6 research outputs found
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
Towards Almost Global Synchronization on the Stiefel Manifold
A graph is referred to as -synchronizing if,
roughly speaking, the Kuramoto-like model whose interaction topology is given
by synchronizes almost globally. The Kuramoto model evolves on
the unit circle, \ie the -sphere . This paper concerns
generalizations of the Kuramoto-like model and the concept of synchronizing
graphs on the Stiefel manifold . Previous work on state-space
oscillators have largely been influenced by results and techniques that pertain
to the -case. It has recently been shown that all connected
graphs are -synchronizing for all . The previous point of
departure may thus have been overly conservative. The -sphere is a special
case of the Stiefel manifold, namely . As such, it is
natural to ask for the extent to which the results on can be
extended to the Stiefel manifold. This paper shows that all connected graphs
are -synchronizing provided the pair satisfies