Most previous works on complete synchronization of chaotic oscillators focused on the one-channel interaction scheme where the oscillators are coupled through only one variable or a symmetric set of variables. Using the standard framework of master-stability function (MSF), we investigate the emergence of complex synchronization behaviors under all possible configurations of two-channel coupling, which include, for example, all possible cross coupling schemes among the dynamical variables. Utilizing the classic Rössler and Lorenz oscillators, we find a rich variety of synchronization phenomena not present in any previously extensively studied, single-channel coupling configurations. For example, in many cases two coupling channels can enhance or even generate synchronization where there is only weak or no synchronization under only one coupling channel, which has been verified in a coupled neuron system. There are also cases where the oscillators are originally synchronized under one coupling channel, but an additional synchronizable coupling channel can, however, destroy synchronization. Direct numerical simulations of actual synchronization dynamics verify the MSF-based predictions. Our extensive computation and heuristic analysis provide an atlas for synchronization of chaotic oscillators coupled through two channels, which can be used as a systematic reference to facilitate further research in this area.

Smart meters, designed for information collection and system monitoring in smart grid, report fine-grained power consumption to utility providers. With these highly accurate profiles of energy usage, however, it is possible to identify consumers' specific activities or behavior patterns, thereby giving rise to serious privacy concerns. This paper addresses these concerns by designing a cost-effective and privacy-preserving energy management technique that uses a rechargeable battery. From a holistic perspective, a dynamic programming framework is designed for consumers to strike a tradeoff between smart meter data privacy and the cost of electricity. In general, a major challenge in solving dynamic programming problems lies in the need for the knowledge of future electricity consumption events. By exploring the underlying structure of the original problem, an equivalent problem is derived, which can be solved by using only the current observations. An online control algorithm is then developed to solve the equivalent problem based on the Lyapunov optimization technique. It is shown that without the knowledge of the statistics of the time-varying load requirements and the electricity price processes, the proposed online control algorithm, parametrized by a positive value V, is within O (1/{V) of the optimal solution to the original problem, where the maximum value of V is limited by the battery capacity. The efficacy of the proposed algorithm is demonstrated through extensive numerical analysis using real data.