Emergent behaviors of the infinite set of Lohe Hermitian sphere oscillators (Q6631580)

The authors consider the system\N\[\N\dot{z}_i=\Omega_i z_i+\lambda_0\sum_{j=1}^\infty \kappa_{ij}\left(\langle z_i,z_i\rangle z_j-\langle z_j,z_i\rangle z_i\right)+\lambda_1\sum_{j=1}^\infty \kappa_{ij}\left(\langle z_i,z_j\rangle-\langle z_j,z_i\rangle \right)z_i\N\]\Nwhere each \(z_i\in\mathbb{C}^d\), \(\lambda_0\) and \(\lambda_1\) are nonnegative real numbers such that \(\lambda_0+\lambda_1=1\), \(\langle\cdot,\cdot\rangle\) denotes the standard inner product in \(\mathbb{C}^d\), the \(\Omega_i\) are \(d\times d\) anti-Hermitian matrices and each \(\kappa_{ij}\), which define the network structure, is positive. When the \(\kappa_{ij}\) are equal and the network is finite, this reduces to the ``Lohe Hermitian sphere model'', which is a generalization of the ``swarm sphere'' model, which is itself a generalization of the Kuramoto model of sinusoidally coupled phase oscillators.\N\NThe authors consider several cases. When the \(\Omega_i\) are anti-symmetric real matrices and the \(z_i\) are real \(d\)-dimensional vectors, they prove ``practical synchronization'' -- in the limit \(t\to\infty\), the maximum distance between any two oscillators, \(|z_j-z_i|\), is bounded. They show that this is also true when the \(\Omega_i\) are anti-Hermitian and the \(z_i\) are in \(\mathbb{C}^d\). The last case is when \(\kappa_{ij}=\kappa_j\), i.e.~a ``sender'' network, and each \(\Omega_i=0\). Here they prove that the only attracting state is either one for which all oscillators have the same state, or a ``bipolar'' state in which some fraction of the oscillators have the same state while the remainder have the negative of that state.