Stretching rates in discrete dynamical systems

In dynamical systems the singular values of the n×n Jacobian matrix J are related to stretching rates of unit vectors in orthogonal directions in . It is shown that these stretching or shrinking factors are maximal values of the quadratic form defined by J. As a consequence, maximal Lyapunov exponents exist in discrete dynamical systems. However, the numerical evaluated exponents are different from these maximal ones if the dynamics is started from arbitrary unit vectors. The maximal stretching rates cannot be calculated by the absolute values of the eigenvalues of Jacobian.