Creative dynamics approach to optimization problems

A type of dynamical system for solving optimization problems is introduced. The approach exploits a novel paradigm in nonlinear dynamics that is based upon the concept of terminal attractors and repellers. A class of dynamical systems-the unpredictable systems-is introduced and analyzed. These systems are represented in the form of coupled activation and learning dynamical equations whose ability to be spontaneously activated is based upon two pathological characteristics: (1) Such systems have zero Jacobian. As a result, they have an infinite number of equilibrium points which occupy curves, surfaces, or hypersurfaces. (2) At all of these equilibrium points, the Lipschitz condition fails, so the equilibrium points become terminal attractors or repellers, depending upon the sign of the periodic excitation. These characteristics result in multichoice response and lead to unpredictable dynamical systems. The systems can be controlled by sign strings which uniquely define the systems´ behavior by specifying the direction of the motions at the critical points. By changing the combinations of signs in the code strings, a system can reproduce any prescribed behavior to a prescribed accuracy, which is why the unpredictable systems driven by sign strings are extremely flexible and can be exploited for solving optimization problems