0
2013
Impact Factor

James Meiss

Boulder, CO 80309-0526
Department of Applied Mathematics, University of Colorado

Publications:

Gonchenko S. V., Meiss J. D., Ovsyannikov I. I.
Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation
2006, vol. 11, no. 2, pp.  191-212
Abstract
We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus 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 three-parameter 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 three-dimensional Hénon-like map. This map possesses, in some parameter regions, a ''wild-hyperbolic'' Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional Hénon maps occupy in the class of three-dimensional quadratic maps with constant Jacobian
Keywords: saddle-focus fixed point, three-dimensional quadratic map, homoclinic bifurcation, strange attractor
Citation: Gonchenko S. V., Meiss J. D., Ovsyannikov I. I.,  Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 191-212
DOI: 10.1070/RD2006v011n02ABEH000345
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.  122-131
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.
Citation: Lenz K. E., Lomeli H. E., Meiss J. D.,  Quadratic volume preserving maps: an extension of a result of Moser, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 122-131
DOI:10.1070/RD1998v003n03ABEH000085

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