PDDE-CONT

PDDE-CONT can continue periodic solutions of autonomous and periodically forced delay equations in the general form

dx(t)/dt = f(t, x(t-τ₀), x(t-τ₁), ..., x(t-τ_r), λ).

To start the continuation, the software needs an initial periodic solution x₀(t) at some parameter value λ₀. This starting periodic solution has to be specified either in the system definition (C++ source file) or it can be loaded from an input file. The periodic solutions can bifurcate in several ways. If one of the three common codimension-one bifurcations (fold, period doubling, Neimark-Sacker) is found along the branch of periodic solutions, the point can be used as a starting point for continuing the branch of bifurcation points in a two parameter space.

Fixed points of autonomous systems can be also handled, as periodic solutions of periodic systems by keeping the period constant. Although it is very inefficient, this capability might be useful, when looking for periodic orbits arising at Hopf bifurcations points. Hopf bifurcations of fixed points are detected as Neimark-Sacker bifurcations, and the periodic solutions emanating from such points can be continued by switching to the periodic solution branch.