The synthesis of arbitrary stable dynamics in non-linear neural networks. II. Feedback and universality

A parametrized family of higher-order, gradient-like neural networks that have known arbitrary equilibria with unstable manifolds of known specified dimension is described. Any system with hyperbolic dynamics is conjugate to one of the systems in a neighborhood of the equilibrium points. Prior work on how to synthesize attractors using dynamical systems theory, optimization, or direct parametric fits to known stable systems is nonconstructive, lacks generality, or has unspecified attracting equilibria. More specifically, a parameterized family of gradient-like neural networks is constructed with a simple feedback rule that will generate equilibrium points with a set of unstable manifolds of specified dimension. Strict Lyapunov functions and nested periodic orbits are obtained for these systems and used as a method of synthesis to generate a large family of systems with the same local dynamics. This work is applied to show how one can interpolate finite sets of data on nested periodic orbits