Abstract
The complex dynamics of generalized Hénon map with nonconstant Jacobian determinant are investigated. The conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos in the sense of Marotto's definition is proved by analytical and numerical methods. The numerical simulations show the consistence with the theoretical analysis and reveal some new complex phenomena which can not be given by theoretical analysis, such as the invariant cycles which are irregular closed graphics, the six and forty-one coexisting invariant cycles, and the two, six, seven, nine, ten, and thirteen coexisting chaotic attractors, and some kinds of strange chaotic attractors.
Highlights
The planar map xn+1 = 1 + ayn − bxn2, (1)yn+1 = xn was first introduced by Henon [1] as a planar diffeomorphism that imitated essential stretching and folding properties of the Poincaremap of the Lorenz system
Feit [2] introduced the characteristic exponents in order to estimate strange attractors numerically
Marotto [3] proved analytically that the map had a transversal homoclinic orbit, which implied the existence of the chaotic behavior for some parameter values
Summary
Yn+1 = xn was first introduced by Henon [1] as a planar diffeomorphism that imitated essential stretching and folding properties of the Poincaremap of the Lorenz system This original Henon map (1) had a strange attractor with fractal structure and had constant Jacobian determinant |J| = −b. Kirchgraber and Stoffer [12] proved this Henon map existing a transversal homoclinic point for a set of parameters which was not small by using shadowing techniques. The conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory [18]; chaotic behavior in the sense of Marotto’s definition [19] is proved.
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