The Henon map is a prototypical 2-D invertible dissipative iterated map with chaotic solutions proposed by the French astronomer Michel Henon (M. Henon, Commun. Math. Phys. Phys.

Since the second equation above can be written as

An interesting question to ask is what are the values of a and b that maximize the largest Lyapunov exponent and the Kaplan-Yorke dimension. The maximum Lyapunov exponent occurs for a = 2 and b = 0, where the map reduces to a one-dimensional quadratic map whose largest Lyapunov exponent is ln(2) = 0.693147181... . The other exponent is minus infinity, which implies that there is infinitely rapid contraction in the direction perpendicular to the one-dimensional parabolic attractor.

Much less trivial is the question of what values of a and b maximize the Kaplan-Yorke dimension. The area contraction for the Henon map is the sum of the two Lyapunov exponents and is given by ln(|b|) which is negative (contractive) for -1 < b < 1 and encompasses the entire region over which the map is bounded as well as chaotic. From the largest Lyapunov exponent L, the Kaplan-Yorke dimension is therefore given by D

A program was written to search ab-plane for the maximum value of D

Note that the attractor nearly touches its basin boundary in numerous places, which is typical for a chaotic system where the maximum chaos often occurs just before the orbit becomes unbounded. Note also that the basin boundary has numerous (perhaps infinitely many) narrow tongues, suggesting a possibly fractal structure that deserves further study. The attractor is not greatly different from the Henon map with the usual parameters of a = 1.4 and b = 0.3 for which L = 0.419222 and D

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