The form invariance and the Lie symmetry of the generalized Hamiltonian system are studied. Firstly, de?nitions and criteria of the form invariance and the Lie symmetry of the system are given. Next, the r...The form invariance and the Lie symmetry of the generalized Hamiltonian system are studied. Firstly, de?nitions and criteria of the form invariance and the Lie symmetry of the system are given. Next, the relation between the form invariance and the Lie symmetry is studied. Finally, two examples are given to illustrate the application of the results.展开更多
In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimen...In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graft and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on submanifolds.展开更多
The nature of infinite-dimensional Hamiltonian systems are studied for the purpose of further study on some generalized Hamiltonian systems equipped with a given Poisson bracket. From both theoretical and practical vi...The nature of infinite-dimensional Hamiltonian systems are studied for the purpose of further study on some generalized Hamiltonian systems equipped with a given Poisson bracket. From both theoretical and practical viewpoints, we summarize a general method of constructing symplectic-like difference schemes of these kinds of systems. This study provides a new algorithm for the application of the symplectic geometry method in numerical solutions of general evolution equations.展开更多
In this paper, we develop a global perturbation technique for the study of periodic orbits in three-dimensional, time dependent and independent, perturbations of generalized Hamiltonian differential equations defined ...In this paper, we develop a global perturbation technique for the study of periodic orbits in three-dimensional, time dependent and independent, perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds. We give existence, stability and bifurcation theorems and illustrate our results with a truncated spectral model of the forced, dissipative quasi-geostrophic flow on a cyclic beta-plane.展开更多
For the constrained generalized Hamiltonian system with dissipation, by introducing Lagrange multiplier and using projection technique, the Lie group integration method was presented, which can preserve the inherent s...For the constrained generalized Hamiltonian system with dissipation, by introducing Lagrange multiplier and using projection technique, the Lie group integration method was presented, which can preserve the inherent structure of dynamic system and the constraint-invariant. Firstly, the constrained generalized Hamiltonian system with dissipative was converted to the non-constraint generalized Hamiltonian system, then Lie group integration algorithm for the non-constraint generalized Hamiltonian system was discussed, finally the projection method for generalized Hamiltonian system with constraint was given. It is found that the constraint invariant is ensured by projection technique, and after introducing Lagrange multiplier the Lie group character of the dynamic system can't be destroyed while projecting to the constraint manifold. The discussion is restricted to the case of holonomic constraint. A presented numerical example shows the effectiveness of the method.展开更多
In this paper we mainly concern the persistence of invariant tori in generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle...In this paper we mainly concern the persistence of invariant tori in generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, system under consideration can be odd dimensional. Under the Riissmann type non-degenerate condition, we proved that the majority of the lower-dimension invariant tori of the integrable systems in generalized Hamiltonian system are persistent under small perturbation. The surviving lower-dimensional tori might be elliptic, hyperbolic, or of mixed type.展开更多
This paper deals with observer design for generalized Hamiltonian systems and its applications. First, by using the systems' structural properties, a new observer design method called Augment Plus Feedback is prov...This paper deals with observer design for generalized Hamiltonian systems and its applications. First, by using the systems' structural properties, a new observer design method called Augment Plus Feedback is provided and two kinds of observers are obtained: non-adaptive and adaptive ones. Then, based on the obtained observer, H∞ control design is investigated for generalized Hamiltonian systems, and an observer-based control design is proposed. Finally, as an application to power systems, an observer and an observer-based H∞ control law are designed for single-machine infinite-bus systems. Simulations show that both the observer and controller obtained in this paper work very well.展开更多
It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-d...It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.展开更多
The generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices.In this paper,we extend these results and present the generating...The generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices.In this paper,we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices.In particular,some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems(such as generalized Lotka-Volterra systems,Robbins equations and so on).展开更多
Adaptive H∞ synchronization of chaotic systems via linear and nonlinear feedback control is investigated. The chaotic systems are redesigned by using the generalized Hamiltonian systems and observer approach. Based o...Adaptive H∞ synchronization of chaotic systems via linear and nonlinear feedback control is investigated. The chaotic systems are redesigned by using the generalized Hamiltonian systems and observer approach. Based on Lya-punov＇s stability theory, linear and nonlinear feedback control of adaptive H∞ synchronization is established in order to not only guarantee stable synchronization of both master and slave systems but also reduce the effect of external disturbance on an Hoe-norm constraint. Adaptive H∞ synchronization of chaotic systems via three kinds of control is investigated with applications to Lorenz and Chen systems. Numerical simulations are also given to identify the effectiveness of the theoretical analysis.展开更多
