Global Orbits Research Articles

Global orbit of a complicated nonlinear system with the global dynamic frequency method

Global orbits connect the saddle points in an infinite period through the homoclinic and heteroclinic types of manifolds. Different from the periodic movement analysis, it requires special strategies to obtain expression of the orbit and detect the associated profound dynamic behaviors, such as chaos. In this paper, a global dynamic frequency method is applied to detect the homoclinic and heteroclinic bifurcation of the complicated nonlinear systems. The so-called dynamic frequency refers to the newly introduced frequency that varies with time t, unlike the usual static variable. This new method obtains the critical bifurcation value as well as the analytic expression of the orbit by using a standard five-step hyperbolic function-balancing procedure, which represents the influence of the higher harmonic terms on the global orbit and leads to a significant reduction of calculation workload. Moreover, a new homoclinic manifold analysis maps the periodic excitation onto the target global manifold that transfers the chaos discussion of non-autonomous systems into the orbit computation of the general autonomous system. That strategy unifies the global bifurcation analysis into a standard orbit approximation procedure. The numerical simulation results are shown to compare with the predictions.

Neimark\u2013Sacker bifurcation and the generate cases of Kopel oligopoly model with different adjustment speed

In this paper, bifurcations and chaotic behaviours of Kopel oligopoly model with different adjustment speed are discussed. The results imply that the Kopel oligopoly model undergoes flip bifurcation, Neimark–Sacker bifurcation, 1:3 and 1:4 resonances, which could induce complex dynamics, especially global behaviours between different orbits. The conditions for the occurrence of three different kinds of bifurcation are derived. Furthermore, the numerical simulations provide us the case study of theoretical analysis and the corresponding dynamical behaviours, especially the occurrence of global orbits.

Topological analysis of stability regions of power systems considering global orbits

In this paper, the stability region of power systems is studied from a geometrical point of view. A new method is proposed to obtain the global orbits (GOs) which have the significance for making the stability region be more complete. This method is based on the theory of invariant manifolds and focuses on dynamical behavior at infinity. The boundary of the stability region (BSR) with respect to the stable manifolds of dimension-1 and dimension-2 are given to describe the topological characteristics of the stability region of a multi-machine system. The effect of damping coefficient on the stability region is discussed through the variation of the singularities at infinity. The GOs of the power system when subjected to large disturbance are analyzed with the time domain simulation. The validity of the proposed approach is demonstrated on a 16-machine 68-bus system. The simulation results reveal the types and characteristics of the GOs, which play an important role in the prediction of stability/instability of the postfault system.

Editors' introduction: Chaucer's global orbits and global communities

AbstractEncompassing perspectives beyond what Braj B. Kachru terms the “Inner Circle” of Anglophone hegemony, this collection of essays presents a vivid and distinct opportunity to appreciate how Chaucer's The Canterbury Tales adapts to life across disparate languages (Persian, Brazilian Portuguese, Japanese, Danish, Spanish, Turkish, American Sign Language, and internal varieties of London English) while also moving across cultures (Shiite Iran, Brazilian Gauchoria, Tokyo academia, rural Denmark, Mexican universities, Ottoman Turkey, US Deaf culture, and London's multiethnic East End communities). The term compaignye, a quintessentially Chaucerian keyword absorbed into Middle English through Anglo‐French, suggests an intimate multitude of people sharing a common interest or purpose. To this end, our textual compaignye incorporates contributors who are academics in the traditional sense as well as authors and artists who operate alongside normative structures of the academy. Many of the compaignye's so‐called “nonacademics” have garnered prestigious awards for their creative work, and “amateur” cultural artifacts reveal that a deep love of Chaucerian material characterizes both academic and nonacademic endeavors. Such a compaignye, with its many forms of expertise, can expand our knowledge by creating “a vibrant resonance,” as Carolyn Dinshaw explains, between “amateur [popular] medievalism and professional [academic] medievalism.” As the fields of comparative literary analysis, translation theory, and medievalism studies increasingly move away from models of textual “fidelity” that reinforce static distinctions between an “original” and a “derivative,” we offer this cluster of essays as an example of the benefits gained by pluralizing the modes through which we understand processes of linguistic transfer and cultural adaptation.

