Bifurcation Cascades and Self-Similarity of Periodic Orbits with Analytical Scaling Constants in Hénon–Heiles Type Potentials

We investigate the isochronous bifurcations of the straight-line librating orbit in the Hénon–Heiles and related potentials. With increasing scaled energy e, they form a cascade of pitchfork bifurcations that cumulate at the critical saddle-point energy e=1. The stable and unstable orbits created at these bifurcations appear in two sequences whose self-similar properties possess an analytical scaling behavior. Different from the standard Feigenbaum scenario in area preserving two-dimensional maps, here the scaling constants and corresponding to the two spatial directions are identical and equal to the root of the scaling constant that describes the geometric progression of bifurcation energies e_n in the limit n. The value of is given analytically in terms of the potential parameters.     

Transient chaos in a generalized Hénon map on the torus

A generalized Hénon system on a torus is considered to investigate some phenomena of transient chaos in weakly dissipative systems. A simple relationship is numerically verified between the life time of chaos and other parameters of the model.     

On a reduction of the generalized Darboux–Halphen system

The equations for the general Darboux–Halphen system obtained as a reduction of the self-dual Yang–Mills can be transformed to a third-order system which resembles the classical Darboux–Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux–Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.     

Stochastic period-doubling bifurcation analysis of a R?ssler system with a bounded random parameter

This paper aims to study the stochastic period-doubling bifurcation of the three-dimensional R?ssler system with an arch-like bounded random parameter. First, we transform the stochastic R?ssler system into its equivalent deterministic one in the sense of minimal residual error by the Chebyshev polynomial approximation method. Then, we explore the dynamical behaviour of the stochastic R?ssler system through its equivalent deterministic system by numerical simulations. The numerical results show that some stochastic period-doubling bifurcation, akin to the conventional one in the deterministic case, may also appear in the stochastic R?ssler system. In addition, we also examine the influence of the random parameter intensity on bifurcation phenomena in the stochastic R?ssler system.     

Stochastic period-doubling bifurcation analysis of a Rssler system with a bounded random parameter

This paper aims to study the stochastic period-doubling bifurcation of the three-dimensional Rssler system with an arch-like bounded random parameter. First, we transform the stochastic Rssler system into its equivalent deterministic one in the sense of minimal residual error by the Chebyshev polynomial approximation method. Then, we explore the dynamical behaviour of the stochastic Rssler system through its equivalent deterministic system by numerical simulations. The numerical results show that some stochastic period-doubling bifurcation, akin to the conventional one in the deterministic case, may also appear in the stochastic Rssler system. In addition, we also examine the influence of the random parameter intensity on bifurcation phenomena in the stochastic Rssler system.     

Dynamical response of a neuron–astrocyte coupling system under electromagnetic induction and external stimulation

Previous studies have observed that electromagnetic induction can seriously affect the electrophysiological activity of the nervous system. Considering the role of astrocytes in regulating neural firing, we studied a simple neuron–astrocyte coupled system under electromagnetic induction in response to different types of external stimulation. Both the duration and intensity of the external stimulus can induce different modes of electrical activity in this system, and thus the neuronal firing patterns can be subtly controlled. When the external stimulation ceases, the neuron will continue to fire for a long time and then reset to its resting state. In this study, delay is defined as the delayed time from the firing state to the resting state, and it is highly sensitive to changes in the duration or intensity of the external stimulus. Meanwhile, the self-similarity embodied in the aforementioned sensitivity can be quantified by fractal dimension. Moreover, a hysteresis loop of calcium activity in the astrocyte is observed in the specific interval of the external stimulus when the stimulus duration is extended to infinity, since astrocytic calcium or neuron electrical activity in the resting state or during periodic oscillation depends on the initial state. Finally, the regulating effect of electromagnetic induction in this system is considered. It is clarified that the occurrence of delay depends purely on the existence of electromagnetic induction. This model can reveal the dynamic characteristics of the neuron–astrocyte coupling system with magnetic induction under external stimulation. These results can provide some insights into the effects of electromagnetic induction and stimulation on neuronal activity.     

