Universality and self-similarity in the bifurcations of circle maps

The bifurcation structure in a two-parameter family of circle maps is considered. These maps have a (topological) degree that may be different from one. A generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described. Continuity arguments are used to establish the existence of periodic orbits. By plotting the locus of parameter values associated with superstable cycles, self-similar bifurcations are found. These bifurcations are a generalization of the familiar period-doubling cascade in maps with one extrema, to two-parameter maps with two extrema. Finally, a scheme for the global organization of bifurcation in these maps is proposed.     

微扰法解由正弦波驱动的非线性漂移波的分岔

在以驱动波相速度运动的坐标系中,用微扰法讨论。在正弦波驱动下的非线性漂移波的分波方程。结果表明,在文献[1]中观察到的波包能量的滞后分岔和由定态向周期态的分岔可以统一地解析描述,它们分别对应某一非线性共振模式在时间维上的鞍结点分岔和Hopf分岔。波包能量失稳的频率是该模式的本征频率,除多普勒移动外,它的大小还因非线性效应而不同于其在实验室坐标系中的线性值。 关键词：     

Unstable behaviors of classical solutions in spinor-type conformal invariant fermionic models

It is well known that instantons are classical topological solutions existing in the context of quantum field theories that lie behind the standard model of particles. To provide a better understanding for the dynamical nature of spinor-type instanton solutions, conformal invariant pure spinor fermionic models that admit particle-like solutions for the derived classical field equations are studied in this work under cosine wave forcing. For this purpose, the effects of external periodic forcing on two systems that have different dimensions and quantum spinor numbers and have been obtained under the use of Heisenberg ansatz are investigated by constructing their Poincaré sections in phase space. As a result, bifurcations and chaos are observed depending on the excitation amplitude of the external forcing in both pure spinor fermionic models.     

拓扑绝缘体椭球粒子的电磁散射

利用麦克斯韦方程组,用球矢量波函数展开了长椭球坐标系中的电磁场分量; 根据拓扑绝缘体的本构关系,修正了椭球内外的电磁场;利用拓扑绝缘体的边界条件, 推导了散射系数和散射电磁场.模拟结果表明:当时间反演对称打破时,拓扑磁电参数对散射截面有明显影响.     

Symmetry breaking via global bifurcations of modulated rotating waves in hydrodynamics

The combined experimental and numerical study finds a complex mechanism of Z(2) symmetry breaking involving global bifurcations for the first time in hydrodynamics. In addition to symmetry breaking via pitchfork bifurcation, the Z(2) symmetry of a rotating wave that occurs in Taylor-Couette flow is broken by a global saddle-node-infinite-period (SNIP) bifurcation after it has undergone a Neimark-Sacker bifurcation to a Z(2)-symmetric modulated rotating wave. Unexpected complexity in the bifurcation structure arises as the curves of cyclic pitchfork, Neimark-Sacker, and SNIP bifurcations are traced towards their apparent merging point. Instead of symmetry breaking due to a SNIP bifurcation, we find a more complex mechanism of Z(2) symmetry breaking involving nonsymmetric two-tori undergoing saddle-loop homoclinic bifurcations and complex dynamics in the vicinity of this global bifurcation.     

Quantum entanglement and classical bifurcations in a coupled two-component Bose-Einstein condensate

We investigate the relation between quantum states and classical fixed-point bifurcations in a coupled two-component Bose-Einstein condensate (BEC). It is shown that the classical bifurcations are closely related to a topological change of the corresponding quantum levels, and such a structure change can be manifested in the entanglement properties of the corresponding quantum states. That is, the entanglement of the quantum states displays some peaks near the classical bifurcation points.     

Effect of various periodic forces on duffing oscillator

Bifurcations and chaos in the ubiquitous Duffing oscillator equation with different external periodic forces are studied numerically. The external periodic forces considered are sine wave, square wave, rectified since wave, symmetric saw-tooth wave, asymmetric saw-tooth wave, rectangular wave with amplitude-dependent width and modulus of sine wave. Period doubling bifurcations, chaos, intermittency, periodic windows and reverse period doubling bifurcations are found to occur due to the applied forces. A comparative study of the effect of various forces is performed.     

