Generalized Henon-Heiles System Research Articles

Separation of variables for the generalized Henon–Heiles system and system with quartic potential

We consider two well-known integrable systems on the plane using the concept of natural Poisson bivectors on Riemannian manifolds. A geometric approach to construction of variables of separation and separated relations for the generalized Henon–Heiles system and the generalized system with quartic potential is discussed in detail.

Adaptive synchronization between two different hyperchaotic systems

This work presents chaos synchronization between two different hyperchaotic systems using adaptive control. The sufficient conditions for achieving synchronization of two high dimensional chaotic systems are derived based on Lyapunov stability theory, and an adaptive control law and a parameter update rule for unknown parameters are given such that generalized Henon–Heiles system is controlled to be hyperchaotic Chen system. Theoretical analysis and numerical simulations are shown to verify the results.

A connection between the Henon-Heiles system and the Garnier system

A remarkable map between the generalized Henon-Heiles system and the Garnier system is obtained by means of a detailed comparison between two finite-dimensional reduction methods for soliton equations: thestationary flows and therestricted flows. The role of the Gelfand-Dickey polynomials and of the KdV Poisson pencil in this construction is emphasized.

A generalized Henon–Heiles system and related integrable Newton equations

A detailed description is given of integrable cases of the generalized Henon–Heiles systems which differs from the standard H–H ones by the term α/q22. Their connection with fifth-order one-component soliton equations is discussed. Lax representations are constructed, and the bi-Hamiltonian formulation of dynamics is given. It is also shown that the gH–H system can be mapped onto another system of Newton equations with a nonstandard Hamiltonian structure.

Elliptic Baker–Akhiezer functions and an application to an integrable dynamical system

An ansatz for elliptic Baker–Akhiezer functions is used to find new solutions for various spectral problems—the Sturm–Liouville problem with four and five-gap Lamé potentials and two third-order problems. One of the third-order problems is the classical Halphen equation with a spectral parameter. The eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized Hènon–Heiles system.

Improved accuracy of the Birkhoff-Gustavson normal form and its convergence properties

The authors study the generalized Henon-Heiles system by systematic exploration and analysis of the Birkhoff-Gustavson normal form and the associated formal integral of motion up to and including the 14th order, which is one order higher than published before. At low energies the formal integral of motion is still an excellent approximation to the exact integral. The convergence properties of the formal integral have been analysed in regular and irregular regions. Strictly convergent behaviour is found in some regions of chaotic motion. No obvious example of divergent behaviour is found. Regions of strict convergence correspond either to regular motion, or to weakly unstable chaotic motion with short-time clustering, characterized by a small value of the finite-time analogue of the Lyapunov exponent.

Mixmaster cosmological models as disturbed Toda lattices

We show that Mixmaster cosmological models belong to the class of disturbed Toda lattices. The original method of investigating chaos in Hamiltonian systems is applied to Mixmaster models by truncation at orders i⩽7. The third order system is equivalent to the generalized Henon-Heiles system. We also show that already the third order approximation exhibits all the qualitative features of Mixmaster models.