Chaotic behaviour of nonlinear coupled reaction–diffusion system in four-dimensional space

In recent years, nonlinear coupled reaction–diffusion (CRD) system has been widely investigated by coupled map lattice method. Previously, nonlinear behaviour was observed dynamically when one or two of the three variables in the discrete system change. In this paper, we consider the chaotic behaviour when three variables change, which is called as four-dimensional chaos. When two parameters in the discrete system are unknown, we first give the existing condition of the chaos in four-dimensional space by the generalized definitions of spatial periodic orbits and spatial chaos. In addition, the chaotic behaviour will vary with the parameters. Then we propose a generalized Lyapunov exponent in four-dimensional space to characterize the different effects of parameters on the chaotic behaviour, which has not been studied in detail. In order to verify the chaotic behaviour of the system and the different effects clearly, we simulate the dynamical behaviour in two- and three-dimensional spaces.     

Stability and transient dynamics of a propeller–shaft system as induced by nonlinear friction acting on bearing–shaft contact interface

This paper investigates the friction-induced instability and the resulting self-excited vibration of a propeller–shaft system supported by water-lubricated rubber bearing. The system under consideration is modeled with an analytical approach by involving the nonlinear interaction among torsional vibrations of the continuous shaft, tangential vibrations of the rubber bearing and the nonlinear friction acting on the bearing–shaft contact interface. A degenerative two-degree-of-freedom analytical model is also reasonably developed to characterize system dynamics. The stability and vibrational characteristics are then determined by the complex eigenvalues analysis together with the quantitative analysis based on the method of multiple scales. A parametric study is conducted to clarify the roles of friction parameters and different vibration modes on instabilities; both the graphic and analytical expressions of instability boundaries are obtained. To capture the nature of self-excited vibrations and validate the stability analysis, the nonlinear formulations are numerically solved to calculate the transient dynamics in time and frequency domains. Analytical and numerical results reveal that the nonlinear coupling significantly affects the system responses and the bearing vibration plays a dominant role in the dynamic behavior of the present system.     

Mathematical properties of the Navier–Stokes dynamical system for incompressible Newtonian fluids

The connection between fluid dynamics and classical statistical mechanics has motivated in the past mathematical investigations of the incompressible Navier–Stokes (NS) equations (INSE) by means of an asymptotic kinetic theory. This feature has suggested the search for possible alternative exact approaches, based on the construction of a suitable inverse kinetic theory (IKT), which can avoid the asymptotic character and the intrinsic mathematical difficulty of direct kinetic theories. In this paper the fundamental mathematical properties of the NS phase-space dynamical system underlying INSE and determined by IKT are investigated. In particular, an equivalence theorem with the INSE problem and a global existence theorem are proved to hold for the NS dynamical system.     

The effect of finite response–time in coupled dynamical systems

The paper investigates synchronization in unidirectionally coupled dynamical systems wherein the influence of drive on response is cumulative: coupling signals are integrated over a time interval τ. A major consequence of integrative coupling is that the onset of the generalized and phase synchronization occurs at higher coupling compared to the instantaneous (τ?=?0) case. The critical coupling strength at which synchronization sets in is found to increase with τ. The systems explored are the chaotic Rössler and limit cycle (the Landau–Stuart model) oscillators. For coupled Rössler oscillators the region of generalized synchrony in the phase space is intercepted by an asynchronous region which corresponds to anomalous generalized synchronization.     

Upstream ion wave excitation in an ion-beam–plasma system

Plasma normal modes in ion-beam–plasma systems were experimentally investigated previously only for the waves propagating in the downstream(along the beam) direction. In this paper, the ion wave excitation and propagation in the upstream(against the beam) direction in an ion-beam–plasma system were experimentally studied in a double plasma device. The waves were launched by applying a ramp voltage to a negatively biased excitation grid. Two kinds of wave signals were detected, one is a particle signal composed of burst ions and the other is an ion-acoustic signal arising from the background plasma. These signals were identified by the dependence of the signal velocities on the characteristics of the ramp voltage. The velocity of the burst ion signal increases with the decrease of the rise time and the increase of the peak-to-peak amplitude of the applied ramp voltage while that of the ion-acoustic signal is independent of these parameters.By adjusting these parameters such that the burst ion velocity approaches to the ion-acoustic velocity, the wave–particle interaction can be observed.     