Soliton interaction with an external traveling wave

The dynamics of soliton pulses in the nonlinear Schrodinger equation (NLSE) driven by an external traveling wave is studied analytically and numerically. The Hamiltonian structure of the system is used to show that, in the adiabatic approximation for a single soliton, the problem is integrable despite the large number of degrees of freedom. Fixed points of the system are found, and their linear stability is investigated. The fixed points correspond to a Doppler shifted resonance between the external wave and the soliton. The structure and topological changes of the phase space of the soliton parameters as functions of the strength of coupling are investigated. A physical derivation of the driven NLSE is given in the context of optical pulse propagation in asymmetric, twin-core optical fibers. The results can be applied to soliton stabilization and amplification.     

On the theory of wave packets

In this paper we discuss some aspects of the theory of wave packets. We consider a popular noncovariant Gaussian model used in various applications and show that it predicts too slow a longitudinal dispersion rate for relativistic particles. We revise this approach by considering a covariant model of Gaussian wave packets, and examine our results by inspecting a wave packet of arbitrary form. A general formula for the time dependence of the dispersion of a wave packet of arbitrary form is found. Finally, we give a transparent interpretation of the disappearance of the wave function over time due to the dispersion—a feature often considered undesirable, but which is unavoidable for wave packets. We find, starting from simple examples, proceeding with their generalizations and finally by considering the continuity equation, that the integral over time of both the flux and probability densities are asymptotically proportional to the factor 1/|x|² in the rest frame of the wave packet, just as in the case of an ensemble of classical particles.     

Surface waves at the interface between a dielectric and a topological insulator

Surface waves that propagate along the interface between an isotropic linear or nonlinear (of the Kerr type) dielectric and a topological insulator have been studied theoretically. A dispersion relation for surface waves, which are represented by superpositions of TE and TM waves, has been obtained. This hybridization occurs because, upon passage through the interface, the polarization of a surface wave changes, which is caused by an induced surface current (which is transverse to the electric field vector of the wave). The surface current of this kind is characteristic of topological insulators. Expressions for the energy flux transferred by a surface wave have been given.     

Topological nodal line semimetals

We review the recent,mainly theoretical,progress in the study of topological nodal line semimetals in three dimensions.In these semimetals,the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone,and any perturbation that preserves a certain symmetry group(generated by either spatial symmetries or time-reversal symmetry) cannot remove this crossing line and open a full direct gap between the two bands.The nodal line(s) is hence topologically protected by the symmetry group,and can be associated with a topological invariant.In this review,(ⅰ) we enumerate the symmetry groups that may protect a topological nodal line;(ⅱ) we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface,establishing a topological classification;(ⅲ) for certain classes,we review the proposals for the realization of these semimetals in real materials;(ⅳ) we discuss different scenarios that when the protecting symmetry is broken,how a topological nodal line semimetal becomes Weyl semimetals,Dirac semimetals,and other topological phases;and(ⅴ) we discuss the possible physical effects accessible to experimental probes in these materials.     

The effect of Lagrangian chaos on locking bifurcations in shear flows

The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.     

Coupling induced topological transition in a ring of chaotic oscillators

A ring of diffusively coupled R?ssler oscillators, which can develop the conventional rotating wave from high-dimensional chaos by increasing the coupling ɛ continuously is studied. The chaotic generator for the rotating wave emerges around ɛ = ɛ, where the topological transition induced by the coupling not only changes the topological structure of all the oscillators, which share a common strange attractor, but also changes them into being different from each other. Starting from this transition, infinitely long range temporal correlation and spatial order in the style of antiphase state are established gradually, which gives rise to the chaotic generator of the rotating wave. Received 15 March 2001     

