Isochronous System Research Articles

The quantitative analysis of homoclinic orbits from quadratic isochronous systems

In this paper, we investigate a pair of symmetric homoclinic orbits given by a Perturbation-Incremental (PI) method for the quadratic isochronous systems, which can bifurcate from the periodic orbits of the quadratic isochronous centers. The existing researches on the isochronous system are mostly from a qualitative perspective, while we have studied the quadratic isochronous system through the quantitative analysis method. As a quantitative analysis method, the PI method can not only determine the position of the limit cycle and the homoclinic (heteroclinic) orbit, but also provide an analytical expression of the asymptotic solution. At the same time, this method combines the significant characteristics of the perturbation method and the parameter incremental method. By utilizing the perturbation method, we obtain the analytic expressions of the zero-order perturbation solutions of homoclinic orbits when the parameters of systems are small. Then, the approximate analytical expressions of homoclinic orbits can be obtained by using the parameter incremental method when they are iterated from small parameters to large parameters. The numerical results will be compared with those from the Runge–Kutta method. The theoretical results agree very well with numerical simulation.

The number of limit cycles from the perturbation of a quadratic isochronous system with two switching lines

<p style='text-indent:20px;'>In this paper, we give an upper bound (for <inline-formula><tex-math id="M1">\begin{document}$ n\geq3 $\end{document}</tex-math></inline-formula>) and the least upper bound (for <inline-formula><tex-math id="M2">\begin{document}$ n = 1,2 $\end{document}</tex-math></inline-formula>) of the number of limit cycles bifurcated from period annuli of a quadratic isochronous system under the piecewise polynomial perturbations of degree <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula>, respectively. The results improve the conclusions in [<xref ref-type="bibr" rid="b19">19</xref>].</p>

Sedimentary Evolution Characteristics of Fine-Grained Lithofacies under the High-Resolution Isochronous Shelf System: Insights from the Wufeng-Longmaxi Shales in the Sichuan Basin

Abstract The deposition and evolution of fine-grained sediments is a hot topic in fine-grained sedimentary rock studies and is important for accurately evaluating shale gas sweet spots. In this paper, the fine-grained deposition and evolution characteristics of the Wufeng-Longmaxi shales, major targets for Chinese shale gas exploration, were studied by using core observations, thin section analyses, scanning electron microscopy, geochemical analysis, and fossil identification. This work accurately identified six typical lithofacies; among them, the organic matter-rich siliceous shale facies (OMRSSF), the high-organic matter siliceous argillaceous shale facies (HOMSASF), and the medium-high organic matter low calcareous siliceous shale facies (M-HOMLCSASF) are favorable facies for shale gas exploration. The high-resolution isochronous unit in the shelf fine-grained sedimentary system was established, and the differential evolution of lithofacies in the system tract was discussed. The lithofacies deposition and differentiation in the transgressive system tract were controlled by the transgressive scale and tectonics under increasingly shallow water conditions. The lithofacies deposition and differentiation in the regressive system tract were controlled by tectonics and the preexisting lithofacies. The lithofacies in the regressive system tract had more frequent facies transitions and greater differentiation than those in the transgressive system tract, and they exhibited significant spatiotemporal inheritance. Sequential differential sedimentary sequences and symmetric differential sedimentary sequences were distinguished in the continental shelf sedimentary system. The lithofacies depocenters and subsidence centers were consistent in the transgressive system tract, while the tectonically active paleocontinent was important in the regression system tract. This study is of great significance for further high-resolution exploration of marine shale and improvement of the theory of shelf fine-grained sedimentary systems.

Uniformly Isochronous Quartic Systems

We show that the order of a fine focus of a uniformly isochronous quartic system is at most 8. Moreover, the system has a center if and only if it satisfies the composition condition. In particular, we prove a recent conjecture about these systems.

Fast Drift and Diffusion in a Class of Isochronous Systems with the Windows Method

We study the problem of Arnold's diffusion in an example of isochronous system by using a geometrical method known as Windows Method. Despite the simple features of this example, we show that the absence of an anisochrony term leads to several substantial difficulties in the application of the method, requiring some additional devices as non-equally spaced transition chains and variable windows. In this way we are able to obtain a set of fast orbits whose drifting time matches, up to a constant, the time obtained via variational methods.

An Application of Nekhoroshev Theory to the Study of the Perturbed Hydrogen Atom

We return to the Keplerian or n-shell approximation to the hydrogen atom in the presence of weak static electric and magnetic fields. At the classical level, this is a Hamiltonian system with the phase space S2 × S2. Its principal order Hamiltonian H0 was known already to Pauli in 1926. H0 defines an isochronous system with a linear flow on S2 × S2 and with frequencies depending on the external fields. Small perturbations of H0 due to higher order terms can be studied by further normalization, either resonant or nonresonant. We study the question, raised previously, of how to decide for given parameters of the fields what normalization should be used and with regard to which resonances. We base this analysis on the Nekhoroshev theory—a branch of the Hamiltonian perturbation theory that complements the Kolmogorov-Arnold-Moser theorem. Our answer depends on the a priori choice of the maximal order N of resonances that are going to be taken into account (a cutoff). For any given N, there is a decomposition of the parameter space into resonant and nonresonant zones, and a normal form with a remainder of order \(\exp (-N)\) may be consistently constructed in each of such zones.

