Lyapunov Exponent of The 2D Henon Map Eui-Sun Lee Department of Physics Kangwon National University 2D Henon Map: The Lyapunov exponent characterize the exponential divergence of a nearby orbit. In the 2D Map, attractor with one or two positive Lyapunov exponent is said to be chaotic attractor. Two Lyapunov exponents The two Lyapunov exponents( ) is allowed along the two eigen-directions in the 2D Henon Map. The total summation of Lyapunov exponent is the constant for the given b .

Gram –Schmidt Reorthonormalization(GSR) Process The Introduction to GSR Method. There are two problem to calculate the Lyapunov exponents. 1.The two evolved vector by the linearized map tends to align with each other to the rapid growth direction. Orthogonalization process . 2.The limited storage of computer can’t store up the exponential growth of the evolved vector. Normalization process . Reorthonormalization algorithm in the 2D Map 1 step: Set the two initial vector , where 2 step: Measure the largest Lyapunov exponent 3 step: Reorthonormalization process. Reorthonormalizated sets: Lyapunov exponent: , where

Period Doubling Route to Chaos Bifurcation and Lyapunov exponent diagram Transition to chaos occurs when the Lyapunov exponent becomes positive. In the case of the 2D Henon Map having a constant Jacobian , the Lyapunov exponent spectra exhibits symmetry about the Bifurcation diagram Lyapunov exponent b=0.3