Phase-plane analysis of test particle orbits in regular black holes

We investigate phase-plane analysis of general relativistic orbits in a gravitational field of the Reissner–Nordstr?m-type regular black hole spacetime. We employ phase-plane analysis to obtain different phase-plane diagrams of the test particle orbits by varying charge q and dimensionless parameter β, where β contains angular momentum of the test particle. We compute numerical values of radii for the innermost stable orbits and corresponding values of energy required to place the test particle in orbits. Later on, we employ similar analysis on an Ayón–Beato–García(ABG) regular black hole and a comparison regarding key results is also included.     

Double quantization of CP type orbits by generalized Verma modules

It is known that symmetric orbits in g^* for any simple Lie algebra g are equipped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the “canonical” R-matrix. We realize quantization of the Poisson pencil CPⁿ type orbits (i.e. orbits in sl(n + 1)^* whose real compact form is CPⁿ) by means of q-deformed Verma modules.     

谐振子系统的量子-经典轨道、Berry相及Hannay角

从量子-经典轨道和几何相两方面, 研究了二维旋转平移谐振子系统的量子-经典对应. 通过广义规范变换得到了Lissajous经典周期轨道和Hannay角. 另外, 使用含时规范变换解析推导了旋转平移谐振子系统Schrödinger方程的本征波函数和Berry相, 得出结论: 原规范中的非绝热Berry相是经典Hannay角的-n倍. 最后, 使用SU(2)自旋相干态叠加, 构造一稳态波函数, 其波函数的概率云很好地局域于经典轨道上, 满足几何相位和经典轨道同时对应.     

On the control of complex dynamic systems

A method is described for the limited control of the dynamics of systems which generally have several dynamic attractors. associated either with maps or first order ordinary differential equations (ODE) in ⁿ. The control is based on the existence of ‘convergent’ regions, C_C(k = 1,2,…), in the phase space of such systems, where there is ‘local convergence’ of all nearby orbits. The character of the control involves the ‘entrainment’ and subsequent possible ‘migration’ of the experimental system from one attractor to another. Entrainment means that lim_{t > → ∞} |x(t) − g(t)| = 0, where is the system's controlled dynamics, and the goal-dynamics, g(t) ε G_k, has any topological form but is limited dynamically and to regions of phase space, G_k, contained in some C_k, G_k C_k. The control process is initiated only when the system enters a ‘basin of entrainment’, BE_k G_k, associated with the goal-region G_k. Aside from this ‘macroscopic’ initial-state information about the system, no further feedback of dynamic information concerning the response of the system is required. The experimental reliability of the control requires that the regions, BE_k, be convex regions in the phase space, which can apparently be assured if G_k C_k. Simple illustrations of these concepts are given, using a general linear and a piecewise-linear ODE in . In addition to these entrainment-goals, ‘migration-goal’ dynamics is introduced, which intersects two convergent regions G ∩ C_j ≠ , G ∩ C_j ≠ (i ≠ j), and permits transferring the dynamics of a system from one attractor to another, or from one convergent region to another. In the present study these concepts are illustrated with various one-dimensional maps involving one or more attractors and convergent regions. Several theorems concerning entrainment are derived for very general, continuous one-dimensional maps. Sufficient conditions are also established which ensure ‘near-entrainment’ for a system, when the dynamic model of the system is not exactly known. The applications of these concepts to higher dimensional maps and flows will be presented in subsequent studies.     

Diffusive dynamics and periodic orbits of dynamical systems

We show how to organize the set of periodic orbits of dynamical systems exhibiting deterministic diffusion into a cycle expansion giving the diffusion coefficient D. This expansion is used to derive analytically D for a class of such dynamical systems.     

On Spin(7) holonomy metric based on SU(3)/U(1): II

We continue the investigation of Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U(1). A special choice of U(1) embedding in SU(3) allows more general metric ansatz with five metric functions. There are two possible singular orbits in the first-order system of Spin(7) instanton equation. One is the flag manifold SU(3)/T² also known as the twistor space of CP(2) and the other is CP(2) itself. Imposing a set of algebraic constraints, we find a two-parameter family of exact solutions which have SU(4) holonomy and are asymptotically conical. There are two types of asymptotically locally conical (ALC) metrics in our model, which are distinguished by the choice of S¹ circle whose radius stabilizes at infinity. We show that this choice of M theory circle selects one of the possible singular orbits mentioned above. Numerical analyses of solutions near the singular orbit and in the asymptotic region support the existence of two families of ALC Spin(7) metrics: one family consists of deformations of the Calabi hyper-Kähler metric, the other is a new family of metrics on a line bundle over the twistor space of CP(2).     

