Relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork–Hopf bifurcation

Abstract

The traditional geometric singular perturbation theory and the slow–fast analysis method cannot be directly used to explore the mechanism of the relaxation oscillations in the forced dynamical system in which the exciting frequency is far less than the natural frequency. Furthermore, higher codimensional bifurcations may result in more complicated alternations between the large-amplitude oscillations (spiking states, SPs) and small-amplitude oscillations or at rest (quiescent states, QSs) on the trajectories of the bursting oscillations, called also the mixed mode oscillations. For the purpose, here we present a method to explore the mechanism of the bursting oscillations in the excited oscillator. Without loss of generality, we apply the proposed method to explore the dynamics of the normal form of the vector field with codimension-two pitchfork–Hopf bifurcation at the origin. When the slow-varying parametric excitation is introduced, with the increase in exciting amplitude, different types of bursting oscillations can be observed, the mechanism of which can be obtained by employing the overlap of the transformed phase portraits and the equilibrium branches as well as the bifurcations of the generalized autonomous system.

Appendix

$$\begin{aligned} \kappa&=1/[(ZS^2+ZS-1) w],\nonumber \\ b_2&=-\kappa [(5 ZS^6+10 ZS^5+(-3 w-10) ZS^4 \nonumber \\&\qquad +(-w-20) ZS^3 +(-w^2+6 w+15) ZS^2 \nonumber \\&\qquad +(-w^2-w+10) ZS+w^2-w-10)],\nonumber \\ b_1&=-\kappa ^3 [-7 ZS^{12}-28 ZS^{11}+12 ZS^{10} w \nonumber \\&\qquad +(32 w+112) ZS^9+(2 w^2-28 w+42) ZS^8 \nonumber \\&\qquad +(12 w^2-104 w-224) ZS^7\nonumber \\&\qquad +(-3 w^3+9 w^2+48 w-56) ZS^6\nonumber \\&\qquad +(-5 w^3-26 w^2+136 w+280) ZS^5\nonumber \\&\qquad +(6 w^3-12 w^2-76 w-7) ZS^4 \nonumber \\&\qquad +(7 w^3+28 w^2-72 w-196) ZS^3\nonumber \\&\qquad +(-6 w^3-6 w^2+52 w+56) ZS^2\nonumber \\&\qquad +(-w^3-10 w^2+8 w+56) ZS\nonumber \\&\qquad +w^3+5 w^2-8 w-28],\nonumber \\ b_0&=-\kappa ^3 [3 ZS^{18}+18 ZS^{17}-9 ZS^{16} w+18 ZS^{16}\nonumber \\&\qquad -43 ZS^{15} w-ZS^{14} w^2-84 ZS^{15}-14 ZS^{14} w\nonumber \\&\qquad -17 ZS^{13} w^2+3 ZS^{12} w^3-153 ZS^{14}+201 ZS^{13} w\nonumber \\&\qquad -52 ZS^{12} w^2+11 ZS^{11} w^3+198 ZS^{13}+167 ZS^{12} w \nonumber \\&\qquad +8 ZS^{11} w^2-5 ZS^{10} w^3+3 ZS^9 w^4+462 ZS^{12}\nonumber \\&\qquad -482 ZS^{11} w+180 ZS^{10} w^2-50 ZS^9 w^3+9 ZS^8 w^4\nonumber \\&\qquad -360 ZS^{11}-396 ZS^{10} w+44 ZS^9 w^2-3 ZS^8 w^3\nonumber \\&\qquad -ZS^7 w^4-819 ZS^10+794 ZS^9 w-299 ZS^8 w^2\nonumber \\&\qquad +95 ZS^7 w^3-18 ZS^6 w^4+582 ZS^9+433 ZS^8 w\nonumber \\&\qquad -28 ZS^7 w^2-6 ZS^6 w^3+918 ZS^8-907 ZS^7 w\nonumber \\&\qquad +296 ZS^6 w^2-99 ZS^5 w^3+14 ZS^4 w^4-756 ZS^7\nonumber \\&\qquad -174 ZS^6 w-61 ZS^5 w^2+25 ZS^4 w^3-3 ZS^3 w^4\nonumber \\&\qquad -603 ZS^6+641 ZS^5 w-143 ZS^4 w^2+46 ZS^3 w^3\nonumber \\&\qquad -3 ZS^2 w^4+666 ZS^5-67 ZS^4 w+76 ZS^3 w^2\nonumber \\&\qquad -17 ZS^2 w^3+ZS w^4+162 ZS^4-232 ZS^3 w\nonumber \\&\qquad +13 ZS^2 w^2-7 ZS w^3-336 ZS^3+72 ZS^2 w\nonumber \\&\qquad -22 ZS w^2+3 w^3+36 ZS^2+28 ZS w+6 w^2\nonumber \\&\qquad +72 ZS-12 w-24]. \end{aligned}$$

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