Dynamics of a generalized Rayleigh system (Q6568973)

Consider the generalized Rayleigh system\N\[\N\frac{{dx}}{{dt}} = y+a(x^{2n}+1)x, \quad \frac{{dy}}{{dt}} = -y,\tag{1}\N\]\Nwhere \( n\) is a positive integer and \(a\) is a real number. The authors prove that system (1) has a unique limit cycle for \( a \neq 0 \) which is stable (unstable) for \( a < 0\) (\(a > 0\) ). Additionally they determine the global phase portrait in the Poincaré disk.