A Higher-Order Period Function and Its Application

This paper derives explicit formulas of the q th period bifurcation function for any perturbed isochronous system with a center, which improve and generalize the corresponding results in the literature. Based on these formulas to the perturbed quadratic and quintic rigidly isochronous centers, we prove that under any small homogeneous perturbations, for ε in any order, at most one critical period bifurcates from the periodic orbits of the unperturbed quadratic system. For ε in order of 1, 2, 3, 4 and 5, at most three critical periods bifurcate from the periodic orbits of the unperturbed quintic system. Moreover, in each case, the upper bound is sharp. Finally, a family of perturbed quintic rigidly isochronous centers is shown, which has three, for ε in any order, as the exact upper bound of the number of critical periods.

The MR-TOF-MS isobar separator for the TITAN facility at TRIUMF

At TRIUMF’s Ion Trap for Atomic and Nuclear Science (TITAN) a multiple-reflection time-of-flight mass spectrometer (MR-TOF-MS) will extend TITAN’s capabilities and facilitate mass measurements and in-trap decay spectroscopy of exotic nuclei that so far have not been possible due to strong isobaric contamination. This MR-TOF-MS will also enable mass measurements of very short-lived nuclides (half-life > 1 ms) that are produced in very low quantities (a few detected ions overall). In order to allow the installation of an MR-TOF-MS in the restricted space on the platform, on which the TITAN facility is located, novel mass spectrometric methods have been developed. Transport, cooling and distribution of the ions inside the device is done using a buffer gas-filled RFQ-based ion beam switchyard. Mass selection is achieved using a dynamic retrapping technique after time-of-flight analysis in an electrostatic isochronous reflector system. Only due to the combination of these novel methods the realization of an MR-TOF-MS based isobar separator at TITAN has become possible. The device has been built, commissioned off-line and is currently under installation at TITAN.

Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters

This article investigates a class of near-Hamiltonian systems and obtains some new conditions for the existence of multiple limit cycles with the help of the first order Melnikov function. As applications to the obtained main results, a cubic reversible isochronous system under cubic polynomial perturbations is studied.

Limit Cycles of Perturbed Cubic Isochronous Center via the Second Order Averaging Method

In this paper, we bound the number of limit cycles for a class of cubic reversible isochronous system inside the class of all cubic polynomial differential systems. By applying the averaging method of second order to this system, it is proved that at most eight limit cycles can bifurcate from the period annulus. Moreover, this bound is sharp.

On a nonlinear isochronous system

A system of second-order nonlinear ordinary differential equations is considered. It is shown analytically that the solutions to this system are isochronous, which is not typical for nonlinear systems. It is also shown that a periodic delta function is a limit of the solution if the amplitude tends to infinity.

Construction of rare-RI ring at RIKEN RI Beam Factory

An isochronous mass spectroscopy system using a newly constructed storage ring named the “rare-RI ring” is expected to be implemented at the RIKEN Nishina Center to determine the masses of short-lived rare nuclei including those in the r-process region with a relative precision of the order of 10-6 even for only one particle. In an isochronous storage ring, the mass is determined by measuring the revolution time of each nucleus. Our rare-RI ring consists of six magnetic sectors, and each sector consists of four bending magnets. To precisely optimize the isochronous conditions of the circulating particles for large acceptance, we install ten trim coils to half of the bending magnets. A fast-response and fast-charging kicker system enables selective and efficient injection of the produced rare nuclei into the ring one by one, along with facilitating efficient extraction of the circulating particles for time-of-flight measurement. Construction of the rare-RI ring was begun in the middle of the fiscal year 2012 at the RIKEN RI Beam Factory, and the ring is expected to be fully functional by 2015.

Amplitude Independent Frequency Synchroniser for a Cubic Planar Polynomial System

The problem of local linearizability of the planar linear center perturbed by cubic non- linearities in all generalities on the system parameters (14 parameters) is far from being solved. The synchronization problem (as noted in Pikovsky, A., Rosenblum, M., and Kurths, J., 2003, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, UK, and Blekhman, I. I., 1988, Synchronisation in Science and Technology, ASME Press Translations, New York) consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lie periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a five parameters non isochronous Chouikha cubic system (Chouikha, A. R., 2007, “Isochronous Centers of Lienard Type Equations and Applications,” J. Math. Anal. Appl., 331, pp. 358–376) we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system. The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb (J. C. Faugère, “FGb Salsa Software,” http://fgbrs.lip6.fr) for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts are different since our object is the synchronisation rather than the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers, (in particular Bardet, M., and Boussaada, I., 2011, “Compexity Reduction of C-algorithm,” App. Math. Comp., in press; Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(1), pp. 89–112; Bardet, M., Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(2), pp. 230–249; and furthermore, it can be considered as an adaptation of a qualitative theory method to a synchronization problem.

