Dynamical Systems, Optimization, and Chaos

Much of engineering is concerned with the topic of optimization, and at the heart of much of our optimization is dynamical systems. Dynamical systems can be thought of as either non-linear continuous-time differential equations or difference equations. Chaos occurs in dynamical systems, and frequently in engineering we seek to avoid chaos. At times chaos becomes the central fascination. This paper first introduces a situation in signal processing for neural systems in which chaos is the perhaps unexpected phenomena and the object of study. The focus then shifts to the topic of optimization of systems via dynamical systems, where traditionally chaos is avoided as much as possible. The essential dynamical system should converge in a very smooth manner to an optimal solution to some problem of interest. Our technical approach is summarized to optimization via dynamical systems is illustrated by an application in the area of robotics. The key questions motivating this research are: Does the human brain exploit chaos for generating intelligence? Can our computing machines and control systems enhance their intelligence by a clever introduction of chaos?