Lohe M A
Department of Physics, The University of Adelaide, Adelaide 5005, Australia.
Chaos. 2024 Aug 1;34(8). doi: 10.1063/5.0216910.
We consider systems of N particles interacting on the unit circle through 2π-periodic potentials. An example is the N-rotor problem that arises as the classical limit of coupled Josephson junctions and for various energies is known to have a wide range of behaviors such as global chaos and ergodicity, together with families of periodic solutions and transitions from order to chaos. We focus here on selected initial values for generalized systems in which the second order Euler-Lagrange equations reduce to first order equations, which we show by example can describe an ensemble of oscillators with associated emergent phenomena such as synchronization. A specific case is that of the Kuramoto model with well-known synchronization properties. We further demonstrate the relation of these models to field theories in 1+1 dimensions that allow static kink solitons satisfying first order Bogomolny equations, well-known in soliton physics, which correspond to the first order equations of the generalized N-rotor models. For the nonlinear pendulum, for example, the first order equations define the separatrix in the phase portrait of the system and correspond to kink solitons in the sine-Gordon equation.
我们考虑在单位圆上通过 2π 周期势相互作用的 N 粒子系统。一个例子是 N 转子问题，它作为耦合约瑟夫森结的经典极限出现，并且已知在各种能量下具有广泛的行为，如全局混沌和遍历性，以及周期解族和从有序到混沌的转变。我们在此关注广义系统的选定初始值，其中二阶欧拉 - 拉格朗日方程简化为一阶方程，我们通过示例表明这些一阶方程可以描述具有同步等相关涌现现象的振子系综。一个具体的例子是具有著名同步特性的仓本模型。我们进一步证明了这些模型与 1 + 1 维场论的关系，该场论允许满足一阶博戈莫尔尼方程的静态扭结孤子，这在孤子物理学中是众所周知的，它对应于广义 N 转子模型的一阶方程。例如，对于非线性摆，一阶方程定义了系统相图中的分界线，并且对应于正弦 - 戈登方程中的扭结孤子。