Onset of Rayleigh-Bénard convection in a ternary liquid system and the Onsager law with generalized thermal diffusion

For the ternary system isopropanol-ethanol-water, a model was proposed to estimate the cross diffusion coefficients from measurements of the critical parameters for the onset of Rayleigh-Bénard convection, together with the consideration of the reciprocity laws of Onsager. Among the three forms of the chemical potentials used, not a single chemical potential was found to provide a correlation between the Onsager laws and the experimental data. The present study shows that consideration of a generalized thermal diffusion term taking account of all components is adequate to estimate quantitative values of the cross diffusion coefficients.     

Existence of a homoclinic point for the Hénon map

We prove analytically that for the Hénon map of the plane into itself (s, t)(t+1–1.4a ², 0.3s), there exists a transversal homoclinic point.     

Soliton solutions and conservation laws for a generalized Ablowitz–Ladik system

In this paper, a generalized Ablowitz–Ladik system is systemically investigated via the Darboux transformation method. Soliton solutions and conservation laws are presented. Depending on the choices of parameters, the dynamic behaviors are discussed graphically.     

Berry phase in a generalized nonlinear two-level system

In this paper, we investigate the behaviour of the geometric phase of a more generalized nonlinear system composed of an effective two-level system interacting with a single-mode quantized cavity field. Both the field nonlinearity and the atom-field coupling nonlinearity are considered. We find that the geometric phase depends on whether the index k is an odd number or an even number in the resonant case. In addition, we also find that the geometric phase may be easily observed when the field nonlinearity is not considered. The fractional statistical phenomenon appears in this system if the strong nonlinear atom-field coupling is considered. We have also investigated the geometric phase of an effective two-level system interacting with a two-mode quantized cavity field.     

Controlled synthesis and analysis of He–H+3 in a 3.7 K ion trap

Complexes of the triatomic hydrogen ion with helium were synthesised in a low-temperature 22-pole rf ion trap at He number densities of up to 10¹⁶ cm^?3. Absolute ternary rate coefficients for sequentially attaching He atoms have been determined from the growth of complexes with increasing storage time. The number of helium-tagged ions is significantly reduced when increasing the nominal temperature from 4 to 25 K. Competition between attachment and dissociation via collisions leads to stationary He_n–H⁺₃ (n up to 9) distributions. State-specific excitation of the trapped H⁺₃ ions via IR transitions significantly reduces the formation of complexes. Tuning the laser to Δv₂ = 1 transitions in the range of 2726 cm^?1 leads to LIICG lines, i.e., to spectra caused by laser-induced inhibition of complex growth. In addition, almost 100 lines have been found between 2700 and 2765 cm^?1, which are attributed to laser-induced dissociation of the in situ formed He–H⁺₃ complex ions. These lines are not yet assigned; however, their absorption strength, statistics and predissociation lifetimes provide interesting information on both the stable complexes as well as on scattering resonances in low-energy H⁺₃+He collisions. New calculations of the potential energy surface will help to analyse the dissociation spectrum. There are some indications that para-H⁺₃ is enriched under the conditions of the present experiment.     

A new type of global bifurcation in Hénon map

A simple branch of solution on a bifurcation diagram, which begins at static bifurcation and ends at boundary crisis (or interior crisis in a periodic window), is generally a period-doubling cascade. A domain of solution in parameter space, enclosed by curves of static bifurcation and that of boundary crisis (or the interior of a periodic window), is the trace of branches of solution. A P-n branch of solution refers to the one starting from a period-n (n≥1) solution, and the corresponding domain in parameter space is named the P-n domain of solution. Because of the co-existence of attractors, there may be several branches within one interval on a bifurcation diagram, and different domains of solution may overlap each other in some areas of the parameter space. A complex phenomenon, concerned both with the co-existence of attractors and the crises of chaotic attractors, was observed in the course of constructing domains of steady state solutions of the Hénon map in parameter space by numerical methods. A narrow domain of period-m solutions firstly co-exists with (lies on) a big period-n (m≥3n) domain. Then it enters the chaotic area of the big domain and becomes period-m windows. The co-existence of attractors disappears and is called the landing phenomenon. There is an interaction between the two domains in the course of landing: the chaotic area in the big domain is enlarged, and there is a crisis step near the landing area.     

