Spurious limit-cycles (Q2784722)

Essentially this paper is concerned with the existence of periodic solutions to the averaged form of the differential equation NEWLINE\[NEWLINE\dot x=\varepsilon f(t,x,\varepsilon)= \varepsilon[f_1(t, x)+|\varepsilon|^p f_2(t, x,\varepsilon)],\quad \varepsilon> 0,\;p>0,NEWLINE\]NEWLINE where \(x\in\mathbb{R}^n\) and \(f\) is \(T\)-periodic in \(t\). Under stated conditions, several theorems are proved which determine the parameter ranges for \(\varepsilon\) for the existence of periodic solutions. Several applications to linear and nonlinear equations are included. In particular, it is shown that the quadratic system NEWLINE\[NEWLINE\dot x= y-\varepsilon xz,\quad \dot y= -x-\varepsilon yz,\quad \dot z= \varepsilon(2x^2- 1+ z),NEWLINE\]NEWLINE has a limit cycle for \(0< |\varepsilon|\leq 1/(21\pi)\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00016].