We survey past work and present new algorithms to numerically integrate the trajectories of Hamiltonian dynamical systems.These algorithms exactly preserve the symplectic 2-form, i.e. they preserve all the Poincar6 invariants. The algorithms have been tested on a variety of examples and results are presented for the Fermi-Pasta-Ulam nonlinear string, the Henon-Heiles system, a four-vortex problem, and the geodesic flow on a manifold of constant negative curvature. In all cases the algorithms possess long-time stability and preserve global geometrical structures in phase space.

A theoretical formalism which allows the generation of a large class of exact Vlasov–Maxwell equilibria with sheared magnetic fields is presented. All quantities are assumed to vary in only one spatial direction, x, and the magnetic field has components only in the y and z directions. The Vlasov equations are solved by making the distribution functions depend only on constants of the motion. The Maxwell equations are then reduced to finding the motion of a pseudo-particle in a two-dimensional potential. Three examples corresponding to sheet-like, sheath-like, and wave-like equilibria are presented.

The origin of spike adding in bursting activity is studied in a reduced model of the leech heart interneuron. We show that, as the activation kinetics of the slow potassium current are shifted towards depolarized membrane potential values, the bursting phase accommodates incrementally more spikes into the train. This phenomenon is attested to be caused by the homoclinic bifurcations of a saddle periodic orbit setting the threshold between the tonic spiking and quiescent phases of the bursting. The fundamentals of the mechanism are revealed through the analysis of a family of the onto Poincaré return mappings.

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