Probing the statistics of transport in the Hénon Map

The phase space of an area-preserving map typically contains infinitely many elliptic islands embedded in a chaotic sea. Orbits near the boundary of a chaotic region have been observed to stick for long times, strongly influencing their transport properties. The boundary is composed of invariant “boundary circles.” We briefly report recent results of the distribution of rotation numbers of boundary circles for the Hénon quadratic map and show that the probability of occurrence of small integer entries of their continued fraction expansions is larger than would be expected for a number chosen at random. However, large integer entries occur with probabilities distributed proportionally to the random case. The probability distributions of ratios of fluxes through island chains is reported as well. These island chains are neighbours in the sense of the Meiss-Ott Markov-tree model. Two distinct universality families are found. The distributions of the ratio between the flux and orbital period are also presented. All of these results have implications for models of transport in mixed phase space.     

Shift automorphisms in the Hénon mapping

We investigate the global behavior of the quadratic diffeomorphism of the plane given byH(x,y)=(1+y–Ax ²,Bx). Numerical work by Hénon, Curry, and Feit indicate that, for certain values of the parameters, this mapping admits a strange attractor. Here we show that, forA small enough, all points in the plane eventually move to infinity under iteration ofH. On the other hand, whenA is large enough, the nonwandering set ofH is topologically conjugate to the shift automorphism on two symbols.Partially supported by NSF Grant MCS 77-00430     

Chaotic behavior in the Hénon mapping

In a previous work Hénon investigated a two-dimensional difference equation which was motivated by a hydrodynamical system of Lorenz. Numerically solving this equation indicated for certain parameter values the existence of a strange attractor, i.e., a region in the plane which attracts bounded solutions and in which solutions wander erratically. In the present work it is shown that this behavior is related to the mathematical concept of chaos. Using general methods previously developed, it is proven analytically that for some parameter values the mapping has a transversal homoclinic orbit, which implies the existence of the chaotic behavior observed by Hénon.     

Dynamical stable-jump–stable-jump picture in a non-periodically driven quantum relativistic kicked rotor system

We study a non-periodically driven kicked rotor based on the one-dimensional quantum relativistic kicked rotor(QRKR). In our model, we add a small constant to the interval of the one-dimensional QRKR after each kick process.It is found that the momentum spreading is stable in finite kicked times, it then jumps up or down and becomes stable again.This interesting phenomenon is understood by quantum resonance. Moreover, the stable-jump–stable-jump phenomenon persists, even though the interval of kick process is randomly increased. This result means that the quantum resonance is independent of the periodic perturbation in the QRKR model.     

Optimized electron–optical system of a static mass-spectrometer for simultaneous isotopic and chemical analysis

A new approach to control the linear dimensions of analytical electrophysical systems is suggested. This approach uses the lens properties of electron–optical elements with a curvilinear axis. It is shown that such an approach can be effectively applied, in particular, to synthesize ion–optical systems (IOSs) for static magnetic mass spectrometers and can be implemented owing to off-axis fundamental points, the “poles” of an electron–optical system, introduced earlier by one of the authors. The capabilities of the new approach are demonstrated with the synthesis of the IOS of a static mass spectrometer dedicated for isotopic and chemical analysis with an increased resolution. A new IOS not only provides desired high ion–optical parameters at decreased dimensions of the mass spectrometer as a whole but also makes it possible to loosen requirements for the manufacturing accuracy of IOS main elements.     

Dynamic contact analysis of a tensioned beam with a moving mass–spring system

The dynamic contact problem of a tensioned beam with clamped-pinned ends is analyzed when the beam contacts a moving mass–spring system. The contact and contact loss conditions are expressed in terms of constraint equations after considering the dynamic contact between the beam and the moving mass. Using these constraints and equations of motion for the beam and moving mass, dynamic contact equations are derived and then discretized using the finite element method, which is based on the Lagrange multiplier method. The time responses for the contact forces are computed from these discretized equations. The contact force variations and contact loss are investigated for the variations of the moving mass velocity, the beam tension, the moving mass, and the stiffness of the moving mass–spring system. In addition, the possibility of contact loss and safe contact conditions between the moving mass and the tensioned beam are also studied.