The Hamiltonian function method plays an important role in stability analysis and stabilization. The key point in applying the method is to express the system under consideration as the form of dissipative Hamiltonian...The Hamiltonian function method plays an important role in stability analysis and stabilization. The key point in applying the method is to express the system under consideration as the form of dissipative Hamiltonian systems, which yields the problem of generalized Hamiltonian realization. This paper deals with the generalized Hamiltonian realization of autonomous nonlinear systems. First, this paper investigates the relation between traditional Hamiltonian realizations and first integrals, proposes a new method of generalized Hamiltonian realization called the orthogonal decomposition method, and gives the dissipative realization form of passive systems. This paper has proved that an arbitrary system has an orthogonal decomposition realization and an arbitrary asymptotically stable system has a strict dissipative realization. Then this paper studies the feedback dissipative realization problem and proposes a control-switching method for the realization. Finally, this paper proposes several sufficient conditions for feedback dissipative realization.展开更多
This paper investigates the relationship between state feedback and Hamiltonian realization. First, it is proved that a completely controllable linear system always has a state feedback state equation Hamiltonian real...This paper investigates the relationship between state feedback and Hamiltonian realization. First, it is proved that a completely controllable linear system always has a state feedback state equation Hamiltonian realization. Necessary and sufficient conditions are obtained for it to have a Hamiltonian realization with natural output. Then some conditions for an affine nonlinear system to have a Hamiltonian realization are given. For generalized outputs, the conditions of the feedback, keeping Hamiltonian, are discussed. Finally, the admissible feedback controls for generalized Hamiltonian systems are considered.展开更多
This paper Investigates Hamiltonian realization of time-varying nonlinear （TVN） systems, and proposes a number of new methods for the problem. It is shown that every smooth TVN system can be expressed as a generaliz...This paper Investigates Hamiltonian realization of time-varying nonlinear （TVN） systems, and proposes a number of new methods for the problem. It is shown that every smooth TVN system can be expressed as a generalized Hamiltonian system if the origin is the equilibrium of the system. If the Jacooian matrix of a TVN system is nonsingular, the system has a generalized Hamiltonian realization whose structural matrix and Hamiltonian function are given explicitly. For the case that the Jacobian matrix is sin- gular; this paper provides a constructive decomposition method, and then proves that a TVN system has a generalized Hamiltonian realization if its Jacobian matrix has a non- singular main diagonal block. Furthermore, some sufficient （necessary and sufficient） conditions for dissipative Hamiltonian realization of TVN systems are also presented in this paper.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos.19972010 and 10272021).
文摘The form invariance and the Lie symmetry of the generalized Hamiltonian system are studied. Firstly, de?nitions and criteria of the form invariance and the Lie symmetry of the system are given. Next, the relation between the form invariance and the Lie symmetry is studied. Finally, two examples are given to illustrate the application of the results.
文摘In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graft and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on submanifolds.
基金Acknowledgments. This work was supported by the China National Key Development Planning Project for Ba-sic Research (Abbreviation: 973 Project Grant No. G1999032801), the Chinese Academy of Sciences Key Innovation Direction Project (Grant No. KZCX2208)
文摘The nature of infinite-dimensional Hamiltonian systems are studied for the purpose of further study on some generalized Hamiltonian systems equipped with a given Poisson bracket. From both theoretical and practical viewpoints, we summarize a general method of constructing symplectic-like difference schemes of these kinds of systems. This study provides a new algorithm for the application of the symplectic geometry method in numerical solutions of general evolution equations.
文摘In this paper, we develop a global perturbation technique for the study of periodic orbits in three-dimensional, time dependent and independent, perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds. We give existence, stability and bifurcation theorems and illustrate our results with a truncated spectral model of the forced, dissipative quasi-geostrophic flow on a cyclic beta-plane.
文摘For the constrained generalized Hamiltonian system with dissipation, by introducing Lagrange multiplier and using projection technique, the Lie group integration method was presented, which can preserve the inherent structure of dynamic system and the constraint-invariant. Firstly, the constrained generalized Hamiltonian system with dissipative was converted to the non-constraint generalized Hamiltonian system, then Lie group integration algorithm for the non-constraint generalized Hamiltonian system was discussed, finally the projection method for generalized Hamiltonian system with constraint was given. It is found that the constraint invariant is ensured by projection technique, and after introducing Lagrange multiplier the Lie group character of the dynamic system can't be destroyed while projecting to the constraint manifold. The discussion is restricted to the case of holonomic constraint. A presented numerical example shows the effectiveness of the method.
基金Partially supported by the Talent Foundation (522-7901-01140418) of Northwest A & FUniversity.
