Applications and Theory of Controlling Symbol Dynamics: Communicating with Chaos

9704639 Bollt这个项目解决了控制混沌和控制小扰动符号动力学的问题。最近的工作已经证明了混沌轨道控制的符号动力学表示在混沌信号发生器通信中的应用。混沌动力系统的轨迹演化等价于适当符号空间中的符号动力学。最近使用OGY （Ott， Grebogi和Yorke）技术和各种目标算法控制混沌的出现，以及物理应用，已经证明可以控制混沌，并且固有的不稳定性可以用作允许小的故意扰动引起大信号变化的优势。通过小扰动对混沌的耦合控制与学习相应符号动力学的语法意味着控制扰动实际上是对原始动力学的编码方案。使用基于映射的动态描述来控制符号动态需要解决以下问题：学习映射迭代对控制参数变化的响应，学习吸引子上映射的动态与相应符号动态的语法之间的半共轭或编码函数，以及找到依赖于相空间中分区的适当选择的最小语法。特别地，扩展包括在高维动力系统中正在进行的通信工作。研究包括实际噪声环境控制与带宽权衡的问题，遍历性问题，学习系统对控制的响应，混沌动力系统的符号动力学，只有通过实验数据的时间序列嵌入才能知道，以及实用语法学习算法的进一步发展。这一非线性通信理论的新领域有望发展成为一种新的范式，提供一种通用的信息传输技术，对电子和光学介质都有用，应该在民用和军事通信基础设施中得到广泛应用。近年来在应用动力系统和理论动力系统方面的大量研究都集中在利用混沌动力系统可以被控制这一事实上。混沌的敏感依赖特性实际上有利于建立一个高度敏捷的控制系统，在这个系统中，一个小的故意变化可以引起大的响应；所谓的“蝴蝶效应”允许我们用极小的功率控制来引导系统响应。遍历理论告诉我们，混沌系统可以看作是一个无限的信息源；而对混沌的控制使我们能够用极低功率的控制来操纵信息流。一个例子应用是一个高功率信号发生器（例如一个电子电路），它有意地在混沌状态下工作，这样一个小型的背驮式控制器电路，在微芯片规模上，有能力精确地操纵高功率的消息承载信号。这种方法与标准的线性通信技术相反，在标准的线性通信技术中，高功率的电子电路需要一个同样大的开关设备来影响传输高功率信息所需的大功率变化。如前所述，通过控制混沌进行通信不仅适用于一维动力学，而且也适用于自然界中发现的更广泛的高维混沌动力学的典型类别。通过控制混沌进行通信有许多实际应用前景，包括新的和简单的电子通信设备的工程，以及新的和简单的光通信设备，以及生物学，化学和认知科学中的现象建模。

9704639 Bollt This project addresses problems in controlling chaos and controlling symbol dynamics with small perturbations. Recent work has demonstrated the utilization of symbolic dynamics representation of controlled chaotic orbits for communications with chaotic signal generators. The evolution of trajectories of a chaotic dynamical system is equivalent to symbolic dynamics in an appropriate symbol space. The recent advent of controlling chaos using the OGY (Ott, Grebogi, and Yorke) technique and a variety of targeting algorithms, with physical applications, has demonstrated that chaos can be mastered, and inherent instabilities can be used as an advantage in allowing small deliberate perturbations to cause large signal variations. Coupling control of chaos through small perturbations with learning the grammar of the corresponding symbol dynamics means that the control perturbations are actually a coding scheme on the original dynamics. Controlling symbol dynamics using a map based description of the dynamics requires resolution of the following issues: learning the response of map iterates to variations in the control parameters, learning the semi- conjugacy, or coding function, between the dynamics of the map on the attractor and the grammar of the corresponding symbol dynamics, and finding the minimal grammar which is dependent on the appropriate choice of the partition in phase space. In particular, extensions include on-going work on communicating in higher dimensional dynamical systems. Investigations include issues of practical noisy environment control versus bandwidth trade-off and issues of ergodicity, learning system response to controls, and symbol dynamics of a chaotic dynamical system which is known only by time-series embedding of experimental data, and the further development of practical grammar learning algorithms. This new field of nonlinear communication theory promises to develop into a new paradigm offe ring a general information transmission technique, useful to both electronic and optical media, which should find wide applications in civil and military communications infrastructures. A great deal of recent research in applied and theoretical dynamical systems has been focused on taking advantage of the fact that a chaotic dynamical system can be controlled. The sensitive dependence characteristic of chaos is actually advantageous to building a highly agile control system in which a small deliberate system variations can cause a large response; the so called ``butterfly effect" allows us to steer the system responses with extremely small powered controls. Ergodic theory tells us that a chaotic system can be considered as an unlimited information source; and control of chaos allows us to manipulate this information flow with extremely low-powered controls. An example application is a high-powered signal generator (e.g. an electronic circuit), which operates intentionally in the chaotic regime, so that a small-scaled piggy-back controller circuit, on the micro-chip scale, has the ability to accurately manipulate high-powered message bearing signals. This method is in contrast to standard linear communication techniques, in which a high powered electronic circuit requires an equally large-scaled switching device to affect the large power variations required to transmit a high powered message. Not only is communicating through control of chaos applicable to one-dimensional dynamics, as previously demonstrated, but also applies to the more widely typical class of higher dimensional chaotic dynamics found in nature. Communicating by control of chaos promises numerous practical applications including the engineering of new and simple electronic communications devices, and new and simple optical communications devices, as well as the modeling of phenomenon in biology, chemistry, and cognitive science.