A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles

This article presents a rigorous existence theory for three-dimensional gravity-capillary water waves which are uniformly translating and periodic in one spatial direction x and have the profile of a uni- or multipulse solitary wave in the other z. The waves are detected using a combination of Hamiltonian spatial dynamics and homoclinic Lyapunov-Schmidt theory. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which z is the timelike variable, and a family of points P_k,k+1, k = 1,2,... in its two-dimensional parameter space is identified at which a Hamiltonian 0²0² resonance takes place (the zero eigenspace and generalised eigenspace are respectively two and four dimensional). The point P_k,k+1 is precisely that at which a pair of two-dimensional periodic linear travelling waves with frequency ratio k:k+1 simultaneously exist (Wilton ripples). A reduction principle is applied to demonstrate that the problem is locally equivalent to a four-dimensional Hamiltonian system near P_k,k+1. It is shown that a Hamiltonian real semisimple 1:1 resonance, where two geometrically double real eigenvalues exist, arises along a critical curve R_k,k+1 emanating from P_k,k+1. Unipulse transverse homoclinic solutions to the reduced Hamiltonian system at points of R_k,k+1 near P_k,k+1 are found by a scaling and perturbation argument, and the homoclinic Lyapunov-Schmidt method is applied to construct an infinite family of multipulse homoclinic solutions which resemble multiple copies of the unipulse solutions.     

The frozen-in condition for a direction field,small denominators and chaotization of steady flows of a viscous fluid

A classical theorem of Helmholtz states that vortex lines are frozen into a flow of barotropic ideal fluid in a potential force field. This result leads to the following general problem: it is required to find conditions under which a given dynamical system admits of a direction field frozen into its phase flow. By the rectification theorem for trajectories, a whole family of frozen direction fields always exists locally. It turns out that the problem of the existence of non-trivial frozen direction fields defined in the whole phase space is closely related to the well-known problem of small denominators. Results of a general nature are applied to Hamiltonian systems, and also to steady flows of a viscous fluid.     

大型不正定陀螺系统本征值问题

陀螺动力系统可以导入哈密顿辛几何体系,在哈密顿陀螺系统的辛子空间迭代法的基础上提出了一种能够有效计算大型不正定哈密顿函数的陀螺系统本征值问题的算法．利用陀螺矩阵既为哈密顿矩阵而本征值又是纯虚数或零的特点,将对应哈密顿函数为负的本征值分离开来,构造出对应哈密顿函数全为正的本征值问题,利用陀螺系统的辛子空间迭代法计算出正定哈密顿矩阵的本征值,从而解决了大型不正定陀螺系统的本征值问题,算例证明,本征解收敛得很快．     

Three‐dimensional internal gravity‐capillary waves in finite depth

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ₀) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.     

The Hess–Appelrot case and quantization of the rotation number

This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.     

Stabilizing a Flexible Beam on a Cart: A Distributed Port-Hamiltonian Approach

Motion planning and stabilization of the inverted pendulum on a cart is a much-studied problem in the control community. We focus our attention on asymptotically stabilizing a vertically upright flexible beam fixed on a moving cart. The flexibility of the beam is restricted only to the direction along the traverse of the cart. The control objective is to attenuate the effect of disturbances on the vertically upright profile of the beam. The control action available is the motion of the cart. By regulating this motion, we seek to regulate the shape of the beam. The problem presents a combination of a system described by a partial differential equation (PDE) and a cart modeled as an ordinary differential equation (ODE) as well as a controller which we restrict to an ODE. We set our problem in the port-controlled Hamiltonian framework. The interconnection of the flexible beam to the cart is viewed as a power-conserving interconnection of an infinite-dimensional system to a finite-dimensional system. The energy-Casimir method is employed to obtain the controller. In this method, we look for some constants of motion that are invariant of the choice of controller Hamiltonian. These Casimirs relate the controller states to the states of the system. We finally prove the stability of the equilibrium configuration of the closed-loop system.     

A variational formula for controlled backward stochastic partial differential equations and some application

An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.     

On the existence of a specified cycle in digraphs with constraints on degrees

We prove that Woodall's and Ghouila-Houri's conditions on degrees which ensure that a digraph is Hamiltonian, also ensure that it contains the analog of a directed Hamiltonian cycle but with one edge pointing the wrong way; that is, it contains two vertices that are connected in the same direction by both an edge and a Hamiltonian path.     

