Hopf Bifurcation of a Positive Feedback Delay Differential Equation  

作　　者：陈玉明 黄立宏 

机构地区：[1]Department of Mathematics, Wilfrid Laurier University Waterloo, Ontario, N2L 3C5, Canada 0. Combining this with a unique result of Krisztin and Walther, we know that this periodic orbit is the one branched out from 0 through Hopf bifurcation. Using the normal form theory for delay differential equations, we show the same result under the condition that f ∈C3(R,R) is such that f''(0) = 0 and f'''(0) < 0, which is weaker than those of Krisztin and Walther.'>[2]College of Mathematics and Econometrics, Hunan University, Changsha, 410082nder some minor technical hypotheses, for each T larger than a certain rS > 0, Krisztin, Walther and Wu showed the existence of a periodic orbit for the positive feedback delay differential equation x(t) = -rμx(t) + rf(x(t-1)), where r and μare positive constants and f : R →R satisfies f(0) = 0 and f' > 0. Combining this with a unique result of Krisztin and Walther, we know that this periodic orbit is the one branched out from 0 through Hopf bifurcation. Using the normal form theory for delay differential equations, we show the same result under the condition that f ∈C3(R,R) is such that f''(0) = 0 and f'''(0) < 0, which is weaker than those of Krisztin and Walther.

出　　处：《Northeastern Mathematical Journal》2003年第3期213-223,共11页东北数学（英文版）

基　　金：The start-up funds of Wilfrid Laurier University of Canada, the NNSF (10071016) of China;the Doctor Program Foundation (20010532002) of Chinese Ministry of Education; the Key Project of Chinese Ministry of Education ([2002]78) and the

摘　　要：Under some minor technical hypotheses, for each T larger than a certain rS > 0, Krisztin, Walther and Wu showed the existence of a periodic orbit for the positive feedback delay differential equation x(t) = -rμx(t) + rf(x(t-1)), where r and μ are positive constants and f : R → R satisfies f(0) = 0 and f' > 0. Combining this with a unique result of Krisztin and Walther, we know that this periodic orbit is the one branched out from 0 through Hopf bifurcation. Using the normal form theory for delay differential equations, we show the same result under the condition that f ∈ C3(R,R) is such that f''(0) = 0 and f'''(0) < 0, which is weaker than those of Krisztin and Walther.Under some minor technical hypotheses, for each T larger than a certain rS > 0, Krisztin, Walther and Wu showed the existence of a periodic orbit for the positive feedback delay differential equation x(t) = -rμx(t) + rf(x(t-1)), where r and μ are positive constants and f : R → R satisfies f(0) = 0 and f' > 0. Combining this with a unique result of Krisztin and Walther, we know that this periodic orbit is the one branched out from 0 through Hopf bifurcation. Using the normal form theory for delay differential equations, we show the same result under the condition that f ∈ C3(R,R) is such that f''(0) = 0 and f'''(0) < 0, which is weaker than those of Krisztin and Walther.

关 键 词：delay differential equation positive feedback Hopf bifurcation 

分 类 号：O175[理学—数学]