Subharmonic Solutions with Prescribed Minimal Period of a Class of Nonautonomous Hamiltonian Systems

In this paper, by using a decomposition technique to estimate the energy (the value of the functional associated with the problem) of a solution in terms of the minimal period of the solution, we give new sufficient conditions for the existence of subharmonic solutions with prescribed minimal period of Hamiltonian systems. Our results improve some known results in the literature. Dedicated to Professor Zhifen Zhang in the occasion of her 80th birthday     

Turbulent Spot in Linearly Stable Taylor Couette Flow

The transition from the laminar to the turbulent regime in linearly stable shear flows, for example, pipe and plane Couette flows, occurs abruptly with no precursor. The evolution of turbulent spots has been studied to better understand the dynamics of this transition and the onset of turbulence. These studies have mostly focused on pipe flows for which a finite lifetime of spots was proven. The same conclusion was drawn in the only available study performed in a Taylor Couette setup. Here, the spot lifetime is measured in a different size TC setup. It is shown that the lifetime is indeed finite and also very sensitive to boundary conditions, but not much to perturbation mechanisms. A scaling approach is provided which suggests in addition to the Reynolds number, the aspect and radius ratios are influential parameters on the lifetime. It is found that the spot size varies during its lifetime and increases with the Reynolds number that confirms the rise in turbulence proliferation by approaching the transitional point.     

Quasi-Periodic Solutions with Prescribed Frequency in Reversible Systems

In this paper, we obtain a family of small-amplitude real analytic quasi-periodic solutions for a class of derivative nonlinear Schrödinger equations, subject to Dirichlet boundary conditions, which correspond to infinite-dimensional reversible systems with critical unbounded perturbations. We prove that the frequencies of the quasi-periodic solutions, accordingly, the tangential frequencies of the invariant tori for these reversible systems can be in a fixed direction.     

Optimal Bounded Control of First-Passage Failure of Quasi-Integrable Hamiltonian Systems with Wide-Band Random Excitation

The optimal bounded control of quasi-integrable Hamiltonian systems with wide-band random excitation for minimizing their first-passage failure is investigated. First, a stochastic averaging method for multi-degrees-of-freedom (MDOF) strongly nonlinear quasi-integrable Hamiltonian systems with wide-band stationary random excitations using generalized harmonic functions is proposed. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximizinig reliability and maximizing mean first-passage time are formulated based on the averaged Itô equations by applying the dynamical programming principle. The optimal control law is derived from the dynamical programming equations and control constraints. The relationship between the dynamical programming equations and the backward Kolmogorov equation for the conditional reliability function and the Pontryagin equation for the conditional mean first-passage time of optimally controlled system is discussed. Finally, the conditional reliability function, the conditional probability density and mean of first-passage time of an optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation. The application of the proposed procedure and effectiveness of control strategy are illustrated with an example.     

Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

We study the behaviour of conformally symplectic systems near rotational Lagrangian tori. We recall that conformally symplectic systems appear for example in mechanical models including a friction proportional to the velocity. We show that in a neighborhood of these Lagrangian, invariant, rotational tori, one can always find a smooth symplectic change of variables in which the time evolution becomes just a rotation in some direction and a linear contraction in others. In particular quasi-periodic solutions with $n$ n independent frequencies of contractive (expansive) diffeomorphisms are always local attractors (repellors). We present results when the systems are analytic, $C^r$ C r or $C^\infty $ C ∞ . We emphasize that the results presented here are non-perturbative and apply to systems that are far from integrable; moreover, we do not require any assumption on the frequency and in particular we do not assume any non-resonance condition. We also show that the system of coordinates can be computed rather explicitly and we provide iterative algorithms, which allow to generalize the notion of “isochrones”. We conclude by showing that the above results apply to quasi-periodic conformally symplectic flows.     

Horseshoes in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

We consider a class of two-degree-of-freedom Hamiltonian systems which have saddle-centers with homoclinic orbits and do not take the form of small perturbations of integrable systems. Using a Melnikov-type global perturbation technique, we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle-centers. We apply the theory to systems with potentials and give concrete examples. In particular, we can detect the presence of horseshoes in a class of systems to which previous results did not apply. Accepted January 4, 2000?Published online August 15, 2000     

一类平面Hamilton系统的大范围非线性化近似方法

在非线性动力系统的研究已经进入了占主导地位的时期,对其提出大范围的非线性化近似方法具有特别重要的意义.在本文中,我们主要对于一类典型的Hamilton系统,根据等势线有两个,或者三个交点的不同情形,给出7种不同的大范围最低次非线性化近似系统,并通过积分近似系统给出近似解(轨道).结果表明,近似椭圆周期轨道可通过线性化近似系统得到,而同(异)宿轨道则可通过2、3次非线性化近似系统得到.最后,将近似方法应用于一个具体Hamilton系统的分析.     

