James Meiss
Publications:
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.
