1,663 research outputs found
Existence and emergent dynamics of quadratically separable states to the Lohe tensor model
A tensor is a multi-dimensional array of complex numbers, and the Lohe tensor
model is an aggregation model on the space of tensors with the same rank and
size. It incorporates previously well-studied aggregation models on the space
of low-rank tensors such as the Kuramoto model, Lohe sphere and matrix models
as special cases. Due to its structural complexities in cubic interactions for
the Lohe tensor model, explicit construction of solutions with specific
structures looks daunting. Recently, we obtained completely separable states by
associating rank-1 tensors. In this paper, we further investigate another type
of solutions, namely "{\it quadratically separable states}" consisting of
tensor products of matrices and their component rank-2 tensors are solutions to
the double matrix model whose emergent dynamics can be studied using the same
methodology of the Lohe matrix model
Collective behaviors of the Lohe hermitian sphere model with inertia
We present a second-order extension of the first-order Lohe hermitian
sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed
model incorporates an inertial effect as a second-order extension. The inertia
term can generate an oscillatory behavior of particle trajectory in a small
time interval(initial layer) which causes a technical difficulty for the
application of monotonicity-based arguments. For emergent estimates, we employ
two-point correlation function which is defined as an inner product between
positions of particles. For a homogeneous ensemble with the same frequency
matrix, we provide two sufficient frameworks in terms of system parameters and
initial data to show that two-point correlation functions tend to the unity
which is exactly the same as the complete aggregation. In contrast, for a
heterogeneous ensemble with distinct frequency matrices, we provide a
sufficient framework in terms of system parameters and initial data, which
makes two-point correlation functions close to unity by increasing the
principal coupling strength
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