Here is some information on software that I frequently use. More dynamical systems software can be found at DSWeb.
MatCont & MatCont for Maps
MatCont is a Matlab software project for the numerical continuation and bifurcation analysis of continuous and discrete parameter-dependent dynamical systems. Download available at Sourceforge.
DsTool
DsTool is a toolkit for the interactive investigation of dynamical systems. It includes a software module for the computation of one-dimensional stable manifolds of two-dimensional discrete systems without requiring explicit or approximate knowledge of the inverse map. Examples at www.geom.uiuc.edu/software/dstool, download available at www.math.cornell.edu/~gucken/software.html.
AUTO
AUTO is a software package for continuation and bifurcation problems in ordinary differential equations. The package includes HomCont, algorithms for the bifurcation analysis of homoclinic orbits. Description at cmvl.cs.concordia.ca/auto, download available at Sourceforge.