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

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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.