On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance

The problem of orbital stability of a periodic motion of an autonomous two-degreeof- freedom Hamiltonian system is studied. The linearized equations of perturbed motion always have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure of the system. The other two multipliers are assumed to be complex conjugate numbers with absolute values equal to one, and the system has no resonances up to third order inclusive, but has a fourth-order resonance. It is believed that this case is the critical one for the resonance, when the solution of the stability problem requires considering terms higher than the fourth degree in the series expansion of the Hamiltonian of the perturbed motion.Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability are obtained, which are represented in the form of inequalities depending on the coefficients of series expansion of the Hamiltonian up to the sixth degree inclusive.     

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M₃.     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

一个二流体系统中非线性水波的Hamilton描述

讨论了一个二流体系统中非线性水波的Hamilton描述,该系统由水平固壁之上的两层常密度不可压无粘流体组成,上表面为自由面.文中将速度势函数展开成垂向坐标的幂级数,在浅水长波的假定下,取下层流体的“动厚度”与上层流体的“折合动厚度”为广义位移、界面上和自由面上的速度势为广义动量,根据Hamilton原理并运用Legendre变换导出该系统的Hamilton正则方程,从而将单层流体情形的结果推广到分层流体的情形.     

Non-linear oscillations of a Hamiltonian system with 2:1 resonance

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.     

Resonant periodic motions of Hamiltonian systems with one degree of freedom when the Hamiltonian is degenerate

The motions of a non-autonomous Hamiltonian system with one degree of freedom which is periodic in time and where the Hamiltonian contains a small parameter is considered. The origin of coordinates of the phase space is the equilibrium position of the unperturbed or complete system, which is stable in the linear approximation. It is assumed that there is degeneracy in the unperturbed Hamiltonian when account is taken of terms no higher than the fourth degree (the frequency of the small linear oscillations depends on the amplitude) and, in this case, one of the resonances of up to the fourth order inclusive is realized in the system. Model Hamiltonians are constructed for each case of resonance and a qualitative investigation of the motions of the model system is carried out. Using Poincaré's theory of periodic motions and KAM-theory, a rigorous solution is given of the problem of the existence, bifurcations and stability of the periodic motions of the initial system, which are analytic with respect to fractional powers of the small parameter. The resonant periodic motions (in the case of the degeneracy being considered) of a spherical pendulum with an oscillating suspension point are investigated as an application.     

Eigenvalue problem of a class of fourth-order Hamiltonian operators

The eigenvalue problem of a class of fourth-order Hamiltonian operators is studied. We first obtain the geometric multiplicity, the algebraic index and the algebraic multiplicity of each eigenvalue of the Hamiltonian operators. Then, some necessary and sufficient conditions for the completeness of the eigen or root vector system of the Hamiltonian operators are given, which is characterized by that of the vector system consisting of the first components of all eigenvectors. Moreover, the results are applied to the plate bending problem.     

Stability of equilibrium solutions of Hamiltonian systems under the presence of a single resonance in the non-diagonalizable case

The problem of knowing the stability of one equilibrium solution of an analytic autonomous Hamiltonian system in a neighborhood of the equilibrium point in the case where all eigenvalues are pure imaginary and the matrix of the linearized system is non-diagonalizable is considered. We give information about the stability of the equilibrium solution of Hamiltonian systems with two degrees of freedom in the critical case. We make a partial generalization of the results to Hamiltonian systems with n degrees of freedom, in particular, this generalization includes those in [1].      

Compatible Dubrovin–Novikov Hamiltonian Operators,Lie Derivative,and Integrable Systems of Hydrodynamic Type

We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators.     

Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance

The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application.     

On homoclinic bifurcation emanating from Takens-Bogdanov points in Hamiltonian systems

The paper is devoted to studying the bifurcation of periodic and homoclinic orbits in a 2n-dimensional Hamiltonian system with 1 parameter from a TB-point (Hamiltonian saddle node). In addition to the proof of existence, the paper gives an expansion formula of the bifurcating homoclinic orbits. With the help of center manifold reduction and a blow up transformation, the problem is focused on studying a planar Hamiltonian system, the proof for the perturbed homoclinic and periodic orbits is elementary in the sense that it uses only implicit function arguments. Two applications to travelling waves in PDEs are shown.