Optimization of Hopf bifurcation points

TL;DR

This paper presents a numerical optimization method to precisely control the location, stability, and oscillation frequency of Hopf bifurcations in various dynamical systems, with applications in biology and fluid dynamics.

Contribution

It introduces a novel optimization-based approach for controlling Hopf bifurcation points and their properties in complex systems.

Findings

Successfully controlled bifurcation points in biological and fluid systems.

Demonstrated robustness and flexibility of the method across different models.

Enabled targeted manipulation of oscillation frequencies.

Abstract

We introduce a numerical technique for controlling the location and stability properties of Hopf bifurcations in dynamical systems. The algorithm consists of solving an optimization problem constrained by an extended system of nonlinear partial differential equations that characterizes Hopf bifurcation points. The flexibility and robustness of the method allows us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency with respect to a parameter of the system or the shape of the domain on which solutions are defined. Numerical applications are presented in systems arising from biology and fluid dynamics, such as the FitzHugh--Nagumo model, Ginzburg--Landau equation, Rayleigh--B\'enard convection problem, and Navier--Stokes equations, where the control of the location and oscillation frequency of periodic solutions is of high interest.