At last , the stability switches of equilibrium and the global diagrams of bifurcation for a duffing oscillator with both delayed velocity feedback and delayed displacement feedback are studied by using the similar method as above 最后,研究了時滯位移反饋與時滯速度反饋聯(lián)合作用下duffing系統(tǒng)的平衡點的穩(wěn)定性切換及全局hopf分叉。
Afterwards , a duffing system with delayed displacement feedback is studied by the method of multiple scales and other numerical methods . the uniform formulas for computing the critical values of time delay are given and the global diagrams of bifurcation for the periodic solutions with respect to the time delay are obtained under different parametric combinations . it is shown that the hopf bifurcation and the saddle - node bifurcation are the only two types of bifurcation observed in such a system 接下來用多尺度法及數(shù)值方法研究了一具有時滯位移反饋的duffing系統(tǒng)的動力學(xué),得到了不同參數(shù)下系統(tǒng)平衡點發(fā)生穩(wěn)定性切換時的臨界時滯計算公式及關(guān)于時滯的大范圍hopf分叉圖,并發(fā)現(xiàn)saddle - node分叉及hopf分叉是系統(tǒng)出現(xiàn)周期運動的兩個主要的來源。
Formulas for computing the critical values of time delay are also obtained after a detailed analysis for the system parameters . it is surprisingly shown that the combination of both displacement feedback and velocity feedback seems to induce no more bifurcation characteristics than one type of the feedback only 根據(jù)對系統(tǒng)參數(shù)的適當(dāng)劃分,給出了在不同參數(shù)組合下系統(tǒng)平衡點發(fā)生穩(wěn)定性切換時的臨界時滯計算公式,并發(fā)現(xiàn)系統(tǒng)的分叉只是包含系統(tǒng)僅有其中一種時滯反饋類型時的分叉類型,而并沒有出現(xiàn)新的分叉類型。