论文标题
从周期到准周期再到混乱：被困在偶极子磁场中的带电颗粒的连续光谱
From period to quasi-period to chaos: A continuous spectrum of orbits of charged particles trapped in a dipole magnetic field
论文作者
论文摘要
通过评估Lyapunov指数，我们报告了发现三翼磁场中的带电颗粒的准周期轨道的三个突出的相空间方案。除了经过广泛研究的低能量状态外，在捕获轨道的相位空间的每个维度上覆盖了10％以上，还有两组高能量状态，其中最大的覆盖了捕获轨道相位空间的每个维度中的4％以上。可以在空间中观察到这些高能轨道中的颗粒，并在地球上的血浆实验中实现。众所周知，在具有2个自由度的哈密顿系统中，稳定的周期轨道周围有准周期轨道，这些准周期轨道也稳定。由于周期性的轨道似乎在相空间中具有可忽略的措施，因此它们在自然界中很难实现。另一方面，准周期轨道可能在4维（4D）相空间中占据有限体积，并且很容易被检测到。混沌轨道至少具有一个正lyapunov指数。另一方面，准周期轨道的Lyapunov指数应为零。通过计算偶极磁场中被困的带电颗粒的轨道的Lyapunov指数，我们扫描了相应的相空间，并发现了与赤道平面中与稳定的周期轨道相关的准周期轨道的几种突出的机制。这些方案似乎与与子午线平面中稳定的周期轨道相关的准周期轨道的一些小策略有关。我们的数值结果还表明了这些轨道的连续光谱，从稳定的周期性到准周期性，并消失了Lyapunov指数，最终是具有至少一个正lyapunov指数的混乱，并且具有不稳定的周期轨道，并具有阳性最大lyapunov指数。
Via evaluation of the Lyapunov exponent, we report the discovery of three prominent sets of phase space regimes of quasi-periodic orbits of charged particles trapped in a dipole magnetic field. Besides the low energy regime that has been studied extensively and covers more than 10% in each dimension of the phase space of trapped orbits, there are two sets of high energy regimes, the largest of which covers more than 4% in each dimension of the phase space of trapped orbits. Particles in these high energy orbits may be observed in space and be realized in plasma experiments on the Earth. It is well-known that there are quasi-periodic orbits around stable periodic orbits in Hamiltonian systems with 2 degrees of freedom and these quasi-periodic orbits are stable as well. Since periodic orbits appear to have a negligible measure in the phase space, they are difficult to realize in nature. Quasi-periodic orbits, on the other hand, may occupy a finite volume in the 4 dimensional (4D) phase space and be readily detectable. A chaotic orbit has at least one positive Lyapunov exponent. The Lyapunov exponents of quasi-periodic orbits, on the other hand, should be zero. Via calculation of the Lyapunov exponent of orbits of trapped charged particles in a dipole magnetic field, we scanned the corresponding phase space and found several prominent regimes of quasi-periodic orbits associated with stable periodic orbits in the equatorial plane. These regimes appear to be connected to some small regimes of quasi-periodic orbits associated with stable periodic orbits in the Meridian plane. Our numerical results also show a continuous spectrum of these orbits from stable periodic, to quasi-periodic with vanishing Lyapunov exponents, and eventually to chaotic ones with at least one positive Lyapunov exponent and there are unstable periodic orbits with a positive maximum Lyapunov exponent.
