Lyapunov Exponents and Sensitive Dependence

Precise relationships between Lyapunov exponents and the instability/stability of orbits of scalar discrete dynamical systems are investigated. It is known that positive Lyapunov exponent does not, in general, imply instability, or, equivalently, sensitive dependence on initial conditions. A notion of strong Lyapunov exponent is introduced and it is proved that for continuously differentiable maps on an interval, a positive strong Lyapunov exponent implies sensitive dependence on initial conditions. However, to have a strong Lyapunov exponent an orbit must stay away from critical points. Nevertheless, it is shown for a restricted class of maps that a positive Lyapunov exponent implies sensitive dependence even for orbits which go arbitrarily close to critical points. Finally, it is proved that for twice differentiable maps on an interval negative Lyapunov exponent does imply exponential stability of orbits.