The Poincare subpackage of DEtools is a collection of commands and routines for working with Poincare Sections of Hamiltonian systems, based on the work by E.S. Cheb-Terrab and H.P de Oliveira (see References below).

This subset of commands consists of a plotting command, poincare, for the 2-D and 3-D plotting of the corresponding projections of surfaces-of-section (2PS and 3PS), plus a set of facility commands for a quick setup of the Hamilton equations of motion, the initial conditions for numerical experiments, and the zooming of plots.

All plots are built by numerically integrating Hamilton's equations for a given set of initial conditions. Instead of analytically enforcing the Hamiltonian constraint, the value of the energy of each plotted point is checked against the corresponding initial value of H. Once the plot is realized, all the Maple facilities for handling plots (such as reorientation, perspective, etc.) are available.

Complementary symbolic studies in the same environment can be developed with the commands of the PDEtools package for the analytical solving of scalar PDEs (the corresponding Hamilton-Jacobi equation), and for changing variables.

The following is a list of available commands.

A brief description of each command follows.

hamilton_eqs receives a Hamiltonian and returns Hamilton's equations and a list with the p's and q's involved.

poincare receives a Hamiltonian, a time range, and initial conditions, and returns a 2-D plot of a projection of the surface-of-section (2PS); or a 3-D plot of 2PS embedded in a related 3-D projection of the solution curves (3PS). In both the 2-D and 3-D plots, the 2-D plane over which the 2PS is plotted can be any of the coordinate planes of the phase space, and can optionally be shifted in the positive or negative directions.

generate_ic generates a set of lists of initial conditions for plotting a 2PS/3PS from a given incomplete set of fixed values and/or ranges for the p's, q's, and the energy. This command speeds up the time-consuming task of setting appropriate initial conditions for performing numerical experiments.

zoom allows for changing the ranges of the display of a given 2-D or 3-D plot, without having to recalculate it (thus saving time and memory resources).

Cheb-Terrab, E. S., and de Oliveira, H. P. "Poincare Sections of Hamiltonian Systems." Computer Physics Communications, Vol. 95, (1996): 171.