Lyapunov exponents for the intermittent transition to chaos

The dependence of the Lyapunov exponent on the closeness parameter, epsilon , in tangent bifurcation systems is investigated. We study and illustrate t wo averaging procedures for defining Lyapunov exponents in such systems. Fi rst, we develop theoretical expressions for an isolated tangency channel in which the Lyapunov exponent is defined on single channel passes. Numerical simulations were done to compare theory to measurement across a range of e psilon values. Next, as an illustration of defining the Lyapunov exponent o n many channel passes, a simulation of the intermittent transition in the l ogistic map is described. The modified theory for the channels is explained and a simple model for the gate entrance rates is constructed. An importan t correction due to the discrete nature of the iterative flow is identified and incorporated in an improved model. Realistic fits to the data were mad e for the Lyapunov exponents from the logistic gate and from the full simul ation. A number of additional corrections which could improve the treatment of the gates are identified and briefly discussed.