Rigorous numerics for fast-slow systems with one-dimensional slow variable: topological shadowing approach

We provide a rigorous numerical computation method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system $x' = f(x,y,\epsilon), y' = \epsilon g(x,y,\epsilon)$ with one-dimensional slow variable $y$. Our validation procedure is based on topological tools called isolating blocks, cone condition and covering relations. Such tools provide us with existence theorems of global orbits which shadow singular orbits in terms of a new concept, the covering-exchange. Additional techniques called slow shadowing and $m$-cones are also developed. These techniques give us not only generalized topological verification theorems, but also easy implementations for validating trajectories near slow manifolds in a wide range, via rigorous numerics. Our procedure is available to validate global orbits not only for sufficiently small $\epsilon > 0$ but all $\epsilon$ in a given half-open interval $(0,\epsilon_0]$. Several sample verification examples are shown as a demonstration of applicability.

ON PARTIAL ORBITS AND STABILIZERS

Within the framework of partial actions of groups, we introduce here the partial stabilizer and prove that it coincides with the global stabilizer. In addition, we define the partial orbits and show that they are completely determined by the global orbits. Finally, we use this result to prove that the partial action (X,�) is n − transitive if and only if its enveloping action (T,�) is n − transitive.

Real-time zenith tropospheric delays in support of numerical weather prediction applications

The Geodetic Observatory Pecný (GOP) routinely estimates near real-time zenith total delays (ZTD) from GPS permanent stations for assimilation in numerical weather prediction (NWP) models more than 12years. Besides European regional, global and GPS and GLONASS solutions, we have recently developed real-time estimates aimed at supporting NWP nowcasting or severe weather event monitoring. While all previous solutions are based on data batch processing in a network mode, the real-time solution exploits real-time global orbits and clocks from the International GNSS Service (IGS) and Precise Point Positioning (PPP) processing strategy. New application G-Nut/Tefnut has been developed and real-time ZTDs have been continuously processed in the nine-month demonstration campaign (February–October, 2013) for selected 36 European and global stations. Resulting ZTDs can be characterized by mean standard deviations of 6–10mm, but still remaining large biases up to 20mm due to missing precise models in the software. These results fulfilled threshold requirements for the operational NWP nowcasting (i.e. 30mm in ZTD). Since remaining ZTD biases can be effectively eliminated using the bias-reduction procedure prior to the assimilation, results are approaching the target requirements in terms of relative accuracy (i.e. 6mm in ZTD). Real-time strategy and software are under the development and we foresee further improvements in reducing biases and in optimizing the accuracy within required timeliness. The real-time products from the International GNSS Service were found accurate and stable for supporting PPP-based tropospheric estimates for the NWP nowcasting.

Periodically growing solutions in a class of strongly monotone semiflows

We study the behavior of unbounded global orbits in a class of stronglymonotone semiflows and give a criterion for the existence of orbitswith periodic growth. We also prove the uniqueness and asymptoticstability of such orbits. We apply our results to a certain class ofnonlinear parabolic equations including a weakly anisotropic curvatureflow in a two-dimensional annulus and show the convergence of the solutionsto a periodically growing solution which grows up in infinite timechanging its profile time-periodically.

Global bifurcations and multi-pulse orbits of a parametric excited system with autoparametric resonance

We consider an autoparametric system which consists of an oscillator coupled with a parametrically excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in principal parametric resonance. The system contains the most general type of quadratic and cubic non-linearities. The method of second-order averaging is used to yield a set of autonomous equations of the second-order approximations to the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Shilnikov-type multi-pulse homoclinic orbits in the averaged equations. The results obtained above mean the existence of amplitude-modulated chaos in the Smale horseshoe sense in the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse chaotic motions of the parametric excited system with autoparametric resonance are also found by using numerical simulation.

