Hopf bifurcation of uncertain nonlinear differential equations

It is reasonable to use uncertain nonlinear differential equations to model systems that are subject to noise disturbances of unstable frequency, such as uncertain financial systems, biological population systems, infectious disease systems, and pharmacokinetic systems. Due to the coupling effect of nonlinearity and uncertainty, it is challenging to directly solve the responses of these equations. This paper addressed the challenge of studying the responses of these systems, especially the changes in steady-state behavior caused by parameter variations, known as "bifurcation" phenomena. We defined the concept of Hopf bifurcation in uncertain differential equations using cross-entropy and investigated the bifurcation phenomena in a class of second-order uncertain nonlinear differential equations. An efficient algorithm was designed to verify uncertain Hopf bifurcation and quantify the bifurcation threshold, with the validity of our definition confirmed through numerical simulations. This paper extended the classical Hopf bifurcation of ordinary differential equations to uncertain differential equations via the $\alpha$-path, thereby proposing a theoretical framework for uncertain bifurcation within uncertain dynamics.