameters and network topologies. Moreover, the TCS model does not exhibit rotating motions in the absence of an external force field. In this paper, we show the emergence of periodically rotating one-point cluster for the TCS model in a harmonic potential field using elementary energy estimates and continuity argument. We also provide several numerical simulations and compare them with analytical results.

We study the emergent dynamics and global well-posedness of the matrix-valued integro-differential equation which can be derived from the continuum limit of the lattice Lohe group model. The lattice Lohe group model corresponds to the generalized high-dimensional Kuramoto model. The solution to the lattice Lohe group model can be cast as a simple function-valued solution to the continuum Lohe group model. We first construct a local classical solution to the continuum Lohe group model, and then we find an invariant set and derive a global well-posedness in some sufficient frameworks formulated in terms of initial data, system functions, and system parameters. We also show that phase-locked states can emerge from the admissible class of initial data in a large coupling regime. Moreover, we show that sequence of simple functions obtained from the solutions of the lattice Lohe group model converges to a local classical solution to the continuum Lohe group model in supremum norm.