A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.     

New Exact Periodic Solitary-Wave Solutions for New （2＋1）-Dimensional KdV Equation

The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions （including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions） for the new （2＋1）-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.     

Standing,Periodic and Solitary Waves in （1＋1）-Dimensional Caudry-Dodd-Gibbon-Sawada-Kortera System

In the paper, the variable separation approach, homoclinic test technique and bilinear method are successfully extended to a （1 ＋1）-dimensional Caudry-Dodd-Gibbon-Sawada-Kortera （CDGSK） system, respectively. Based on the derived exact solutions, some significant types of localized excitations such as standing waves, periodic waves, solitary waves are simultaneously derived from the （1＋1）-dimensional Caudry-Dodd-Gibbon-Sawada-Kortera system by entrancing appropriate parameters.     

The fractal structure in the Rydberg lithium atoms in ionization dynamics of 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.     

A Geometric Criterion for Positive Topological Entropy¶II: Homoclinic Tangencies

In a series of important papers [GS1,GS2] Gavrilov and Shilnikov established a topological conjugacy between a surface diffeomorphism having a dissipative hyperbolic periodic point with certain types of quadratic homoclinic tangencies and the full shift on two symbols, thus exhibiting horseshoes near a tangential homoclinic point. In this note, which should be viewed of as an addendum to [BW] we extend this result by showing that such a diffeomorphism with a one-sided isolated homoclinic tangency having any order contact, possible with infinite order contact, possesses a horseshoe near the homoclinic point. Received: 2 March 1999 / Accepted: 14 May 1999     

General existence conditions of homoclinic trajectories in dissipative systems. Lorenz,Shimizu–Morioka,Lu and Chen systems

The general method for proving the existence of homoclinic trajectories in dissipative systems is developed. The applications of this method to Lorenz-like systems: Lorenz, Shimizu–Morioka, Lu and Chen systems are demonstrated. A criterion for the existence of a homoclinic trajectory within a given family of differential equations (Fishing principle) is presented. New numerical algorithm for the approximation of a homoclinic point in parameters space is constructed. The comparison with Kaplan–Yorke and Shilnikov results is made.     

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.     

Shilnikov sense chaos in a simple three-dimensional system

The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative (PID) controller is improved by introducing the quadratic and cubic nonlinearities into the governing equations. For the discussion of chaos, the bifurcate parameter value is selected in a reasonable regime at the requirement of the Shilnikov theorem. The analytic expression of the Shilnikov type homoclinic orbit is accomplished. It depends on the series form of the manifolds surrounding the saddle-focus equilibrium. Then the methodology is extended to research the dynamical behaviours of the simplified solar-wind-driven-magnetosphere-ionosphere system. As is illustrated, the Lyapunov characteristic exponent spectra of the two systems indicate the existence of chaotic attractor under some specific parameter conditions.     

Chaos via Shilnikov's saddle-node bifurcation in a theory of the electroencephalogram

We study the bifurcation diagram of a mesoscopic model of the human cortex. This model is known to exhibit robust chaotic behavior in the space of parameters that model exterior forcing. We show that the bifurcation diagram has an unusual degree of organization. In particular, we show that the chaos is spawned by a codimension-one homoclinic bifurcation that was analyzed by Shilnikov in 1969 but has never before been found in a physical application.     

Multi-pulse homoclinic orbits and chaotic dynamics of a parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities

In this paper,the complicated dynamics and multi-pulse homoclinic orbits of a two-degree-of-freedom parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities are studied.The damping,parametrical excitation and the nonlinearities are regarded as weak.The averaged equation depicting the fast and slow dynamics is derived through the method of multiple scales.The dynamics near the resonance band is revealed by doing a singular perturbation analysis and combining the extended Melnikov method.We are able to determine the criterion for the existence of the multi-pulse homoclinic orbits which can form the Shilnikov orbits and give rise to chaos.At last,numerical results are also given to illustrate the nonlinear behaviors and chaotic motions in the nonlinear nano-oscillator.     

OBSERVATIONS OF HOMOCLINIC CHAOS THROUGH ALTERNATING PERIODIC-CHAOTIC SEQUENCES IN A NONLINEAR CIRCUIT

Homoclinic chaos in the alternating periodic-chaotic sequences is observed in a nonlinear circuit with sinusoidal driving force. In particular, a complete Alternating Periodic-Chaotic sequence is recorded with a high-resolution up to P⁽⁸⁾ state. The experimental results, analyzed by constructing the time of flight and the next maximal amplitude return maps, are in good agreement with the scenario described by Shilnikov. The underlying dynamics of homoclinic chaos is determined from the next amplitude return map, to be that of a unimodal map and thus a strong dissipation case.     

Study of RPC gas composition using Daya Bay RPCs

This study uses Daya Bay RPCs operating in streamer mode to investigate gas mixtures of at least 50% argon, at most 6% isobutane, and with small amounts of SF6. Isobutane is reduced to 2% without degradation of performance, and SF6 reduces the noise rate and current, as well as the signal size. This study provides quantitative relationships between basic RPC operating parameters and various gas compositions.     

