Rigorous verification of period-doubling bifurcations in maps

Abstract

We present a computer-assisted approach for proving period-doubling bifurcations in one-parameter families of maps $x \mapsto f(x,λ)$ on $\mathbb{R}^n$. We solve a square augmented system, called the period-doubling map, whose unknowns are the fixed point, the parameter, the right eigenvector attached to the eigenvalue $-1$, the left eigenvector with its eigenvalue, and the vector which describes the center manifold to second order. A contraction argument gives a solution close to a numerical approximation, at which the Jacobian is invertible. The determinant of this Jacobian factors into a simple expression, so that it is nonzero exactly when $1$ is not an eigenvalue and $-1$ is a simple eigenvalue crossing the unit circle transversally. Apart from the sign of the cubic normal-form coefficient, computed afterwards, it remains to check with Gershgorin disks that no other eigenvalue lies on the unit circle. The method is applied to a $10$-mode Galerkin projection of the Kot--Schaffer growth-dispersal model, for which we prove a supercritical period-doubling bifurcation.