James Meiss
Publications:
Gonchenko S. V., Meiss J. D., Ovsyannikov I. I.
Chaotic dynamics of threedimensional Hénon maps that originate from a homoclinic bifurcation
2006, vol. 11, no. 2, pp. 191212
Abstract
We study bifurcations of a threedimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddlefocus fixed point with multipliers $(\lambda e^{i\varphi}, \lambda e^{i\varphi}, \gamma)$, where $0<\lambda<1<\gamma$ and $\lambda^2 \gamma=1$. We show that in a threeparameter family, $g_\varepsilon$, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near $\varepsilon=0$ where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a threedimensional Hénonlike map. This map possesses, in some parameter regions, a ''wildhyperbolic'' Lorenztype strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these threedimensional Hénon maps occupy in the class of threedimensional quadratic maps with constant Jacobian

Lenz K. E., Lomeli H. E., Meiss J. D.
Quadratic volume preserving maps: an extension of a result of Moser
1998, vol. 3, no. 3, pp. 122131
Abstract
A natural generalization of the Henon map of the plane is a quadratic diffeomorphism that has a quadratic inverse. We study the case when these maps are volume preserving, which generalizes the the family of symplectic quadratic maps studied by Moser. In this paper we obtain a characterization of these maps for dimension four and less. In addition, we use Moser's result to construct a subfamily of in n dimensions.