Nekhoroshev and KAM Stabilities in Generalized Hamiltonian Systems

We present some Nekhoroshev stability results for nearly integrable, generalized Hamiltonian systems, which can be odd dimensional and admit a distinct number of action and angle variables. Using a simultaneous approximation technique due to Lochak, Nekhoroshev stabilities are shown for various cases of quasi-convex generalized Hamiltonian systems along with concrete estimates on stability exponents. Discussions on KAM metric stability of generalized Hamiltonian systems are also given.     

Periodic Orbits in Hamiltonian Systems with Involutory Symmetries

We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetry. In both classes, the involution reverses the sign of the Hamiltonian function, and the system is in 1:1 resonance. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R, and furthermore that there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian (correcting a minor error in their paper). In the second problem we study an equivariant Hamiltonian system with a symmetry S that acts anti-symplectically. Generically, there is no S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we prove the existence of at least 2 and at most 12 families of non-symmetric periodic solutions. We conclude with a brief study of systems with both forms of symmetry, showing they have very similar structure to the system with symmetry R.     

Discrete Models of a Class of Isolation Systems

In this paper, a class of isolation systems with rigid limiters has been considered. For this class of systems, some general discrete-time models described by means of some impact Poincaré maps have been established. Two examples: a simple isolation system of one-stage and a real isolation system of two-stages have been investigated. The calculated results show that those models can reveal complex nonlinear behaviors. And even a small random perturbation may change the dynamical character of the system.     

Dynamical Spectrum in Random Dynamical Systems

Dynamical spectrum is a concept in terms of exponential dichotomy. The theory of dynamical spectrum, due to Sacker and Sell, plays important roles in many fields of dynamical systems and differential equations. Noticing its significance and importance, we study in this paper the theory of dynamical spectrum for some general random dynamical systems. More precisely, after introducing a random version of the concept of exponential dichotomy, by using some methods and techniques from dynamical systems and ergodic theory, under some general integrability conditions, we establish the dynamical spectral decomposition theorem in framework of random dynamical systems, which can be regarded as a random version of the deterministic dynamical systems due to Sacker and Sell. In our result, the dynamical spectral intervals and the corresponding spectral subbundles will be given.     

Realization of Period Maps of Planar Hamiltonian Systems

We consider the set of 2π-periodic solutions of the ordinary differential equation u′′ + g(u) = 0 for a nonlinearity , satisfying a dissipative condition of the form for , and under the generic assumption that the potential G, given by , is a Morse function. Under these assumptions, we characterize the period maps realizable by planar Hamiltonian systems of the form . Considering the Morse type of G, the set of periodic orbits in the phase space is decomposed into disks and annular regions. Then, the realizable period maps are described in terms of sets of sequences of positive integers corresponding to the lap numbers of the 2π-periodic solutions. This leads to a characterization of the classes of Morse–Smale attractors that are realizable by dissipative semilinear parabolic equations of the form defined on the circle, .      

线性哈密顿系统的本征摄动法

利用哈密顿算子辛自共轭的特点讨论了保守哈密顿体系的摄动问题,给出了哈密顿矩阵的本征值与本征向量的二阶摄动分析方法。即当系统在哈密顿框架下进行较小修改时,不重复求解大型哈密顿矩阵的本征问题,只需在原系统的模态参数基础上进行模态分析即可,这种矩阵摄动法给出了修改后矩阵的二阶本征值和本征向量,为一般线性保守体系的本征摄动求解提出了一个新方法。     

Affine Transformation in Random Iterated Function Systems

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Random Vibration of Strongly Nonlinear Systems

Procedures developed recently to extend the quasi-conservativeaveraging method to the case of non-white broad-band excitations arefurther extended and presented in a unified framework. The excitationscan be stationary or nonstationary with known evolutionary spectra, andthey can be additive, or multiplicative, or both. Relative merits ofthese procedures are compared, and their accuracies are substantiated byMonte Carlo simulations.     

Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems

A method is sketched to determine the presence of non-degenerate Hamiltonian Hopf bifurcations in three-degree-of-freedom systems by putting the bifurcation into standard form. Detailed computations are performed for the non-trivial example of the 3D Hénon–Heiles family. After a careful formulation of the local once reduced system in terms of properly chosen invariants the system can be compared to the standard form by the application of singularity theoretic results.     

Hamilton系统的一类新型守恒律

研究Hamilton系统的Lie对称性与守恒律。根据微分方程在无限小群变换下的不变性理论，建立了Hamilton系统仅依赖于正则变量的无限小群变换的Lie对称变换，给出了Lie对称性的确定方程，并直接由系统的Lie对称性得到了系统的一类新型定恒律。文末，举例说明结果的应用。