A numerical method for determining bifurcation curves of mappings

A shooting type iterative method for determining bifurcation curves of mappings is developed in this paper. By using this method we are able to calculate bifurcation curves for orbits of periods ≤ 5 of a Hénon-like map in all details. We find an interesting phenomenon that two or more stable periodic orbits of the same or different periods may coexist for some combinations of parameter values. Two stable p5 orbits and their attracting regions are shown for (μ, b) = (1.536, 0.153).     

Basic structures of the Shilnikov homoclinic bifurcation scenario

We find numerically small scale basic structures of homoclinic bifurcation curves in the parameter space of the Chua circuit. The distribution of these basic structures in the parameter space and their geometrical properties constitute a complete homoclinic bifurcation scenario of this system. Furthermore, these structures and the scenario are theoretically demonstrated to be generic to a large class of dynamical systems that presents, as the Chua circuit, Shilnikov homoclinic orbits. We classify the complexity of primary and subsidiary homoclinic orbits by their order given by the number of their returning loops. Our results confirm previous predictions of structures of homoclinic bifurcation curves and extend this study to high order primary orbits. Furthermore, we identify accumulations of bifurcation curves of subsidiary homoclinic orbits into bifurcation curves of both primary and subsidiary orbits.     

Ising model on mixed two-dimensional lattices

We inform results on physical and topological magnitudes related to the ground level of Ising model on mixed two-dimensional lattices of coordination numbers 4 (Kagomé lattices) and 5 (five-point star lattices). We consider little clusters of size N, where N represents the total number of spins, subject to periodic boundary conditions. On these systems we randomly distribute ±J nearest-neighbor interactions (+J: antiferromagnetic, −J: ferromagnetic (F)). Concentration x of F interactions is varied in the interval (0,1). Two different methods are used to obtain results reported here. First, a numerical method related to multiple replicas. Second, an analytical method based on probabilistic analysis of flat and curved plaquettes. Both methods are complementary to each other. Initially, this study is restricted to calculate frustration of plaquettes and bonds, energy and bond order parameter at T=0. The results of magnitudes informed here are compared with the similar ones obtained for honeycomb, square and triangular lattices.     

阶矩阵及其在传统预处理方法中的应用

本文应用矩阵元素阶和阶矩阵概念,讨论了ICCG和MICCG这两种传统的预处理方法在实用中的一些问题。为什么ICCG(s,t)在s+t固定时取(s,t)=(1,1),(1,2),(1,3),(2,4),(3,5),…有较高的收敛速度?为什么MICCG(m)当m>3时迭代次数不变?ICCG和MICCG的填入方式如何系统化?MICCG是否总比ICCG收敛速度高?本文拟作一个初步的讨论。通过LU分解的阶矩阵,本文给出了按阶递增的填入原则,将ICCG和MICCG系统化为P阶ICCG和P阶MICCG,并讨论了MICCG原有填入方式存在的问题。应用误差阵的阶矩阵,本文讨沦了MICCG迭代参数选取中存在的问题,给出了合理的参数选取方法。通过不同算例,本文还比较了ICCG和MICCG的计算效率。     

Control of chaos in Frenkel–Kontorova model using reinforcement learning

It is shown that we can control spatiotemporal chaos in the Frenkel–Kontorova(FK) model by a model-free control method based on reinforcement learning. The method uses Q-learning to find optimal control strategies based on the reward feedback from the environment that maximizes its performance. The optimal control strategies are recorded in a Q-table and then employed to implement controllers. The advantage of the method is that it does not require an explicit knowledge of the system, target states, and unstable periodic orbits. All that we need is the parameters that we are trying to control and an unknown simulation model that represents the interactive environment. To control the FK model, we employ the perturbation policy on two different kinds of parameters, i.e., the pendulum lengths and the phase angles. We show that both of the two perturbation techniques, i.e., changing the lengths and changing their phase angles, can suppress chaos in the system and make it create the periodic patterns. The form of patterns depends on the initial values of the angular displacements and velocities. In particular, we show that the pinning control strategy, which only changes a small number of lengths or phase angles, can be put into effect.     

Calculating low-altitude trapped particle fluxes with the NASA models AP-8 and AE-8

Accessing the NASA trapped radiation models AP-8 and AE-8 with (B,L) values obtained with modern geomagnetic field models causes an unrealistic secular increase of the predicted flux over low altitude orbits. We show the secular variation of the orbit-averaged particle flux along the LDEF orbit and the mission dose, obtained using the AP-8 trapped radiation models with an epoch-dependent magnetic field. The artificially increased epoch-dependent fluxes and doses are compared to the flux and dose obtained with a consistent and more correct procedure for predicting fluxes with the NASA models. This procedure has been implemented in the radiation analysis software package UNIRAD developed and distributed by BIRA-IASB.     

