Hopf bifurcation phenomenon in catalytic reaction (Q1059764)

In catalytic reaction, the flickering equation is \[ (1)\quad du/dt=\gamma u-\omega v+\epsilon \delta_ 1\sin \omega_ 0t+\epsilon ru \sin \omega_ 0t-2u^ 3 \] \[ dv/dt=\omega u+\gamma v+\epsilon \delta_ 2\sin \omega_ 0t+\epsilon \lambda v \sin \omega_ 0t+2u^ 2v. \] In the non-resonance case (\(\omega\) is far from \(\omega_ 0)\), the frequency and amplitude of the perturbation term do not cause any effect on Hopf bifurcation at \(\gamma =0\). In the resonance case (\(\omega\) is close to \(\omega_ 0)\), by means of the multi-scales method, equation (1) can be reduced to \[ (2)\quad dR/d\tau =\sigma R- R^ 3-\beta \sin \theta,\quad R(d\theta /d\tau)=R-\beta \cos \theta. \] We carefully analyse (2) and draw some conclusions as follows: (1) Because of the existence of the s term, Hopf bifurcation at \(\gamma =0\) removes \(O(\epsilon^{2/3})\) towards the right side. (2) When \(\gamma <\gamma_ 0\), the solution of (1) is a periodic solution and its magnitude is \(O(\epsilon^{1/3})\); when \({}^ 4\sqrt{64/27}<\beta <\sqrt{2}\), perhaps there are two equilibrium periodic solutions. (3) When \(\gamma >\gamma_ 0\), the solution of (1) is an almost periodic solution and its magnitude is \(O(\epsilon^{1/3})\).