Effective computation of periodic orbits and bifurcation diagrams in delay equations

Abstract

Summary. By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differentialdifference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

“…The software package XPPAUT [13] has some capabilities for delay equations, but this does not include continuation of periodic solutions. Certain numerical continuation schemes have been developed, see for example [15,10,28] and a first generally available continuation package has recently appeared [11]. For more progress in this direction see [14,24,12].…”

Oscillatory modes in a nonlinear second-order differential equation with delay

“…These periodic solutions are of simple type, i.e., they have exactly one maximum per period. Analytic results (Walther, 1981a) and numerical simulations (Mackey and Glass, 1977;Hadeler, 1980;Saupe, 1982;Longtin and Milton, 1989a) support evidence that these periodic solutions are stable limit cycles. On the other hand, when f is a nonmonotone function, (3) can exhibit very complex dynamics including period-doubling bifurcations and chaos, as has been shown both numerically (Mackey and Glass, 1977;Glass and Mackey, 1979) and analytically (Peters, 1980;Walther, 1981b;an der Heiden and Mackey, 1982;an der Heiden and Walther, 1983).…”

Effective approaches to explore rich dynamics of delay-differential equations

“…(E*) was observed in [11] whereas Hopf bifurcation to the left with an unstable limit cycle (Backwards [2,1] or subcritical Bifurcation) and a unique stable limit cycle then only a stable limit cycle without breaking the special symmetry (x(t + b) = Àx(t)) for Eq. (1.1) and much richer dynamical behavior for Eq.…”