Orbit correction using an eigenvector method with constraints for synchrotron radiation sources

An eigenvector method with constraints (EVC) is proposed as a new orbit correction scheme for synchrotron light sources. EVC efficiently corrects the global orbit in a storage ring, and can simultaneously perform exact correction of local orbits without deterioration of the global orbit. To demonstrate the advantages of EVC over the ordinary eigenvector method (EV), we carried out experimental studies at the Photon Factory storage ring (PF-ring) and the Photon Factory Advanced Ring (PF-AR) at the High Energy Accelerator Research Organization (KEK). The performance of EVC was systematically examined at PF-ring and PF-AR. The experimental results agreed well with the simulated ones. Consequently, we confirmed that EVC easily realized orbit correction for both global and local orbits, and that it was very effective for the beam stabilization of synchrotron radiation (SR) sources.

Symbolic Vector Dynamics Approach to Initial Condition and Control Parameters Estimation of Coupled Map Lattices

In this paper, we extend symbolic dynamics, a standard analytical method for 1-D chaotic map, and initiate a solution to the problem of estimation in coupled map lattices (CMLs) by introducing the symbolic vector dynamics. We develop a novel technique for estimating initial conditions. We also expand the applicable scope of a word-lifting technique for parameter estimation from a 1-D chaotic map to CMLs. Both theoretical and experimental results show that those algorithms can construct one-to-one correspondence between the set of global orbits and the set of admissible codes. Therefore, we provide novel analytical techniques for understanding turbulences in CMLs.

Computing global orbits of the forced spherical pendulum

Orbits of the spherical pendulum with time-periodic forcing are considered. A numerical framework is developed, which allows orbits to explore the entire “globe”: the spherical pendulum is considered as an invariant manifold in an ambient six-dimensional Euclidean space. The numerical integrator is the second-order Störmer–Verlet method coupled with the Shake–Rattle algorithm. The algorithm preserves numerically the phase space of the sphere, which is a manifold, to machine accuracy. Poincaré sections, restricted to the configuration space, are used to illustrate the transition from oscillatory behavior to chaotic solutions, as the amplitude of the pivot motion is changed. The qualitative change in the Poincaré sections from regular to chaotic behavior is in excellent qualitative agreement with corresponding computations of the Lyapunov exponents (LEs). The LEs are also computed using a novel variant of the Shake–Rattle algorithm. The results show that irregular behavior can explore the entire sphere—even at low forcing amplitudes—and therefore local methods which parameterize only part of the sphere are inadequate in general, and may lead to spurious dynamics. The numerical framework provides a tool for detailed investigation of the symmetric chaos of the forced spherical pendulum.

Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime.

We give experimental and numerical evidence for a new dynamical regime in the operation of semiconductor lasers subject to delayed optical feedback occurring for short delay times. This short cavity regime is dominated by a striking dynamical phenomenon: regular pulse packages forming a robust low-frequency state with underlying fast, regular intensity pulsations. We demonstrate that these regular pulse packages correspond to trajectories moving on global orbits comprising several destabilized fixed points within the complicated phase space structure of this delay system.

Global stability and periodic orbits for a two-patch diffusion predator–prey model with time delays

Global stability and periodic orbits for a two-patch diffusion predator–prey model with time delays

Accuracy of absolute precipitable water vapor estimates from GPS observations

We present GPS, radiosonde and microwave radiometer (MWR) estimates of precipitable water vapor (PW) at Cape Grim, Tasmania, during November and December 1995. The rms differences between GPS and radiosonde, MWR and radiosonde and GPS and MWR estimates of PW were 1.5 mm, 1.3 mm and 1.4 mm, respectively, whilst the biases between the three systems were ∼0.2 mm. However, there are occasions when the amount of PW was underestimated by GPS whilst at other times was over‐estimated by MWR. The average overlap error of the GPS estimates of PW between adjacent daily solutions is related to the orbit overlap error and we removed a 2 mm bias introduced using International GPS Service orbits by estimating more accurate global orbits. The discrepancies of up to 3–4 mm between the MWR and GPS systems are not caused by rain, waveguide losses, varying waveguide temperature, detector non‐linearity or inaccurate estimates of the mean radiating temperature of the atmosphere. However, small differences between mapping functions at low elevations can produce biases comparable with the bias between the two systems. Consequently, we suspect that the biases arise because the mapping functions do not represent the localized atmospheric conditions at Cape Grim. The most accurate GPS estimates are achieved when the GPS analysis contains station separations of more than 2000 km, an elevation cutoff angle of 12° is used and the CFA2.2 wet mapping function is used to map the wet delay at any angle to the delay in the zenith.

