Emergence of Complexity and Chaos via Symmetry Breaking in Coupled Engineering Systems.

研究了一种三维对称的无源耦合网络。初始条件的变化表明，具有完全D3对称的动力学以及对称破缺到Z2和O（不对称）的动力学也发生了。对于三维对称解，所有振子的行为相同。对于z2对称，两个振子的行为相同，而3^<rd>振子的行为与其他两个不同。在最后一种情况下，每个振子的行为与其他两个不同。重要的发现是，复杂动力学是由原始系统的对称性决定的。最近研究了一种主动协作机器人网络。机器人行为可以用马尔可夫决策过程（MDP）来描述。在存在对称性的情况下，最优价值函数也是对称的。所有由对称相关的状态都映射到同一个单一状态。同样地，对称的动作也与相同的动作相关。因此，有效的民主党可能会大幅下降。在z2对称状态下，MDP的大小减小了一半。利用这种对称性，可以提高学习速度。多个相同（对称）的机器人通过交换信息来提高学习速度。研究了一个双机器人系统。这个系统是z2对称的。在一个问题中识别对称性是相当复杂的。在我们的分析中，机器人被设计为“固有地”识别环境中的对称性。相同的机器人也表明机器人之间的对称性。我们的研究结果证实，对称环境中的对称机器人会导致对称的MDP。更重要的是，它表明，考虑对称性，机器人学习（在对称环境的导航中）在考虑对称性时要快得多。

A passive coupled network having D3-Symmetry was previously studied. Changes in initial conditions showed that dynamics with the full D3 symmetry as well as symmetry breaking to Z2 and O (no symmetry) also occurred. For D3-symmetric solutions all oscillators behave identically. For Z2-symmetry, two oscillators behave identically which the 3^<rd> oscillator behaves differently from the other two. In the last case, each oscillator behaved differently from the other two. The important finding was that the complex dynamics are determined by the symmetry of the original system. Recently an active network of cooperating robots is studied. Robot behavior may be described by a Markov Decision Process (MDP). In the presence symmetry, the Optimal Value Function is also Symmetrical. All states related by symmetry are mapped to the same single state. Similarly, symmetrical actions are related to the same action. Thus, the effective MDP can be drastically reduced. In the case of Z2-symmetrical states, the MDP is reduced half its original size. Exploiting such symmetry, learning speeds can be increased. Multiple identical (symmetrical) robots increase learning speed by exchanging information. A 2-robot system has been studied. This system has Z2-symmetry. Identifying symmetry in a problem is quite complex. In our analysis, a robot is designed to 'inherently' identify symmetry in the environment. Identical robots also indicate inter-robot symmetry. Our results confirm that symmetrical robots in a symmetrical environment lead to a symmetrical MDP. More importantly it is shown that considering symmetry, Robot learning (in the navigation of a symmetrical environment) is much faster when symmetry is considered.