Integrability and ergodicity of classical billiards in a magnetic field

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not trivially integrable can be described by a bouncing map. We compute a general expression for the Jacobian matrix of this map, which allows us to determine stability and bifurcation values of specific periodic orbits. In some cases, the bouncing map is a twist map and admits a generating function. We give a general form for this function which is useful to do perturbative calculations and to classify periodic orbits. We prove that billiards in convex domains with sufficiently smooth boundaries possess invariant tori corresponding to skipping trajectories. Moreover, in strong field we construct adiabatic invariants over exponentially large times. To some extent, these results remain true for a class of nonconvex billiards. On the other hand, we present evidence that the billiard in a square is ergodic for some large enough values of the magnetic field. A numerical study reveals that the scattering on two circles is essentially chaotic.     

On the Integrability of Geodesic Flows of Submersion Metrics

Letters in Mathematical Physics - Suppose we are given a compact Riemannian manifold (Q,g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by...     

Recurrence of particles in static and time varying oval billiards

Dynamical properties are studied for escaping particles, injected through a hole in an oval billiard. The dynamics is considered for both static and periodically moving boundaries. For the static boundary, two different decays for the recurrence time distribution were observed after exponential decay for short times: A changeover to: (i) power law or; (ii) stretched exponential. Both slower decays are due to sticky orbits trapped near KAM islands, with the stretched exponential apparently associated with a single group of large islands. For time dependent case, survival probability leads to the conclusion that sticky orbits are less evident compared with the static case.     

Generalized Lie Symmetries and the Integrability of Generalized Emden-Fowler Equations

Generalized Lie symmetries and the integrability of generalized Emden-Fowler equations (GEFEs) are considered. It is shown that the constraint which the variable-coefficient functions must satisfy for the GEFEs to have infinite-dimensional symmetry algebras is precisely the same as this in order that the equation may be transformed into the integrable Emden-Fowler equation. fiom the nature of the symmetry vector fields one can write down the integrals of motion for the above systems. The structure of the symmetry algebras is also presented.     

Similarity Reduction and Integrability for the Nonlinear Wave Equations from EPM Model

Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plastic-microstructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou.As a result,the nonlinear wave equation is not integrable.     

Integrability and the variational formulation of non‐conservative mechanical systems

It is shown that one can obtain canonically‐defined dynamical equations for non‐conservative mechanical systems by starting with a first variation functional, instead of an action functional, and finding their zeroes. The kernel of the first variation functional, as an integral functional, is a 1‐form on the manifold of kinematical states, which then represents the dynamical state of the system. If the 1‐form is exact then the first variation functional is associated with the first variation of an action functional in the usual manner. The dynamical equations then follow from the vanishing of the dual of the Spencer operator that acts on the dynamical state. This operator, in turn, relates to the integrability of the kinematical states. The method is applied to the modeling of damped oscillators.     

Homoclinic tangle of the primary separatrix in the compact and closed versus open and unbounded magnetic topologies for divertor tokamaks

The simple map and the symmetric quartic map are the generic representation of the open-unbounded and closed-compact magnetic topologies for single-null divertor tokamaks, respectively. The map parameter represents the magnetic asymmetries. The two maps are used to calculate the homoclinic tangles in the two topologies in divertor tokamaks. It is found that as the magnetic asymmetries become larger, the homoclinic tangle of the simple map becomes extremely elongated radially as opposed to the homoclinic tangle of the symmetric quartic map which becomes wider poloidally.     

A note on the harmonic plus inverse-harmonic potential in quantum mechanics

The ground state of a class of potentials described byV(x)=a ₂₀ x ²+a ₁/x ² is investigated. This potential possesses some distinct features like that of a symmetric double-well potential, and can be used in modelling the fusion process of two identical nuclei as well.     

Degenerate Integrability of the Spin Calogero–Moser Systems and the Duality with the Spin Ruijsenaars Systems

It is shown that spin Calogero–Moser systems are completely integrable in a sense of degenerate integrability. Their Liouville tori have dimension less than half of the dimension of the phase space. It is also shown that rational spin Ruijsenaars systems are degenerately integrable and dual to spin Calogero–Moser systems in a sense that action-angle variables of one are angle-action variables of the other.     

Integrability Test and Spatiotemporal Feature of Breather-Wave to the (2+1)-Dimensional Boussinesq Equation

Painlev integrability has been tested for (2+1)D Boussinesq equation with disturbance term using the standard WTC approach after introducing the Kruskai's simplification. New breather solitary solutions depending on constant equilibrium solution are obtained by using Extended Homoclinic Test Method. Moreover, the spatiotemporal feature of breather solitary wave is exhibited.     

