Numerical study of some bifurcation diagrams arising from ordinary differential equations

Abstract

Loud systems are a family of reversible quadratic centers depending on a bidimensional parameter µ = (D, F ) ∈ R2 . Determining the bifurcation diagram of the period function by stating the exact number of critical periods for some regions of the space of parameters is still a matter of conjecture. As opposed to an analytical study, this work aims to tackle this question numerically with a view to provide solid evidence about the exact number of critical periods arising within these regions as well as analysing bifurcation curves. By compactifying the vector field of Loud systems to RP2 , the period annulus remains bounded for all values of µ within the regions of interest, avoiding periodic orbits to escape to infinity, which may propagate numerical errors. Combining both 4th-order Runge-Kutta methods and other strategies of numerical integration, the period function and, subsequently, the number of critical periods will be determined for each value of µ with the intention of sketching the bifurcation diagram and comparing the results obtained with those proven analytically. Thus, the notion of criticality -number of critical periods in the period function- resembles Hilbert’s 16th problem and the concept of cyclicity. Three different types of bifurcations will be identified: at the inner boundary of the period annulus, at the outer boundary and at the interior; and some conjectures will be verified in accordance with the limitation imposed by the computational cost of discretizing the space of parameters.