Numerical bifurcation analysis of a class of nonlinear renewal equations

• Main Authors: Dimitri Breda, Odo Diekmann, Davide Liessi, Francesca Scarabel
• Format: Article
• Language: English
• Published: University of Szeged 2016-09-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: renewal equations; structured populations; stability of periodic solutions; period doubling cascade; numerical continuation and bifurcation; pseudospectral and collocation methods; kaplan-yorke periodic orbits
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=5273

We show, by way of an example, that numerical bifurcation tools for ODE yield reliable bifurcation diagrams when applied to the pseudospectral approximation of a one-parameter family of nonlinear renewal equations. The example resembles logistic- and Ricker-type population equations and exhibits transcritical, Hopf and period doubling bifurcations. The reliability is demonstrated by comparing the results to those obtained by a reduction to a Hamiltonian Kaplan-Yorke system and to those obtained by direct application of collocation methods (the latter also yield estimates for positive Lyapunov exponents in the chaotic regime). We conclude that the methodology described here works well for a class of delay equations for which currently no tailor-made tools exist (and for which it is doubtful that these will ever be constructed).