Wave packet dynamics of nonlinear Gazeau–Klauder coherent states of a position-dependent mass system in a Coulomb-like potential

AD = 1 position-dependent mass approach to constructing nonlinear quantum states for a modified Coulomb potential is used to generate Gazeau–Klauder coherent states. It appears that their energy eigenvalues are scaled down by the quantum number and the nonlinearity coefficient. We study the basic properties of these states, which are found to be undefined on the whole complex plane, and some details of their revival structure are discussed.     

Linear and nonlinear optical susceptibilities in a laterally coupled quantum-dot–quantum-ring system

Linear and nonlinear optical susceptibilities in a laterally coupled quantum-dot–quantum-ring system have been theoretically studied. In general, we find that the structure parameters of the coupled system significantly affect the optical susceptibilities. The enhancement of the coupling effects between the dot and ring is found to increase considerably the optical susceptibilities and redshift drastically the transition energies. Comparing to the linear susceptibility, the nonlinear optical susceptibility is found to be more sensitive to the variation of the structure parameters. A comprehensive analysis of the electron probability density movement with respect to the modification of the structure parameters is provided, which offers a unique perspective of the ground-state localization.     

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.     

Multicritical dynamical phase diagrams of the kinetic Blume–Emery–Griffiths model with repulsive biquadratic coupling in an oscillating field

We study, within a mean-field approach, the stationary states of the kinetic Blume–Emery–Griffiths model with repulsive biquadratic coupling under the presence of a time-varying (sinusoidal) magnetic field. We employ the Glauber-type stochastic dynamics to construct set of dynamic equations of motion. The behavior of the time dependence of the order parameters and the behavior of the average order parameters in a period, which is also called the dynamic order parameters, as functions of the reduced temperature are investigated. The dynamic phase transition points are calculated and phase diagrams are presented in the reduced magnetic field amplitude and reduced temperature plane. The dynamical transition from one regime to the other can be of first- or second order depending on the region in the phase diagram. According to the values of the crystal field interaction or single-ion anisotropy constant and biquadratic exchange constant, we find 20 fundamental types of phase diagrams which exhibit many dynamic critical points, such as tricritical points, zero-temperature critical points, double critical end points, critical end point, triple point and multicritical point. Moreover, besides a disordered and ordered phases, seven coexistence phase regions exist in the system.     

The kinetics of the formation of a solid solution in an Ag–Pd polycrystalline film system

The kinetics of homogenization of an Ag–Pd film system with a total thickness of 120 nm and a grain size of 5–10 nm has been studied by means of in situ TEM heating. The film system has been formed by the sequential deposition of components in a vacuum on the substrate at room temperature. It has been shown that diffusion processes are activated, starting from the temperature 453 K, resulting in complete homogenization of the film system at 573 K with preservation of its fine-grained structure. The effective diffusion coefficient in the Ag–Pd system was measured as 10^?17–10^?18 m²/s at 553 K. A possible mechanism of homogenization is discussed.     

Calculation of the vibration–rotational transition intensities of water molecules trapped in an argon matrix: stretching O–H vibrations spectral region

The frequencies and intensities of vibration–rotational transitions of water molecules in an argon matrix were calculated for temperatures of 6 and 30 K. The rigid asymmetric top approximation was used with available literature values of the effective rotational constants in the ground and excited vibrational states. The calculations were carried out by taking into account the existence of a non-equilibrium population distribution between the rotational levels of ortho- and para-water isomers. It was assumed that the temperature relaxation of the population of rotational levels is independent of the ortho- and para-isomers. Comparison of the results of the theoretical calculations with experimental literature data shows good agreement for the majority of the rotational structure lines for symmetric and antisymmetric stretching vibrations both in the frequency values and in the values of the relative intensities.     

Discontinuity Structures of Solutions to Equations Describing Longitudinal–Torsional Waves in Elastic Rods

Doklady Physics - Discontinuity structures (taking into account viscosity) in solutions to hyperbolic systems describing coupled longitudinal–torsional waves in elastic rods are studied. The...     

