## Dynamical Systems: Morris W. Hirsch

In this chapter we look at neural nets as dynamical systems in the mathematical sense.

The following notations will be used. Rn denotes n-dimensional Euclidean space comprising all n-tuples of real numbers. We denote the inner or dot product of two vectors in Rn by x â€¢ y = (x, y) E xi y1. The Euclidean norm of a vector x is VE(.02 =

A map H : Rn rn is smooth, or Câ€™, provided it has continuous partial derivatives. In this case we denote the Jacobian matrix at p E Rn by DH(p) = [(a Hilaxj)(p)]. We say H is C2 if the partial derivatives are Câ€™. For integers k > 1 we recursively define H to be Ck if the partial derivatives are C" . If f is a smooth real-valued function on /V, the gradient vector grad f (p) of f at p E Rn is the column vector of partial derivatives of f at p.