一个三维四翼自治混沌系统的拓扑马蹄分析

基于拓扑马蹄映射理论,验证了一个三维四翼自治的混沌系统的拓扑马蹄的存在.由于该混沌系统是连续系统,首先选取了一个Poincaré截面,并在该截面下定义了该混沌系统的一个一次回归Poincaré映射.通过利用计算机辅助证明方法,得出了该映射与一个2移位映射拓扑半共扼,说明该三维四翼自治系统的拓扑熵大于或等于ln2,进而证明了该系统的混沌行为. 关键词： 四翼混沌系统 拓扑马蹄 Poincaré映射 拓扑熵     

Construction of constant curvature punctured Riemann surfaces with particle-scattering interpretation

A class of punctured constant curvature Riemann surfaces, with boundary conditions similar to those of the Poincaré half plane, is constructed. It is shown to describe the scattering of particle-like objects in two Euclidian dimensions. The associated time delays and classical phase shifts are introduced and connected to the behaviour of the surfaces at their punctures. For each such surface, we conjecture that the time delays are partial derivatives of the phase shift. This type of relationship, already known to be correct in other scattering problems, leads to a general integrability condition concerning the behaviour of the metric in the neighbourhood of the punctures. The time delays are explicitly computed for three punctures, and the conjecture is verified. The result, reexpressed as a product of Riemann zeta-functions, exhibits an intringuing number-theoretic structure: a p-adic product formula holds and one of Ramanujan's identities applies. An ansatz is given for the corresponding exact quantum S-matrix. It is such that the integrability condition is replaced by a finite difference relation only involving the exact spectrum already derived, in the associated Liouville field theory, by Gervais and Neveu.     

Group theory of the massless spin 2 field and gravitation

The simplest manifestly covariant unitary representation of the Poincaré group for zero mass and spin 2 is constructed. This representation is carried by fourth rank tensors which satisfy the equations of the Riemann curvature tensor in the linearized theory of gravitation in vacuo. In particular, the requirement of unitarity implies the Bianchi identities.     

From Open Quantum Systems to Open Quantum Maps

For a class of quantized open chaotic systems satisfying a natural dynamical assumption we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories.     

Automorphic Functions,Poincaré Series,and Conformal Fields on Riemann Surfaces

Poincaré series and automorphic functions for SU(1, 1) and a discrete subgroup Γ are studied with harmonic analysis. We consider automorphic functions on the open unit circle with general “spin label” m and their decomposition into irreducible automorphic functions by means of the Plancherel formula. These automorphic functions are bijectively mapped onto automorphic distributions on the boundary of the unit circle by meam of the Poisson kernel. The exponent of convergence of Poincaré series is expressed in representation theory language. The results are applied to two-point functions of conformal fields.     

Statistical characteristics of the Poincaré return times for a one-dimensional nonhyperbolic map

Characteristics of the Poincaré return times are considered in a one-dimensional cubic map with a chaotic nonhyperbolic attractor. Two approaches, local one (Kac’s theorem) and global one related with the AP-dimension estimation of return times, are used. The return times characteristics are studied in the presence of external noise. The characteristics of Poincaré recurrences are compared with the form of probability measure and the complete correspondence of the obtained results with the mathematical theory is shown. The influence of the attractor crisis on the return time characteristics is also analyzed. The obtained results have a methodical and educational significance and can be used for solving a number of applied tasks.     

The fractal structure in the ionization dynamics of Rydberg lithium atoms in a static electric field

The ionization rate of Rydberg lithium atoms in a static electric field is examined within semiclassical theory which involves scattering effects off the core. By semiclassical analysis, this ionization process can be considered as the promoted valence electrons escaping through the Stark saddle point into the ionization channels. The resulting escape spectrum of the ejected electrons demonstrates a remarkable irregular electron pulse train in time-dependence and a complicated nesting structure with respect to the initial launching angles. Based on the Poincaré} map and homoclinic tangle approach, the chaotic behaviour along with its corresponding fractal self-similar structure of the ionization spectra are analysed in detail. Our work is significant for understanding the quantum-classical correspondence.     

Poincaré-sphere equivalent for light beams containing orbital angular momentum

The polarization state of a light beam is related to its spin angular momentum and can be represented on the Poincaré sphere. We propose a sphere for light beams in analogous orbital angular momentum states. Using the Poincaré-sphere equivalent, we interpret the rotational frequency shift for light beams with orbital angular momentum [Phys. Rev. Lett. 80, 3217 (1998)] as a dynamically evolving geometric phase.     

Bifurcation analysis of the logistic map via two periodic impulsive forces

The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincare′ map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.     

Towards a quasi-periodic mean field theory for globally coupled oscillators

We show how a quasi-periodic mean field theory may be used to understand the chaotic dynamics and geometry of globally coupled complex Ginzburg-Landau equations. The Poincaré map of the mean field equations appears to have saddlenode-homoclinic bifurcations leading to chaotic motion, and the attractor has the characteristic ρ shape identified by numerical experiments on the full equations.     

