Zero-Hopf Periodic Orbit of a Quadratic System of Differential Equations Obtained from a Third-Order Differential Equation

Abstract

We study the zero-Hopf bifurcation of the third-order differential equations $$\\begin{aligned} x^{\\prime \\prime \\prime }+ (a_{1}x+a_{0})x^{\\prime \\prime }+ (b_{1}x+b_{0})x^{\\prime }+x^{2} =0, \\end{aligned}$$x″′+(a1x+a0)x″+(b1x+b0)x′+x2=0,where $$a_{0}$$a0, $$a_{1}$$a1, $$b_{0}$$b0 and $$b_{1}$$b1 are real parameters. The prime denotes derivative with respect to an independent variable t. We also provide an estimate of the zero-Hopf periodic solution and their kind of stability. The Hopf bifurcations of these differential systems were studied in [5], here we complete these studies adding their zero-Hopf bifurcations.