Asymptotic behaviour and existence of a limit cycle of cubic autonomous systems (Q2757027)

The two-dimensional real autonomous system NEWLINE\[NEWLINE \dot{x}_1 =\alpha(a_0+a_1 x_1 +bx_2+ a_2 x_1^2 +a_3 x_1^3),\qquad \dot{x}_2=\beta(c_0 +c x_1 +d x_2),NEWLINE\]NEWLINE is studied. Here, \(\alpha,\beta >0\) are real parameters and \(a_0,a_1,b,a_2,a_3,c_0,c,d\) are real coefficients satisfying the following assumptions: \(b<0\), \(a_3<0\), \(c>0\), \(d<0\) and the quadratic equation \(a_1+2a_2x +3a_3x^2 =0\) has two distinct real roots. NEWLINENEWLINENEWLINEThe Hopf bifurcation is analyzed and the existence of a limit cycle is proved. A new formula to determine stability or instability of this limit cycle is introduced. A positively invariant set, which is globally attractive, is found. The existence of a stable limit cycle around an unstable critical point is proved and also a sufficient condition for the nonexistence of a closed trajectory in the phase space is given. Global characteristics of the system are also studied. An application to economics is presented.