Applications and Theory of Controlling Symbol Dynamics: Communicating with Chaos

9704639 Bollt这个项目解决了在控制混沌和用小扰动控制符号动力学方面的问题。最近的工作证明了利用受控混沌轨道的符号动力学表示与混沌信号产生器进行通信。混沌动力系统的轨迹演化等价于符号空间中的符号动力学。最近出现的使用OGY(Ott，Grebogi和York ke)技术和各种目标算法控制混沌的物理应用表明，混沌是可以控制的，内在的不稳定性可以被用作允许小的故意扰动引起大的信号变化的优势。通过小扰动控制混沌与学习相应符号动力学的语法相结合，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适当的划分选择的最小语法。特别是，扩展包括正在进行的关于高维动力系统中的通信的工作。研究包括实际噪声环境控制与带宽权衡问题、遍历性问题、学习系统对控制的响应问题、混沌动力系统的符号动力学问题(只有通过实验数据的时间序列嵌入才知道)，以及实用语法学习算法的进一步发展。非线性通信理论的这一新领域有望发展成为一种通用的信息传输技术的新范式，对电子和光学介质都有用，它将在民用和军用通信基础设施中得到广泛应用。最近在应用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是可以控制的这一事实。混沌的敏感依赖特性实际上有利于建立高度灵活的控制系统，在该系统中，细小的有意的系统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操纵系统的响应。遍历理论告诉我们，混沌系统可以被认为是一个无限的信息源；对混沌的控制允许我们用极低功率的控制来操纵这种信息流。一个示例应用是高功率信号生成器(例如，电子电路)，其有意地在混沌区域中操作，从而使得微芯片尺度上的小规模的背负式控制器电路具有准确处理高功率消息承载信号的能力。这种方法与标准线性通信技术不同，在标准线性通信技术中，高功率电子电路需要同样大规模的开关设备来影响传输高功率消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典型高维混沌动力学。通过控制混沌进行通信具有广泛的实际应用前景，包括设计新的简单的电子通信设备、设计新的简单的光通信设备以及对生物、化学和认知科学中的现象进行建模。

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.