Poincaré recurrences and Ulam method for the Chirikov standard map

We study numerically the statistics of Poincaré recurrences for the Chirikov standard map and the separatrix map at parameters with a critical golden invariant curve. The properties of recurrences are analyzed with the help of a generalized Ulam method. This method allows us to construct the corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the powerful Arnoldi method. We also develop a new survival Monte Carlo method which allows us to study recurrences on times changing by ten orders of magnitude. We show that the recurrences at long times are determined by trajectory sticking in a vicinity of the critical golden curve and secondary resonance structures. The values of Poincaré exponents of recurrences are determined for the two maps studied. We also discuss the localization properties of eigenstates of the Ulam matrix and their relation with the Poincaré recurrences.     

Illusion and reality in the quantum world

Often considered as the last ‘encyclopedist’, Henri Poincaré died one hundred years ago. If he was a prominent man in 1900 French Society, his heritage is not so clearly recognised, particularly in France. Among his too often misunderstood works is his contribution to the theory of relativity, mainly because it is almost never presented within Poincaré's general approach to science, including his philosophical writings. Our aim is therefore to provide an historical account of the main steps (experimental as well as theoretical) which led Poincaré to contribute to the theory of relativity. Starting from the optical experiments which led to the inconsistency of the classical (Galilean) composition law for velocities to explain light propagation, we introduce the FitzGerald and Lorentz contraction which was viewed as the ‘sole hypothesis’ to explain the Michelson and Morley experiment. We then show that Poincaré's contribution starts with a discussion of the principles governing the mechanics and was built step by step up to express in all its generality the principle of relativity. Poincaré thus showed the invariance of the Maxwell equations under the Lorentz transformation. In doing so, he also discovered the right composition law for velocities. Poincaré's approach to philosophy is detailed to help the reader to understand what a theory meant to him.     

拉曼效应对低双折射光纤偏振特性的影响

在低双折射光纤中,利用线偏振光满足的包含拉曼效应的非线性耦合模传输方程,通过引入斯托克斯参量,导出了斯托克斯参量所满足的耦合模传输方程.利用庞加莱球图示法,描述了拉曼增益效应作用下光波偏振态的演化,研究分析了拉曼效应对低双折射光纤中光波偏振态演化规律的影响.结果表明,当输入功率与运动常量满足一定关系时,拉曼增益效应改变了光波传输时其偏振态演化周期和偏振态的椭圆率.     

On the Poincaré Gauge Theory of Gravitation

We present a compact, self-contained review of the conventional gauge theoretical approach to gravitation based on the local Poincaré group of symmetry transformations. The covariant field equations, Bianchi identities and conservation laws for angular momentum and energy-momentum are obtained.     

Poincaré gauge theory from self-coupling

Poincaré gauge theory is derived from a linear theory by the method suggested by Gupta for deriving Einstein’s general relativity from the linear theory of a spin-2 field. Non-linearity is introduced by requiring that a set of tensor fields be coupled to the Noether currents of the Poincaré group (energy-momentum and spin).     

Physical principles in quantum field theory and in covariant harmonic oscillator formalism

It is shown that both covariant harmonic oscillator formalism and quantum field theory are based on common physical principles which include Poincaré covariance, Heisenberg's space-momentum uncertainty relation, and Dirac's “C-number” time-energy uncertainty relation. It is shown in particular that the oscillator wave functions are derivable from the physical principles which are used in the derivation of the Klein-Nishina formula.     

Poincaré’s Relativistic Physics: Its Origins and Nature

Henri Poincaré (1854–1912) developed a relativistic physics by elevating the empirical inability to detect absolute motion, or motion relative to the ether, to the principle of relativity, and its mathematics ensured that it would be compatible with that principle. Although Poincaré’s aim and theory were similar to those of Albert Einstein (1879–1955) in creating his special theory of relativity, Poincaré’s relativistic physics should not be seen as an attempt to achieve Einstein’s theory but as an independent endeavor. Poincaré was led to advance the principle of relativity as a consequence of his reflections on late nineteenth-century electrodynamics; of his conviction that physics should be formulated as a physics of principles; of his conventionalistic arguments on the nature of time and its measurement; and of his knowledge of the experimental failure to detect absolute motion. The nonrelativistic theory of electrodynamics of Hendrik A.Lorentz (1853–1928) of 1904 provided the means for Poincaré to elaborate a relativistic physics that embraced all known physical forces, including that of gravitation. Poincaré did not assume any dynamical explanation of the Lorentz transformation, which followed from the principle of relativity, and he did not seek to dismiss classical concepts, such as that of the ether, in his new relativistic physics. Shaul Katzir teaches in the Graduate Program in History and Philosophy of Science, Bar Ilan University.     

On the difference between Poincaré and Lorentz gravity

The Poincaré invariance of GR is usually interpreted as Lorentz invariance plus diffeomorphism invariance. In this paper, by introducing the local inertial coordinates (LIC), it is shown that a theory with Lorentz and diffeomorphism invariance is not necessarily Poincaré invariant. Actually, the energy–momentum conservation is violated there. On the other hand, with the help of the LIC, the Poincaré invariance is reinterpreted as an internal symmetry. In this formalism, the conservation law is derived, which has not been sufficiently explored before.