Extended symbolic dynamics in bistable CML: Existence and stability of fronts

We consider a diffusive coupled map lattice (CML) for which the local map is piecewise affine and has two stable fixed points. By introducing a spatio-temporal coding, we prove the one-to-one correspondence between the set of global orbits and the set of admissible codes. This relationship is applied to the study of the (uniform) fronts' dynamics. It is shown that, for any given velocity in [−1, 1], there is a parameter set for which the fronts with that velocity exist and their shape is unique. The dependence of the fronts' velocity on the local map's discontinuity is proved to be a Devil's staircase. Moreover, the linear stability of the global orbits which do not reach the discontinuity follows directly from our simple map. For the fronts, this statement is improved and as a consequence, the velocity of all the propagating interfaces is computed for any parameter. The fronts' are shown to be also nonlinearly stable under some restrictions on the parameters. Actually, these restrictions follow from the co-existence of uniform fronts and non-uniformly travelling fronts for strong coupling. Finally, these results are extended to some C ∞ local maps.

On the global orbits in a bistable CML.

In an infinite one-dimensional coupled map lattice (CML) for which the local map is piecewise affine and bistable, we study the global orbits using a spatiotemporal coding introduced in a previous work. The set of all the fixed points is first considered. It is shown that, under some restrictions on the parameters, the latter is a Cantor set, and we introduce an order to study the fixed points' existence. This also involves the proof of the coexistence of propagating fronts and stationary structures. In the second part, we analyze the global orbits which occur for strong coupling using the splitting of the dynamics into two independent (sub-)lattices, and emphasize the description of various traveling structures. (c) 1997 American Institute of Physics.

Calibration of the TOPEX/POSEIDON altimeters at Lampedusa: Additional results at Harvest

The Italian Lampedusa island in the Mediterranean Sea was chosen by the Centre National d'Etudes Spatiales (CNES) to support the verification of altimeters on board the TOPEX/POSEIDON satellite. The orbit was phased in such a way that a descending pass overflies a small islet called Lampione, 20 km west of Lampedusa. During the TOPEX/POSEIDON verification phase, until mid‐December 1992, the satellite passed nine times over Lampedusa. The Lampedusa site was equipped with all the required instrumentation to provide independent and accurate knowledge of the altimeter system components: sea level, orbit, and geophysical corrections. In addition, a reprocessing of the altimeter data was performed to better adjust the waveform parameters estimation especially when quasi‐specular echoes due to flat sea conditions have disturbed the on‐board estimation. Three different local orbits computed by expert groups (Centre d'Etudes et de Recherches Geodynamiques et Astronomiques, Delft University of Technology, and Jet Propulsion Laboratory), plus CNES and NASA global orbits were considered. The bias of the CNES altimeter (SSALT) was then calculated, based on six overflights, giving a mean value of 0.7 cm. A SSALT bias also was calculated at Harvest, the NASA calibration site, using six overflights. The deduced bias is 1.3 cm. Integrating both site results gives a mean SSALT bias of 1 cm with an uncertainty of 2.4 cm. The NASA altimeter (ALT) bias computed at Lampedusa is −18.5 cm with an uncertainty of 3.4 cm (the negative sign meaning that the altimeter is measuring short), but this result must be carefully considered because it was obtained with only two overflights. Considering the Lampedusa SSALT calibration results and those derived from statistical repeat track analysis giving an estimation of the relative bias between SSALT and ALT (e.g., Le Traon et al., this issue) a bias of −14.8 cm with an uncertainty of 2.6 cm was found for the ALT altimeter.