Existence and Multiplicity of Periodic solutions for the Non-autonomous Second-order Hamiltonian Systems

In this paper,we study the existence and multiplicity of periodic solutions of the non-autonomous second-order Hamiltonian systems■where T> 0.Under suitable assumptions on F,some new existence and multiplicity theorems are obtained by using the least action principle and minimax methods in critical point theory.

HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS

In this paper,we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems ż=J H_(z)(t,z),where the Hamiltonian function H possesses the form H(t,z)=1/2L(t)z⋅z+G(t,z),and G(t,z)is only locally defined near the origin with respect to z.Under some mild conditions on L and G,we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense,which is essentially forced by the subquadraticity of G near the origin with respect to z.

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit,K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge–Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida’s method, the optimized partitioned Runge–Kutta and Runge–Kutta–Nystr ¨om explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge–Kutta and Runge–Kutta–Nystr ¨om extended phase space symplectic-like methods with the midpoint permutation.

Crossing Limit Cycles of Planar Piecewise Hamiltonian Systems with Linear Centers Separated by Two Parallel Straight Lines

In this paper, we have studied several classes of planar piecewise Hamiltonian systems with three zones separated by two parallel straight lines. Firstly, we give the maximal number of limit cycles in these classes of systems with a center in two zones and without equilibrium points in the other zone (or with a center in one zone and without equilibrium points in the other zones). In addition, we also give examples to illustrate that it can reach the maximal number.

Some results on entropy dimension for non-autonomous systems

In this paper,the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced.Moreover,formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given.Finally,examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.

Sensitivity of the Product System of Set-Valued Non-Autonomous Discrete Dynamical Systems

This paper is concerned with the sensitivity of set-valued discrete systems. Firstly, this paper obtained the equivalence between or and the product system in sensitivity, infinite sensitivity, F-sensitivity, (F1, F2)-sensitivity. Then, the relation between (X, f1,∞) or (Y, g1,∞) and in ergodic sensitivity is obtained. Where is the set-valued dynamical system induced by a non-autonomous discrete dynamical system (X, f1,∞).

Multiple Periodic Solutions for Some Classes of First-Order Hamiltonian Systems

Considering a decomposition R2N=A⊕B of R2N , we prove in this work, the existence of at least (1+dimA) geometrically distinct periodic solutions for the first-order Hamiltonian system Jx'(t)+H'(t,x(t))+e(t)=0 when the Hamiltonian H(t,u+v) is periodic in (t,u) and its growth at infinity in v is at most like or faster than |v|a, 0≤ae is a forcing term. For the proof, we use the Least Action Principle and a Generalized Saddle Point Theorem.

Noether conserved quantities and Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices

The Noether conserved quantities and the Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices are studied. The generalized Hamiltonian equations of the systems are given on the basis of the transformation operators in the space of discrete Hamiltonians. The Lie transformations acting on the lattice, as well as the equations and the determining equations of the Lie symmetries are obtained for the nonholonomic Hamiltonian systems. The discrete analogue of the Noether conserved quantity is constructed by using the Lie point symmetries. An example is discussed to illustrate the results.

THE EXISTENCE OF NONTRIVIAL SOLUTIONS OF HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS

Some theorems are obtained for the existence of nontrivial solutions of Hamiltonian systems with Lagrangian boundary conditions by the minimax methods.

HOMOCLINIC ORBITS FOR A CLASS OF THE SECOND ORDER HAMILTONIAN SYSTEMS

The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q（t） ＋ ↓△V（t, q（t）） = 0, where V（t, x） = -K（t, x） ＋ W（t, x）, K（t, x） is neither a quadratic form in x nor periodic in t and W（t, x） is superquadratic in x.

Robust Adaptive Control for Robotic Systems With Input Time-Varying Delay Using Hamiltonian Method

This paper addresses the problem of robust adaptive control for robotic systems with model uncertainty and input time-varying delay. The Hamiltonian method is applied to develop the stabilization results of the robotic systems. Firstly, with the idea of shaping potential energy and the pre-feedback skill, the n degree-of-freedom(DOF) uncertain robotic systems are realized as an augmented dissipative Hamiltonian formulation with delay.Secondly, based on the obtained Hamiltonian system formulation and by using of the Lyapunov-Krasovskii(L-K) functional method, an adaptive controller is designed to show that the robotic systems can be asymptotically stabilized depending on the input delay. Meanwhile, some sufficient conditions are spelt out to guarantee the rationality and validity of the proposed control law. Finally, study of an illustrative example with simulations shows that the controller obtained in this paper works very well in handling uncertainties and input delay in the robotic systems.

Symmetries and variational calculation of discrete Hamiltonian systems

We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.

NONTRIVIAL SOLUTIONS FOR ASYMPTOTICALLY LINEAR HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS

In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory developed by the first author.

Passive robust control for uncertain Hamiltonian systems by using operator theory

In this study,the passivity-based robust control and tracking for Hamiltonian systems with unknown perturbations by using the operator-based robust right coprime factorisation method is concerned.For the system with unknown perturbations,a design scheme is proposed to guarantee the uncertain non-linear systems to be robustly stable while the equivalent non-linear systems is passive,meanwhile the asymptotic tracking property of the plant output is discussed.Moreover,the design scheme can be also used into the general Hamiltonian systems while the simulation is used to further demonstrate the effectiveness of the proposed method.

Proto col Design for Output Consensus of Port-controlled Hamiltonian Multi-agent Systems

这份报纸调查控制港口的 Hamiltonian (PCH ) 多代理人系统的产量一致问题与修理并且切换的拓扑学。第一，一个分布式的组产量一致协议经由塑造方法全球性到达稳定性和组产量一致的能量被设计。第二，一个新分布式的控制协议被使用 PCH 系统的结构的性质建议。这个协议的优点是它能由构造一种虚拟邻居转变指导的图到未受指导的图。第三，一个控制协议被设计，扩大 LaSalle0s 不变性原则在联合连接的拓扑学条件下面为交换系统发展了让所有代理人当拓扑学正在切换时，到达输出一致。最后，有模拟的一些解说性的例子被提供表明在这份报纸设计的协议的有效性。

A CLASS OF QUADRATIC HAMILTONIAN SYSTEMS UNDER QUADRATIC PERTURBATION

This paper deals with a class of quadratic Hamiltonian systems with quadratic perturbation. The authors prove that if the first order Melnikov function M1 (h) 0 and the second order Melnikov function M2(h) 0, then the origin of the Hamiltonian system with small perturbation is a center.

The Poincaré Bifurcation of a Class of Planar Hamiltonian Systems

In this paper, we discuss the Poincaré bifurcation of a class of Hamiltonian systems having a region consisting of periodic cycles bounded by a parabola and a straight line. We prove that the system can generate at most two limit cycles and may generate two limit cycles after a small cubic polynomial perturbation.

Hamiltonian Systems in Elasticity and Their Variational Principles

As an inverse problem of Hamiltonian mechanics, a new Hamiltonian system inelasticity and its variational principle are derived from the basic equations of elasticity.

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.