Complex oscillations and waves of calcium in pancreatic acinar cells

We perform a bifurcation analysis of a model of Ca²⁺ wave propagation in the basal region of pancreatic acinar cells. The model we consider was first presented in Sneyd et al. [J. Sneyd, K. Tsaneva-Atanasova, J.I.E. Bruce, S.V. Straub, D.R. Giovannucci, D.I. Yule, A model of calcium waves in pancreatic and parotid acinar cells, Biophys. J. 85 (2003) 1392–1405], where a partial bifurcation analysis was given of the model in the absence of diffusion. We obtain more complete information about bifurcations of the diffusionless model via numerical studies, then analyse the spatially extended model by numerical investigation of the travelling wave equations and direct numerical solution of the model equations. We find solitary waves in the model equations arising from homoclinic bifurcations in the travelling wave equations. The solitary waves exist and appear to be stable for a significant interval of the primary bifurcation parameter (i.e., the concentration of inositol trisphosphate) but are eventually replaced by irregular spatio-temporal behaviour. The homoclinic bifurcations are related to a number of complicated mathematical structures in the travelling wave equations, including an anomalous homoclinic-Hopf bifurcation, heteroclinic bifurcations between an equilibrium and a periodic orbit, and homoclinic bifurcations of periodic orbits.     

Noninertial Effects on a Dirac Neutral Particle Inducing an Analogue of the Landau Quantization in the Cosmic String Spacetime

We discuss the behavior of external fields interacting with a Dirac neutral particle with a permanent electric dipole moment in order to achieve relativistic bound state solutions in a noninertial frame and in the presence of a topological defect spacetime. We show that the noninertial effects of the Fermi?CWalker reference frame induce a radial magnetic field even in the absence of magnetic charges, which is influenced by the topology of the cosmic string spacetime. We then discuss the conditions that the induced fields must satisfy to yield the relativistic bound states corresponding to the Landau?CHe?CMcKellar?CWilkens quantization in the cosmic string spacetime. Finally, we obtain the Dirac spinors for positive-energy solutions and the Gordon decomposition of the Dirac probability current.     

Bifurcations and Dynamics of Traveling Wave Solutions to a Fujimoto-Watanabe Equation

In this paper, we study the bifurcations and dynamics of traveling wave solutions to a Fujimoto-Watanabe equation by using the method of dynamical systems. We obtain all possible bifurcations of phase portraits of the system in different regions of the parametric space. Then we show the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, compactions and kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions and periodic wave solutions are implicitly given,while the expressions of kink-like and antikink-like wave solutions are explicitly shown. The dynamics of these new traveling wave solutions will greatly enrich the previews results and further help us understand the physical structures and analyze the propagation of the nonlinear wave.     

Travelling Waves in a Chain¶of Coupled Nonlinear Oscillators

In a chain of nonlinear oscillators, linearly coupled to their nearest neighbors, all travelling waves of small amplitude are found as solutions of finite dimensional reversible dynamical systems. The coupling constant and the inverse wave speed form the parameter space. The groundstate consists of a one-parameter family of periodic waves. It is realized in a certain parameter region containing all cases of light coupling. Beyond the border of this region the complexity of wave-forms increases via a succession of bifurcations. In this paper we give an appropriate formulation of this problem, prove the basic facts about the reduction to finite dimensions, show the existence of the ground states and discuss the first bifurcation by determining a normal form for the reduced system. Finally we show the existence of nanopterons, which are localized waves with a noncancelling periodic tail at infinity whose amplitude is exponentially small in the bifurcation parameter. Received: 10 September 1999 / Accepted: 15 December 1999     

Rotating vortex dipoles in ferromagnets

Vortex-antivortex pairs are spontaneously created in magnetic elements. In the case of opposite vortex polarities the pair has a nonzero topological charge, and it behaves as a rotating vortex dipole. We find theoretically and confirm numerically its energy as a function of angular momentum and the associated rotation frequencies. The annihilation process of the pair changes the topological charge while the energy is monotonically decreasing. The change of topological charge affects the dynamics profoundly. We finally discuss the implications of our results for Bloch point dynamics.