In this letter we deal with the Hopf bifureation from the equilibrium x=0 for Hamiltonian wquations x=JH’（x）,x∈R2n,where J=■ A well-known fact is that for existing bifurcation of periodic orbits from the x=0, it ...In this letter we deal with the Hopf bifureation from the equilibrium x=0 for Hamiltonian wquations x=JH’（x）,x∈R2n,where J=■ A well-known fact is that for existing bifurcation of periodic orbits from the x=0, it is necessary to have eigenvalue of JH'（0） on the imaginary axis. If det H'（0）≠0, under additional condition about the odevity of the purely imaginary eigenvalues, （H） has periodic orbits bifurcating from x=0 as has been proved by J. C. Alexander. Without the restriction on the odevity we can reach the same end.展开更多
§ 1. IntroductionIn the present paper, we study following Hamiltonian system of Second-order:(1) with the boundary conditionx (0) = x (2jr) , x’ (0) = x’ (2w) ,(2)where p(t) ￡O(R, Rn"), G(t, x) ￡O(RxR...§ 1. IntroductionIn the present paper, we study following Hamiltonian system of Second-order:(1) with the boundary conditionx (0) = x (2jr) , x’ (0) = x’ (2w) ,(2)where p(t) ￡O(R, Rn"), G(t, x) ￡O(RxR", R). P(-) and &( , x) are 25F-periodio functions. Vj, or G’x(t, a/) will denote the gradient with respect to x. Moreover, we shall always assume that G’x(t, x) is continuous.展开更多
文摘In this letter we deal with the Hopf bifureation from the equilibrium x=0 for Hamiltonian wquations x=JH’（x）,x∈R2n,where J=■ A well-known fact is that for existing bifurcation of periodic orbits from the x=0, it is necessary to have eigenvalue of JH'（0） on the imaginary axis. If det H'（0）≠0, under additional condition about the odevity of the purely imaginary eigenvalues, （H） has periodic orbits bifurcating from x=0 as has been proved by J. C. Alexander. Without the restriction on the odevity we can reach the same end.
文摘§ 1. IntroductionIn the present paper, we study following Hamiltonian system of Second-order:(1) with the boundary conditionx (0) = x (2jr) , x’ (0) = x’ (2w) ,(2)where p(t) ￡O(R, Rn"), G(t, x) ￡O(RxR", R). P(-) and &( , x) are 25F-periodio functions. Vj, or G’x(t, a/) will denote the gradient with respect to x. Moreover, we shall always assume that G’x(t, x) is continuous.
