标题： 能量自平衡作为轨道量子化的物理基础 显示英文标题

标题： Energy self-balance as the physical basis of orbit quantization

摘要： 我们证明了非保守力在一个耗散动力系统的稳定极限环吸引子上所做的功总是等于零。因此，机械能在周期轨道上平均来说是守恒的。这种能量获得与能量损失之间的平衡负责了这些系统中量子化轨道的存在。 此外，我们展示了沿量子化轨道瞬时保存投影相空间面积描述了相位的中性动态，从而允许我们从这个方程推导出类似于威尔逊-索末菲的量子化条件。 我们将我们的一般结果应用于近哈密顿系统，将克里洛夫-博戈利乌布夫径向方程的不动点与梅尔尼科夫函数的零点相对应。 此外，我们将沿量子化轨道的相空间面积的瞬时保存与描述相位动态的第二个克里洛夫-博戈利乌布夫方程联系起来。 我们在流体力学量子模拟的背景下测试这两个量子化条件，在那里最近发现了一个巨大的量子化轨道光谱。 具体地，我们使用了一个广义的引导波模型来描述限制在一个谐波势中的行走液滴，并找到了一个可数无限的嵌套极限环吸引子集，这代表了量子化轨道的经典模拟。 我们计算了这个自激系统的能谱和本征函数。 显示英文摘要

摘要： We show that work done by the non conservative forces along a stable limit cycle attractor of a dissipative dynamical system is always equal to zero. Thus, mechanical energy is preserved on average along periodic orbits. This balance between energy gain and energy loss along different phases of the self sustained oscillation is responsible for the existence of quantized orbits in such systems. Furthermore, we show that the instantaneous preservation of projected phase space areas along quantized orbits describes the neutral dynamics of the phase, allowing us to derive from this equation the Wilson Sommerfeld like quantization condition. We apply our general results to near Hamiltonian systems, identifying the fixed points of Krylov Bogoliubov radial equation governing the dynamics of the limit cycles with the zeros of the Melnikov function. Moreover, we relate the instantaneous preservation of the phase space area along the quantized orbits to the second Krylov Bogoliubov equation describing the dynamics of the phase. We test the two quantization conditions in the context of hydrodynamic quantum analogs, where a megastable spectra of quantized orbits have recently been discovered. Specifically, we use a generalized pilot wave model for a walking droplet confined in a harmonic potential, and find a countably infinite set of nested limit cycle attractors representing a classical analog of quantized orbits. We compute the energy spectrum and the eigenfunctions of this self excited system.