On the periodic orbits of the third-order differential equation \(x^{\prime\prime\prime}-\mu x^{\prime\prime}+x^{\prime }-\mu x=\varepsilon F(x, x^{\prime },x^{\prime\prime})\) (Q1940681)

Consider the differential equation \[ x'''-\mu x''+ x'-\mu x=\varepsilon F(x,x',x''),\tag{\(*\)} \] where \(\mu\) and \(\varepsilon\) are real parameters, \(|\varepsilon|\) small, \(F\) is a \(C^2\)-function. After some transformations and introducing cylindrical coordinates, equation \((*)\) is equivalent to the system \[ r'=\varepsilon G(r,\theta,Z)\cos\theta,\quad \theta'= 1-\varepsilon G(r,\theta,Z)/r,\quad Z'= \mu Z-\varepsilon G(r,\theta,Z) \] which is for sufficiently small \(\varepsilon\) equivalent to the non-autonomous system \[ \begin{aligned} {dr\over d\theta} &=\varepsilon G(r,\theta,Z)\cos\theta+ \theta(\varepsilon^2),\\ {dZ\over d\theta} &=\mu Z+\varepsilon {\mu Z\sin\theta- r\over r} G(r,\theta,Z)+ 0(\varepsilon^2).\end{aligned}\tag{\(**\)} \] Applying averaging theory to \((**)\), the authors derive results on the existence of periodic solutions for sufficiently small \(\varepsilon\) bifurcating from special periodic solutions of the unperturbed \((\varepsilon= 0)\) system. They distinguish the cases \(\mu= 0\) and \(\mu\neq 0\) in the unperturbed system.