Dyonic membranes

We present dyonic multi-membrane solutions of the N = 2 D = 8 supergravity theory that serves as the effective field theory of the T²-compactified type II superstring theory. The ‘electric’ charge is fractional for generic asymptotic values of an axion field, as for D = 4 dyons. These membrane solutions are supersymmetric, saturate a Bogomolnyi bound, fill out orbits of an subgroup of the type II D = 8 T-duality group, and are non-singular when considered as solutions of T³-compactified D = 11 supergravity. On K₃ compactification to D = 4, the conjectured type II/heterotic equivalence allows the group to be reinterpreted as the S-duality group of the toroidally compactified heterotic string and the dyonic membranes wrapped around homology two-cycles of K₃ as S-duals of perturbative heterotic string states.     

Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems

In this paper we study the dynamics near resonant elliptic equilibria in three-degree-of-freedom Hamiltonian systems. The resonances we consider have multiplicity two, and the corresponding local normal form for the equilibrium is integrable at cubic order. We prove the existence of families of 3-tori and whiskered 2-tori with nearby chaotic dynamics in the quartic normal form. The whiskers of the 2-tori intersect in a non-trivial way giving rise to multi-pulse homoclinic and heteroclinic connections. These connections survive in the full system as orbits homoclinic to invariant 3-spheres.     

Superperiodicity,chaos and coexisting orbits of ion-acoustic waves in a four-component nonextensive plasma

Superperiodicity, chaos and coexisting orbits of ion-acoustic waves (IAWs) are studied in a multi-component plasma consisting of fluid ions, q -nonextensive cold and hot electrons and Maxwellian hot positrons. The significant impacts of the system parameters on superperiodic and nonlinear periodic IAWs are presented. Considering an external periodic perturbation various types of quasiperiodic and chaotic features for IAWs are studied in different parametric ranges through time series’ plots, phase spaces and Lyapunov exponents. It has been observed that there exist some coexisting orbits for IAWs. Coexisting orbits for IAWs in a classical electron–positron–ion plasma system are reported.     

CP violation in multi-Higgs supersymmetric models

We consider supersymmetric extensions of the standard model with two pairs of Higgs doublets. We study the possibility of spontaneous CP violation in these scenarios and present a model where the origin of CP violation is soft, with all the complex phases in the Lagrangian derived from complex masses and vacuum expectation values (VEVs) of the Higgs fields. The main ingredient of the model is an approximate global symmetry, which determines the order of magnitude of Yukawa couplings and scalar VEVs. We assume that the terms violating this symmetry are suppressed by powers of the small parameter ε_PQ = O(m_b/m_t). The tree-level flavor-changing interactions are small due to a combination of this global symmetry and a flavor symmetry, but they can be the dominant source of CP violation. All CP-violating effects occur at order ε_PQ² as the result of exchange of almost decoupled extra Higgs bosons and/or through the usual mechanisms with an almost real CKM matrix. On dimensional grounds, the model gives ε_K ≈ ε_PQ² and predicts for the neutron electric dipole moment (and possibly also for ε_K¹) a suppression of order ε_PQ² with respect to the values obtained in standard and minimal supersymmetric scenarios. The predicted CP asymmetries in B decays are generically too small to be seen in the near future. The mass of the lightest neutral scalar, the strong CP problem, and possible contributions to the Z decay into b quarks (the R_b puzzle) are also briefly addressed in the framework of this model.     

Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbits

We investigate how the transition to chaos with multiple positive Lyapunov exponents can be characterized by the set of infinite number of unstable periodic orbits embedded in the chaotic invariant set. We argue and provide numerical confirmation that the transition is generally accompanied by a nonhyperbolic behavior: unstable dimension variability. As a consequence, the Lyapunov exponents, except for the largest one, pass through zero continuously.     

Analytical study of chaos and applications

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Hénon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.     

多点边值问题的插值矩阵法及误差分析

插值矩阵法是求解多点边值问题的数值法。本文给出的该法的误差分析,论证了插值矩阵法解得的y(x),y'(x),…,y^(m)(x)有相同的精度,并对二阶方程,给出该法的稳定性证明和收敛阶。     

Asymptotics of Rydberg States for the Hydrogen Atom

The asymptotics of Rydberg states, i.e., highly excited bound states of the hydrogen atom Hamiltonian, and various expectations involving these states are investigated. We show that suitable linear combinations of these states, appropriately rescaled and regarded as functions either in momentum space or configuration space, are highly concentrated on classical momentum space or configuration space Kepler orbits respectively, for large quantum numbers. Expectations of momentum space or configuration space functions with respect to these states are related to time-averages of these functions over Kepler orbits. Received: 8 November 1996 / Accepted: 13 January 1997