Thermal buckling and natural vibration of the beam with an axial stick–slip–stop boundary

As a first attempt to study the dynamics of a heated structure with complicated boundaries, this paper deals with the thermal buckling and the natural vibration of a simply supported slender beam, which is subject to a uniformly distributed heating and has a frictional sliding end within a clearance. This sliding end is initially at a stick status under the friction force, but may be slightly slipping due to the thermal expansion of the beam until the sliding end contacts a stop, i.e., the bound of the clearance. The material properties of the beam are temperature-independent for low temperature, but temperature-dependent for high temperature. For each case, the analytic solutions for the critical buckling temperature and the natural frequencies of the heated beam are derived first. Then, discussions are made to reveal the effects of beam parameters, such as the ratio of beam length to beam thickness, the ratio of clearance to beam length and the temperature-dependent material properties, on the critical buckling temperature and the fundamental natural frequency of the heated beam. The study shows that both friction force and clearance have significant influences on the critical buckling temperature and the fundamental natural frequency of the beam. When the friction force is not very large, the clearance can greatly increase the critical buckling temperature. These conclusions enable one to properly design the stick–slip–stop boundary so as to improve the mechanical performance of the beam in thermal environments.     

The efficient computation of the nonlinear dynamic response of a foil–air bearing rotor system

The foil–air bearing (FAB) enables the emergence of oil-free turbomachinery. However, its potential to introduce undesirable nonlinear effects necessitates a reliable means for calculating the dynamic response. The computational burden has hitherto been alleviated by simplifications that compromised the true nature of the dynamic interaction between the rotor, air film and foil structure, introducing the potential for significant error. The overall novel contribution of this research is the development of efficient algorithms for the simultaneous solution of the state equations. The equations are extracted using two alternative transformations: (i) Finite Difference (FD); and (ii) a novel arbitrary-order Galerkin Reduction (GR) which does not use a grid, considerably reducing the number of state variables. A vectorized formulation facilitates the solution in two alternative ways: (i) in the time domain for arbitrary response via implicit integration using readily available routines; and (ii) in the frequency domain for the direct computation of self-excited periodic response via a novel Harmonic Balance (HB) method. GR and FD are cross-verified by time domain simulations which confirm that GR significantly reduces the computation time. Simulations also cross-verify the time and frequency domain solutions applied to the reference FD model and demonstrate the unique ability of HB to correctly accommodate structural damping.     

Resistance oscillations in junctions of superconductor–magnetic system

Resistance oscillations as a function of magnetic field were observed in superconductor–magnetic tunnel junctions of Nb–Fe–FeO_x–SiO₂–Au–Nb. Junctions involving superconductor–magnetic layer superconductor system are exciting because for certain regime of ferromagnetic layer thickness, a Josephson coupling with an intrinsic phase difference of π might be stabilized. For fabrication of the tunnel junctions the thin films were deposited by RF/DC magnetron sputtering. Using photolithography and reactive ion etching, square junctions of size varying from 50 μm to 250 μm were defined. I–V characteristics and R vs. H characteristics were studied at 4.2 K. When the magnetic field is applied parallel to the junction plane, measurements of the junction resistance as a function of magnetic field at a fixed temperature show resistance peaks whenever the total magnetic flux through the junction equals an integral multiple of flux quantum. The penetration depth of the superconducting electrodes was estimated from the positions of the resistance peaks.     

Low frequency electrostatic nonlinear structures in an inhomogeneous magnetized non-Maxwellian electron–positron–ion plasma

The properties of low frequency (coupled acoustic and drift wave) nonlinear structures including solitary waves and double layers in an inhomogeneous magnetized electron–positron–ion (EPI) nonthermal plasma with density and temperature inhomogeneities are studied in a simplified way. The nonlinear differential equation derived here for the study of double layers in the inhomogeneous EPI plasma resembles with the modified KdV equation in the stationary frame. But the method used for the derivation of nonlinear differential equation is simple and consistent to give both the stationary solitary waves and double layers. Further, the illustrations show that superthermality κ, drift velocity and temperature inhomogeneity have significant effects on the amplitude, width, and existence range of the structures.