A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of...A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. Bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.展开更多
For the quadratic system: x=-y+δx + lx2 + ny2, y=x(1+ax-y) under conditions -1<l<0,n+l - 1>0 the author draws in the (a, ()) parameter plane the global bifurcationdiagram of trajectories around O(0,0). Notic...For the quadratic system: x=-y+δx + lx2 + ny2, y=x(1+ax-y) under conditions -1<l<0,n+l - 1>0 the author draws in the (a, ()) parameter plane the global bifurcationdiagram of trajectories around O(0,0). Notice that when na2+l < 0 the system has one saddleN(0,1/n) and three anti-saddles.展开更多
By using the generalized PoincarE index theorem it is proved that if the n2 critical points of an n-polynomial system form a configuration of type (2n -1) - (2n - 3) +(2n- 5) -…+ (- 1 )n- 1, and the 2n -1 outmost ant...By using the generalized PoincarE index theorem it is proved that if the n2 critical points of an n-polynomial system form a configuration of type (2n -1) - (2n - 3) +(2n- 5) -…+ (- 1 )n- 1, and the 2n -1 outmost anti-saddles form the venices of a convex (2n -1)-polygon, then among these 2n-1 anti-saddles at least one must be a node.展开更多
基金Project(10672053) supported by the National Natural Science Foundation of ChinaProject(2002AA503010) supported by the National High-Tech Research and Development Program of China
文摘A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. Bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.
基金Project supported by the National Natural Science Foundation of China
文摘For the quadratic system: x=-y+δx + lx2 + ny2, y=x(1+ax-y) under conditions -1<l<0,n+l - 1>0 the author draws in the (a, ()) parameter plane the global bifurcationdiagram of trajectories around O(0,0). Notice that when na2+l < 0 the system has one saddleN(0,1/n) and three anti-saddles.
文摘By using the generalized PoincarE index theorem it is proved that if the n2 critical points of an n-polynomial system form a configuration of type (2n -1) - (2n - 3) +(2n- 5) -…+ (- 1 )n- 1, and the 2n -1 outmost anti-saddles form the venices of a convex (2n -1)-polygon, then among these 2n-1 anti-saddles at least one must be a node.