Isochronous Synchronization Between Chaotic Semiconductor Lasers Over 40-km Fiber Links

A long-distance isochronous chaos synchronization system, composed of a driving chaotic semiconductor laser and twin semiconductor lasers, is demonstrated experimentally. Via a driven chaotic signal which is generated by the driving chaotic semiconductor laser subject to external feedback and then injects into the twin semiconductor lasers by a pair of 20-km single-mode fibers, respectively, isochronous synchronization between the twin semiconductor lasers is experimentally observed. Further results show that high-quality isochronous synchronization can still be maintained even though the correlation coefficient between the driving laser and twin lasers is low. In addition, the bandwidth of synchronized chaotic signals is enhanced to more than 10 GHz although the bandwidth of driving signal is about 5.7 GHz.

Integrability, analyticity, isochrony, equilibria, small oscillations, and Diophantine relations: results from the stationary Burgers hierarchy

An isochronous system is introduced by modifying the Nth ODE of the stationary Burgers hierarchy, and then, by investigating its behaviour near its equilibria, neat Diophantine relations are identified, involving (well-known) polynomials of arbitrary degree having integer zeros, or equivalently matrices the determinants of which yield such polynomials. The basic idea to arrive at such relations is not new, but the specific application reported in this paper is new, and it is likely to open the way to several analogous new findings.

Oscillatory and isochronous rate equations possibly describing chemical reactions

We study a simple mathematical model that can be interpreted as a description of the kinetics of the following four reactions involving the two chemicals U and W: (i) U + U ⟹ U with rate α, (ii) U + W ⟹ U with rate β, (iii) W + W ⟹ U with rate γ and (iv) W + W ⟹ W + W + W with rate δ + 2γ. The model can be generally solved by quadratures, and in the special case β = 2α, explicitly in terms of elementary functions.We focus on the case characterized by the two inequalities γβ2 > αδ2 and 2βγ > δ2, and we show that in this case the solutions vanish asymptotically at large times. But if a constant decay with rate θ of chemical U is added, then a nonvanishing equilibrium configuration arises. Moreover, for arbitrary strictly positive initial conditions, the solutions remain bounded. They either tend asymptotically (in the remote future) to this nonvanishing equilibrium configuration, or are periodic, or tend to a limit cycle. Indeed, we find that this system goes through a standard supercritical Hopf bifurcation at an appropriate value of the parameters. Another interesting case arises when, in addition to the original reaction, a negative constant term is added to the equation corresponding to chemical U, corresponding to siphoning out a constant amount of chemical U per unit time, independent of its concentration. A very remarkable feature of this (possibly not very realistic) model is the following: in the special case β = 2α, we again find an explicit solution in terms of elementary functions, which oscillates at a fixed frequency, independent of the initial condition. In other words, it is an isochronous system. If β ≠ 2α, however, no periodic orbits exist, implying that the nature of the bifurcation at β = 2α is rather peculiar.

Isochronous dynamical systems

An isochronous dynamical system is characterized by the existence of an open domain of initial data such that all motions evolving from it are completely periodic with a fixed period (independent of the initial data). Taking advantage of a recently introduced trick, a (quite large) class of such systems is identified, with equations of motion ‘of Newtonian type.’ Subcases are exhibited in which the equations of motion are Hamiltonian, and/or are interpretable as the equations of motion of a many-body problem, possibly with one- and two-body forces only, possibly invariant under rotations and/or translations, possibly constituting a one-parameter deformation of a classical physical problem (such as the many-body gravitational problem in ordinary three-dimensional space).

Periodic solutions of perturbed isochronous hamiltonian systems at resonance

We look for periodic solutions of planar systems obtained by adding an asymptotically positively homogeneous nonlinear term to an isochronous hamiltonian system. Precise computations of the topological degree are obtained by elementary phase-plane analysis.

Adding of microtron electron bunches for improving of far infrared FEL performance

The method of microtron electron bunch charge accumulation for a far infrared free electron laser operation is studied. Microtron electron bunches are added by the achromatic and isochronous round-trip magnetic system. The electron bunch peak current will grow proportionally to the number of bunches added. The growth of the small signal FEL gain is estimated.

Random Vibrations with Impacts: A Review

Analytical methods and specific results are presented for random vibrations of systems with lumped parameters and “classical” impacts whereby finite relations between impact/rebound velocities are imposed at the impact instants that are not known in advance but rather governed by the equations of motion. Emphasis is placed on the procedures using special piecewise-linear transformation of state variables that exclude the velocity jumps at impacts or makes them small if impact losses are present. In the former case, exact analyses for stationary probability densities of the response to white-noise excitation are possible, whereas the stochastic averaging method is applied in the latter case. Furthermore, the special case of an isochronous system permits a more profound response analysis, such as predicting the spectral density of the response or subharmonic response to narrow-band excitation. The method of direct energy balance is also illustrated, based on direct application of the stochastic differential equation calculus between impacts. Certain two-degree-of-freedom impacting systems are considered, with application to moored systems, as used in ocean engineering.