文摘In this paper we mainly concern the persistence of invariant tori in generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, system under consideration can be odd dimensional. Under the Riissmann type non-degenerate condition, we proved that the majority of the lower-dimension invariant tori of the integrable systems in generalized Hamiltonian system are persistent under small perturbation. The surviving lower-dimensional tori might be elliptic, hyperbolic, or of mixed type.
基金This work was supported by the National Natural Science Foundation of China(Grant No.G60474001)RFDP of China(Grant No,G20040422059).
文摘This paper deals with observer design for generalized Hamiltonian systems and its applications. First, by using the systems' structural properties, a new observer design method called Augment Plus Feedback is provided and two kinds of observers are obtained: non-adaptive and adaptive ones. Then, based on the obtained observer, H∞ control design is investigated for generalized Hamiltonian systems, and an observer-based control design is proposed. Finally, as an application to power systems, an observer and an observer-based H∞ control law are designed for single-machine infinite-bus systems. Simulations show that both the observer and controller obtained in this paper work very well.
文摘It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.
基金projects NSF of China(11271311)Program for Changjiang Scholars and Innovative Research Team in University of China(IRT1179)the Aid Program for Science and Technology,Innovative Research Team in Higher Educational Institutions of Hunan Province of China,and Hunan Province Innovation Foundation for Postgraduate(CX2011B245).
文摘The generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices.In this paper,we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices.In particular,some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems(such as generalized Lotka-Volterra systems,Robbins equations and so on).
文摘Adaptive H∞ synchronization of chaotic systems via linear and nonlinear feedback control is investigated. The chaotic systems are redesigned by using the generalized Hamiltonian systems and observer approach. Based on Lya-punov＇s stability theory, linear and nonlinear feedback control of adaptive H∞ synchronization is established in order to not only guarantee stable synchronization of both master and slave systems but also reduce the effect of external disturbance on an Hoe-norm constraint. Adaptive H∞ synchronization of chaotic systems via three kinds of control is investigated with applications to Lorenz and Chen systems. Numerical simulations are also given to identify the effectiveness of the theoretical analysis.
基金This work was supported by Project 973 of China(Grant Nos.G1998020307,G1998020308)China Postdoctoral Science Foundation.
文摘The Hamiltonian function method plays an important role in stability analysis and stabilization. The key point in applying the method is to express the system under consideration as the form of dissipative Hamiltonian systems, which yields the problem of generalized Hamiltonian realization. This paper deals with the generalized Hamiltonian realization of autonomous nonlinear systems. First, this paper investigates the relation between traditional Hamiltonian realizations and first integrals, proposes a new method of generalized Hamiltonian realization called the orthogonal decomposition method, and gives the dissipative realization form of passive systems. This paper has proved that an arbitrary system has an orthogonal decomposition realization and an arbitrary asymptotically stable system has a strict dissipative realization. Then this paper studies the feedback dissipative realization problem and proposes a control-switching method for the realization. Finally, this paper proposes several sufficient conditions for feedback dissipative realization.
基金This research is supported partly by the National Natural Science Foundation of China(No.G59837270)and National 973 Project(No.G
文摘This paper investigates the relationship between state feedback and Hamiltonian realization. First, it is proved that a completely controllable linear system always has a state feedback state equation Hamiltonian realization. Necessary and sufficient conditions are obtained for it to have a Hamiltonian realization with natural output. Then some conditions for an affine nonlinear system to have a Hamiltonian realization are given. For generalized outputs, the conditions of the feedback, keeping Hamiltonian, are discussed. Finally, the admissible feedback controls for generalized Hamiltonian systems are considered.
基金Supported by the National Natural Science Foundation of China (Grant No. 60474001)the Research Fund of the Doctoral Program of ChineseHigher Education (Grant No. 20040422059)the Natural Science Foundation of Shandong Province (Grant No. Y2006G10)
文摘This paper Investigates Hamiltonian realization of time-varying nonlinear （TVN） systems, and proposes a number of new methods for the problem. It is shown that every smooth TVN system can be expressed as a generalized Hamiltonian system if the origin is the equilibrium of the system. If the Jacooian matrix of a TVN system is nonsingular, the system has a generalized Hamiltonian realization whose structural matrix and Hamiltonian function are given explicitly. For the case that the Jacobian matrix is sin- gular; this paper provides a constructive decomposition method, and then proves that a TVN system has a generalized Hamiltonian realization if its Jacobian matrix has a non- singular main diagonal block. Furthermore, some sufficient （necessary and sufficient） conditions for dissipative Hamiltonian realization of TVN systems are also presented in this paper.
