Convergence of analytic gradient-type systems with periodicity and its applications in Kuramoto models
In this paper, the authors analyze convergence properties of two gradient-type systems. Namely, the first-order system \[ \dot x+\nabla f(x)=0,\quad x\in\mathbb R^n, \] and the second-order system \[ M\ddot x+D\dot x+\nabla f(x)=0,\quad x\in\mathbb R^n, \] where $M$ and $D$ are the diagonal matrices of inertia and damping coefficients, respectively. The main result claims that, in case of a real analytic $\pi K$-periodic function $f$ ($K\in\mathbb Z^n$), any solution of each of the above systems tends asymptotically to a critical point $x^*\in\{x\in\mathbb R^n:\nabla f(x)=0\}$. For the second-order system, it is also shown that $\dot x(t)\to 0$. The main feature of the obtained result in the absence of a priory assumption about the boundedness of the trajectories of the above systems. The first-order and second-order Kuramoto models are considered as examples.
scientific article; zbMATH DE number 7026119