Evaluation of the 1077 keV γ-ray emission probability from 6SGa decay

6SGa decays to the excited states of aSzn through the electron capture decay mode. New recommended values for the emission probability of 1077 keV γ-ray given by the ENSDF and DDEP databases all use data from absolute measurements. In 2011, JIANG Li-Yang deduced a new value for 1077 keV γ-ray emission probability by measuring the 69Ga（n,2n） 6SGa reaction cross section. The new value is about 20% lower than values obtained from previous absolute measurements and evaluations. In this paper, the discrepancies among the measurements and evaluations are analyzed carefully and the new values are re-recommended. Our recommended value for the emission probability of 1077 keV γ-ray is （2.72±0.16）%.     

Relativistic treatment of the spin-zero particles subject to the second Poschl-Teller-like potential

The three-dimensional Klein-Gordon equation is solved for the case of equal vector and scalar second Poschl-Teller potential by proper approximation of the centrifugal term within the framework of the asymptotic iteration method. Energy eigenvalues and the corresponding wave function are obtained analytically. Eigenvalues are computed numerically for some values of n and It is found that the results are in good agreement with the findings of other methods for short-range potential.     

Collective dynamics in a non-dissipative two-coupled pendulum system

We study the collective dynamics of a non-dissipative two-coupled pendulum system, including phase synchronization （PS） and measure synchronization （MS）. We find that as the coupling intensity between the two pendulums increases, the PS happens prior to the MS. We also present a three-dimensional phase space representation of MS, from which a more detailed information about evolution can be obtained. Fu~.hermore, the order parameters are introduced to describe the phase transition between PS and MS. Finally, through the analysis of the Poincar6 sections, we show that the system exhibits separatrix crossing behavior right at the MS transition point, and as the total initial energy increases, the Hamiltonian chaos will arise with separatrix chaos at the chaotic MS transition point.     

Generalized Harmonic Oscillator and the SchrSdinger Equation with Position-Dependent Mass

We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties corresponding effective potentials for several mass functions, for the system with PDM are also discussed. We give the the systems with such potentials are isospectral to the usual harmonic oscillator.     

Relativistic symmetries in the Hulth6n scalar-vector-tensor interactions

In the presence of spin and pseudospin （p-spin） symmetries, the approximate analytical bound states of the Dirac equation for scalar-vector-tensor Hulth6n potentials are obtained with any arbitrary spin-orbit coupling number using the Pekeris approximation. The Hulth6n tensor interaction is studied instead of the commonly used Coulomb or linear terms. The generalized parametric Nikiforov-Uvarov （NU） method is used to obtain energy eigenvalues and corresponding wave functions in their closed forms. It is shown that tensor interaction removes degeneracy between spin and p-spin doublets. Some numerical results are also given.     

Na decorated B6 cluster and its hydrogen storage properties

The structures and hydrogen storage properties of sodium atoms decorated B6 clusters are investigated by the B3LYP method with a 6-311＋G （d, p） basis set. For NamB6 （m = 1-3） clusters, Na atoms are always inclined to separate far enough from each other and not cluster together on a B6 cluster surface so that each Na atom has sufficient space to bind hydrogen molecules. The hydrogen storage gravimetric density of a two Na atoms decorated B6 cluster is 17.91 wt% with an adsorption energy per H2 molecule （AAE/H2） of 0.6851 kcal.mo1^-1. The appropriate AAE/H2 and preferable gravimetric density of the two Na atoms decorated B6 cluster complex indicate that it is feasible for hydrogen storage application in ambient conditions.     

Extinction Effects of Multiplicative Non-Gaussian L′evy Noise in a Tumor Growth System with Immunization

The extinction phenomenon induced by multiplicative non-Gaussian Levy noise in a tumor growth model with immune response is discussed. Under the influence of the stochastic immune rate, the model is analyzed in terms of a stochastic differential equation with multiplicative noise. By means of the theory of the infinitesimal generator of Hunt processes, the escape probability, which is used to measure the noise-induced extinction probability of tumor cells, is explicitly expressed as a function of initial tumor cell density, stability index and noise intensity. Based on the numerical calculations, it is found that for different initial densities of tumor cells, noise parameters play opposite roles on the escape probability. The optimally selected values of the multiplicative noise intensity and the stability index are found to maximize the escape probability.     

Formation Mechanism and Binding Energy for Regular Octahedral Structure of Li6 Cluster

The formation mechanism for the regular octahedral structure of Liscluster is proposed. The curve of the total energy versus the separation R between any two neighboring nuclei has been calculated by using the method of Gou＇s modified arrangement channel quantum mechanics （MACQM）. The result shows that the curve has a minimal energy of -44.736 89 a.u. at R = 5.07a0. When R approaches infinity, the total energy of six lithium atoms has the value of -44.568 17 a.u. So the binding energy of Li6 with respect to six lithium atoms is 0.1687 a.u. Therefore, the binding energy per atom for Li6 is 0.028 12 a.u., or 0.7637 eV, which is greater than the binding energy per atom of 0.453 eV for Li2 and the binding energy per atom of 0.494 eV for Li3 calculated in our previous work. This means that the Li6 cluster may be formed in a regular octahedral structure with a greater binding energy.