A hierarchy of multidimensional Hénon-Heiles(M-H-H)systems are constructed via the x-and t_n-higher-order-constrained flows of KdV hierarchy.The Lax representation for the M-H-H hierarchy is determined from the a...A hierarchy of multidimensional Hénon-Heiles(M-H-H)systems are constructed via the x-and t_n-higher-order-constrained flows of KdV hierarchy.The Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV hierarchy.By using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed.展开更多
Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians H = （p1^2 ＋ p2^2）/2 ＋ V（q1, q2） whose some integer power qj^nj （t）...Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians H = （p1^2 ＋ p2^2）/2 ＋ V（q1, q2） whose some integer power qj^nj （t） of the general solution is a singlevalued function of the complez time t. In addition to the well known rational potentials V of Hénon-Heiles, this selects possible cases with a trigonometric dependence of V on qj. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven （three “cubic” plus four “quartic”） rational Hénon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.展开更多
Investigating local dynamics of equilibrium points of nonlinear systems plays an important role in studying the behavior of dynamical systems. There are many different definitions for stable and unstable solutions in ...Investigating local dynamics of equilibrium points of nonlinear systems plays an important role in studying the behavior of dynamical systems. There are many different definitions for stable and unstable solutions in the literature. The main goal to develop stability definitions is exploring the responses or output of a system to perturbation as time approaches infinity. Due to the wide range of application of local dynamical system theory in physics, biology, economics and social science, it still attracts many researchers to play with its definitions to find out the answers for their questions. In this paper, we start with a brief review over continuous time dynamical systems modeling and then we bring useful examples to the playground. We study the local dynamics of some interesting systems and we show the local stable behavior of the system around its critical points. Moreover, we look at local dynamical behavior of famous dynamical systems, Hénon-Heiles system, Duffing oscillator and Van der Pol equation and analyze them. Finally, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples.展开更多
基金Supported by National Research Project "Nonlinear Sciences"
文摘A hierarchy of multidimensional Hénon-Heiles(M-H-H)systems are constructed via the x-and t_n-higher-order-constrained flows of KdV hierarchy.The Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV hierarchy.By using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed.
文摘Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians H = （p1^2 ＋ p2^2）/2 ＋ V（q1, q2） whose some integer power qj^nj （t） of the general solution is a singlevalued function of the complez time t. In addition to the well known rational potentials V of Hénon-Heiles, this selects possible cases with a trigonometric dependence of V on qj. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven （three “cubic” plus four “quartic”） rational Hénon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.
文摘Investigating local dynamics of equilibrium points of nonlinear systems plays an important role in studying the behavior of dynamical systems. There are many different definitions for stable and unstable solutions in the literature. The main goal to develop stability definitions is exploring the responses or output of a system to perturbation as time approaches infinity. Due to the wide range of application of local dynamical system theory in physics, biology, economics and social science, it still attracts many researchers to play with its definitions to find out the answers for their questions. In this paper, we start with a brief review over continuous time dynamical systems modeling and then we bring useful examples to the playground. We study the local dynamics of some interesting systems and we show the local stable behavior of the system around its critical points. Moreover, we look at local dynamical behavior of famous dynamical systems, Hénon-Heiles system, Duffing oscillator and Van der Pol equation and analyze them. Finally, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples.
