James Meiss
Publications:
Bäcker A., Meiss J. D.
Moser’s Quadratic, Symplectic Map
2018, vol. 23, no. 6, pp. 654664
Abstract
In 1994, Jürgen Moser generalized Hénon’s areapreserving quadratic map to obtain
a normal form for the family of fourdimensional, quadratic, symplectic maps. This map has at
most four isolated fixed points. We show that the bounded dynamics of Moser’s six parameter
family is organized by a codimensionthree bifurcation, which we call a quadfurcation, that can
create all four fixed points from none.
The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser’s map there can be two such fixed points: this structure is not what one would expect from dynamics near the cross product of a pair of uncoupled Hénon maps, where there is at most one doubly elliptic point. We visualize the dynamics by escape time plots on 2D planes through the phase space and by 3D slices through the tori. 
Guillery N., Meiss J. D.
Diffusion and Drift in VolumePreserving Maps
2017, vol. 22, no. 6, pp. 700–720
Abstract
A nearlyintegrable dynamical system has a natural formulation in terms of actions, $y$ (nearly constant), and angles, $x$ (nearly rigidly rotating with frequency $\Omega(y)$). We study angleaction maps that are close to symplectic and have a twist, the derivative of the frequency map, $D\Omega(y)$, that is positivedefinite. When the map is symplectic, NekhoroshevЃfs theorem implies that the actions are confined for exponentially long times: the drift is exponentially small and numerically appears to be diffusive. We show that when the symplectic condition is relaxed, but the map is still volumepreserving, the actions can have a strong drift along resonance channels. Averaging theory is used to compute the drift for the case of rank$r$ resonances. A comparison with computations for a generalized Froeschlé map in fourdimensions shows that this theory gives accurate results for the rankone case.

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.
