A HIERARCHY OF LIOUVILLE INTEGRABLE FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

A hierarchy of Liouville integrable finite-dimensional Hamiltonian systems whose Hamiltonian phase flows commute with each other is generated and an infinite number of involutive explicit common integrals of motion and a set of its involutive explicit generators are given.

An Eight Component Integrable Hamiltonian Hierarchy from a Reduced Seventh-Order Matrix Spectral Problem

We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.

A New Finite-dimensional Integrable System Associated to(1+1)-dimensional Soliton Equations

In this paper, a new spectral problem is proposed and the corresponding soliton equations hierarchy are also obtained. Under a constraint between the potentials and the eigenfunctions, the eigenvalue problem is nonlinearized so as to be a new finite-dimensional Hamiltonian system. By resotring to the generating function approach, we obtain conserved integrals and the involutivity of the conserved integrals. The finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Finally, we show the decomposition of the soliton equations.

An integrable Hamiltonian hierarchy and associated integrable couplings system

This paper establishes a new isospectral problem. By making use of the Tu scheme, a new intcgrablc system is obtained. It gives integrable couplings of the system obtained. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.

Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure

By using a Lie algebra, an integrable couplings of the classicai-Boussinesq hierarchy is obtained. Then, the Hamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-form identity.

Multi-component KN Hierarchy and Associated two Integrable Couplings as Well as Their Hamiltonian Structure

Firstly, a vector loop algebra G3 is constructed, by use of it multi-component KN hierarchy is obtained. Further, by taking advantage of the extending vector loop algebras G6 and G9 of G3 the double integrable couplings of the multi-component KN hierarchy are worked out respectively. Finally, Hamiltonian structures of obtained system are given by quadratic-form identity.

STOCHASTIC HOPF BIFURCATION IN QUASIINTEGRABLE-HAMILTONIAN SYSTEMS

A new procedure is developed to study the stochastic Hopf bifurcation in quasi- integrable-Hamiltonian systems under the Gaussian white noise excitation.Firstly,the singular bound- aries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system's energy levels with respect to the stochastic aver- aging method.Secondly,the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones.Lastly,a quasi-integrable- Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure. Moreover,simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure.It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system's parameters.Therefore,one can see that the numerical results are consistent with the theoretical predictions.

A NEW COMPLETELY INTEGRABLE LIOUVILLE' S SYSTEM, ITS LAX REPRESENTATION AND BI-HAMILTONIAN STRUCTURE

A new isospectral problem and the corresponding hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known MKdV equation is obtained. It is shown that the hierarchy of equations is integrable in Liouville' s sense and possesses Bi-Hamiltonian structure. Under the constraint between the potentials and eigenfunctions, the eigenvalue problem can be nonlinearized as a finite dimensional completely integrable system.

FIRST-PASSAGE TIME OF QUASI-NON-INTEGRABLE-HAMILTONIAN SYSTEM

Studies on first-passage failure are extended to the multi-degree-of-freedom quasi-non-integrable-Hamiltonian systems under parametric excitations of Gaussian white noises in this paper. By the stochastic averaging method of energy envelope, the system's energy can be modeled as a one-dimensional approximate diffusion process by which the classical Pontryagin equation with suitable boundary conditions is applicable to analyzing the statistical moments of the first-passage time of an arbitrary order. An example is studied in detail and some numerical results are given to illustrate the above procedure.

STOCHASTIC OPTIMAL CONTROL FOR THE RESPONSE OF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS

A strategy is proposed based on the stochastic averaging method for quasi non- integrable Hamiltonian systems and the stochastic dynamical programming principle.The pro- posed strategy can be used to design nonlinear stochastic optimal control to minimize the response of quasi non-integrable Hamiltonian systems subject to Gaussian white noise excitation.By using the stochastic averaging method for quasi non-integrable Hamiltonian systems the equations of motion of a controlled quasi non-integrable Hamiltonian system is reduced to a one-dimensional av- eraged It stochastic differential equation.By using the stochastic dynamical programming princi- ple the dynamical programming equation for minimizing the response of the system is formulated. The optimal control law is derived from the dynamical programming equation and the bounded control constraints.The response of optimally controlled systems is predicted through solving the FPK equation associated with It stochastic differential equation.An example is worked out in detail to illustrate the application of the control strategy proposed.

The Liouville integrable coupling system of the m-AKNS hierarchy and its Hamiltonian structure

In this paper a type of 9-dimensional vector loop algebra F is constructed, which is devoted to establish an isospectral problem. It follows that a Liouville integrable coupling system of the m-AKNS hierarchy is obtained by employing the Tu scheme, whose Hamiltonian structure is worked out by making use of constructed quadratic identity. The method given in the paper can be used to obtain many other integrable couplings and their Hamiltonian structures.

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedintegrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity.Finally,we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discreteHamiltonian systems.

Nonlinear Super Integrable Couplings of A Super Integrable Hierarchy and Its Super Hamiltonian Structures

Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction,special cases of this nonlinear super integrable couplings were obtained.

New DLW Hierarchy of an Integrable Coupling and Its Hamiltonian Structure

A type of higher-dimensionaJ loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic identity.

Integrable Coupling of KN Hierarchy and Its Hamiltonian Structure

The Hamiltonian structure of.the integrable couplings obtained by our method has not been solved. In this paper, the Hamiltonian structure of the KN hierarchy is obtained by making use of the quadratlc-form identity.

Nonlinear Super Integrable Couplings of Super Yang Hierarchy and Its Super Hamiltonian Structures

Nonlinear super integrable couplings of the super Yang hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of Yang hierarchy were obtained.

NEW METHOD FOR THE CONSTRUCTION OF INTEGRABLE HAMILTONIAN SYSTEMS

A new method.for the construction of integrable Hamiltonian system is proposed.For a given Poisson manifold the present paper constructs new Poisson brackets on it by making use of the Dirac-Poisson structure[1],and obtains .further new integrable Hamiltonian systems The constructed Poisson bracket is usual non-linear, and this new method is also different from usual ones[2-4].Two examples are given.

Integrable Coupling of (2+1)-Dimensional Multi-component DLW Integrable Hierarchy and Its Hamiltonian Structure

A new multi-component Lie algebra is constructed, and a type of new loop algebra is presented. A （2＋1）-dimensional multi-component DLW integrable hierarchy is obtained by using a （2＋1）-dimensional zero curvature equation. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system and its corresponding Hamiltonian structure are worked out.

Lie Algebras for Constructing Nonlinear Integrable Couplings

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti- Johnson （G J） hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their ttamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.

GENERALIZED FRACTIONAL TRACE VARIATIONAL IDENTITY AND A NEW FRACTIONAL INTEGRABLE COUPLINGS OF SOLITON HIERARCHY

Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.