Symmetries,Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations

This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method.     

Integrability and Solutions of the (2+1)-Dimensional Broer-Kaup Equation with Variable Coefficients

The integrability of the(2+1)-dimensional Broer-Kaup equation with variable coefficients(VCBK) is verified by finding a transformation mapping it to the usual(2+1)-dimensional Broer-Kaup equation(BK).Thus the solutions of the(2+1)-dimensional VCBK are obtained by making full use of the known solutions of the usual(2+1)dimensional BK.Two new integrable models are given by this transformation,their dromion-like solutions and rogue wave solutions are also obtained.Further,the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the transformation.     

On the connection between harmonic maps and the self-dual Yang—Mills and the sine-Gordon equations

This is an algebraic paper which demonstrates some algebraic connections between the Yang—Mills and the sine-Gordon equation.     

双色场中高次谐波转化效率提高的数值研究

采用渐近边界条件和辛算法数值求解了双色激光场分别与模型Pschl-Teller势和模型He⁺相互作用的含时Schrdinger方程的无穷空间初值问题,计算了电离概率、电子平均位移、高次谐波与跃迁概率;数值结果显示,添加适当的倍频光的双色激光场使高次谐波转化效率极大提高,并给出定性与定量分析. 关键词： 辛算法 双色激光场 高次谐波 定量分析     

Theoretical and nonlinear behavior analysis of a flexible rotor supported by a relative short herringbone-grooved gas journal-bearing system

This paper considers the bifurcation and nonlinear behavior of a flexible rotor supported by a relative short herringbone-grooved gas journal bearing system. A numerical method is employed to a time-dependent mathematical model. A finite difference method with successive over relation method is employed to solve the Reynolds’ equation. The system state trajectory, Poincaré maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and journal centers in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and quasi-periodic response of the rotor and journal centers. It further shown the dynamic behavior of this type of system varies with changes in bearing number and rotor mass. The results of this study contribute to a better understanding of the nonlinear dynamics of herringbone-grooved gas journal bearing systems.     

Generalized Lyapunov exponents in high-dimensional chaotic dynamics and products of large random matrices

We study the behavior of the generalized Lyapunov exponents for chaotic symplectic dynamical systems and products of random matrices in the limit of large dimensionsD. For products of random matrices without any particular structure the generalized Lyapunov exponents become equal in this limit and the value of one of the generalized Lyapunov exponents is obtained by simple arguments. On the contrary, for random symplectic matrices with peculiar structures and for chaotic symplectic maps the generalized Lyapunov exponents remains different forD , indicating that high dimensionality cannot always destroy intermittency.     

Dynamical quantities and their numerical analysis by saddle periodic orbits

We study estimators for dynamical quantities such as the topological entropy and the topological pressure which are based on numerical computations on periodic orbits. For the particular case of the Hénon family for some parameter ranges we find a reasonable convergence of the entropy, the pressure, and Birkhoff averages of test functions. However, pointing out possible pitfalls of such an analysis, we show that the evaluation by means of saddle orbits alone can be misleading if, for example, chaotic saddles and attractors coexist.     

算法对混沌体系逃逸率的影响

以Hénon-Heiles体系为例，研究算法对混沌体系运动轨道和逃逸率计算结果的影响.比较了新发现的四阶辛算法和一种非辛的高阶算法得到的结果.发现两种算法给出的轨迹之间的距离随时间增大，增加的速度可以作为体系相空间混沌的度量.通过跟踪大数量的粒子轨迹，提取出了逃逸率随体系能量的变化.发现由两种算法得到的逃逸率相互符合得很好. 关键词： 逃逸率 Hénon-Heiles体系 辛算法     

计算7Li2振动能级煌振动-转动能级的辛格式矩阵法

利用辛格式矩阵法计算了双原子分子⁷Li₂在A¹Σ_u⁺态的振动能级和振动-转动能级,并与Ley-Koo等的计算结果作了比较.结果显示,辛格式矩阵法是收敛的和可靠的,是计算双原子分子的振动能级和振动-转动能级的合理的数值方法.     

Deformation Dynamics and the Gauss–Bonnet Topological Term in String Theory

We show that there exists a nontrivial contribution on the Witten covariant phase space when the Gauss–Bonnet topological term is added to the Dirac–Nambu–Goto action describing strings, because the geometry of deformations is modified, and on such space we construct a symplectic structure. Future extensions of the present results are outlined.