Complex Dynamics in Generalized Hénon Map

The complex dynamics of generalizedHénon map with nonconstant Jacobian determinant areinvestigated. The conditions of existence for fold bifurcation,flip bifurcation, and Hopf bifurcation are derived by using centermanifold theorem and bifurcation theory and checked up bynumerical simulations. Chaos in the sense of Marotto's definitionis proved by analytical and numerical methods. The numericalsimulations show the consistence with the theoretical analysis andreveal some new complex phenomena which can not be given bytheoretical analysis, such as the invariant cycles which areirregular closed graphics, the six and forty-one coexistinginvariant cycles, and the two, six, seven, nine, ten, and thirteencoexisting chaotic attractors, andsome kinds of strange chaotic attractors.