Periodic oscillation and fine structure of wedge-induced oblique detonation waves

An oblique detonation wave for a Mach 7 inlet flow over a long enough wedge of 30 turning angle is simulated numerically using Euler equation and one-step rection model.The fifth-order WENO scheme is adopted to capture the shock wave.The numerical results show that with the compression of the wedge wall the detonation wave front structure is divided into three sections：the ZND model-like strcuture,single-sided triple point structure and dual-headed triple point strucuture.The first structure is the smooth straight,and the second has the characteristic of the triple points propagating dowanstream only with the same velocity,while the dual-headed triple point structure is very complicated.The detonation waves facing upstream and downstream propagate with different velocities,in which the periodic collisions of the triple points cause the oscillation of the detonation wave front.This oscillation process has temporal and spatial periodicity.In addition,the triple point trace are recorded to obtain different cell structures in three sections.

On Periodic Oscillation and Its Period of a Circadian Rhythm Model

We theoretically study periodic oscillation and its period of a circadian rhythm model of Neurospora and provide the conditions for the existence of such a periodic oscillation by the theory of competitive dynamical systems.To present the exact expression of the unique equilibrium in terms of parameters of system,we divide them into eleven classes for the Hill coefficient n=1 or n=2,among seven classes of which nontrivial periodic oscillations exist.Numerical simulations are made among the seven classes and the models with the Hill coefficient n=3 or n=4 to reveal the influence of parameter variation on periodic oscillations and their periods.The results show that their periods of the periodic oscillations are approximately 21.5 h,which coincides with the known experiment result observed in constant darkness.

Periodic electron oscillation in coupled two-dimensional lattices

We study the time evolution of electron wavepacket in the coupled two-dimensional(2D)lattices with mirror symmetry,utilizing the tight-binding Hamiltonian framework.We show analytically that the wavepacket of an electron initially located on one atomic layer in the coupled 2D square lattices exhibits a periodic oscillation in both the transverse and longitudinal directions.The frequency of this oscillation is determined by the strength of the interlayer hopping.Additionally,we provide numerical evidence that a damped periodic oscillation occurs in the coupled 2D disordered lattices with degree of disorderW,with the decay time being inversely proportional to the square ofW and the frequency change being proportional to the square of W,which is similar to the case in the coupled 1D disordered lattices.Our numerical results further confirm that the periodic and damped periodic electron oscillations are universal,independent of lattice geometry,as demonstrated in AA-stacked bilayer and tri-layer graphene systems.Unlike the Bloch oscillation driven by electric fields,the periodic oscillation induced by interlayer coupling does not require the application of an electric field,has an ultrafast periodicity much shorter than the electron decoherence time in real materials,and can be tuned by adjusting the interlayer coupling.Our findings pave the way for future observation of periodic electron oscillation in material systems at the atomic scale.

Periodic oscillation of quantum diffusion in coupled one-dimensional systems

We analytically show that quantum diffusion in the coupled system composed of two identical chains exhibits a well-defined periodic oscillation in both transverse and longitudinal directions with a frequency determined by the interchain hopping strength, no matter whether the chains are periodic or non-periodic. We illustrate the result through numerical work on the coupled periodic chains and the quasiperiodic Aubry-Andre-Harper(AAH) chains with various modulations of onsite potentials supporting extended, critical, and localized states. We further numerically show that quantum diffusion in the coupled chains of different degrees of disorder W exhibits an exponential decay oscillation similar to the behavior of an underdamped harmonic oscillator, with a decay time inversely proportional to the square of W and a slight frequency change proportional to the square of W. Moreover, quantum diffusions in the coupled systems composed of two different chains are numerically studied, including periodic/disordered chains, periodic/AAH chains, and two different AAH chains, which exhibit the same behavior of underdamped periodic oscillation if the onsite potential difference between two chains is smaller than the interchain hoping strength.Existence of this universal periodic oscillation is a result of spectral splitting of the iso-spectra of two chains determined by interchain hopping, independent of system size, boundary condition, and intrachain onsite potentials. Because the oscillation frequency and spreading distance of wavepacket can be tuned separately by interchain hopping and intrachain potentials, the periodic oscillation of quantum diffusion in coupled chains is expected to find applications in control of quantum states and in designing nanoscale quantum devices.

Periodic and chaotic oscillations in mutual-coupled mid-infrared quantum cascade lasers

Dynamic states in mutual-coupled mid-infrared quantum cascade lasers(QCLs) were numerically investigated in the parameter space of injection strength and detuning frequency based on the Lang-Kobayashi equations model. Three types of period-one states were found, with different periods of injection time delay τ_(inj), 2τ_(inj), and reciprocal of the detuning frequency. Besides, square-wave, quasi-period, pulse-burst and chaotic oscillations were also observed. It is concluded that external-cavity periodic dynamics and optical modes beating are the mainly periodic dynamics. The interaction of the two periodic dynamics and the high-frequency dynamics stimulated by strong injection induces the dynamic states evolution.This work helps to understand the dynamic behaviors in QCLs and shows a new way to mid-infrared wide-band chaotic laser.

Dynamical response of hyper-elastic cylindrical shells under periodic load

Dynamical responses, such as motion and destruction of hyper-elastic cylindrical shells subject to periodic or suddenly applied constant load on the inner surface, are studied within a framework of finite elasto-dynamics. By numerical computation and dynamic qualitative analysis of the nonlinear differential equation, it is shown that there exists a certain critical value for the internal load describing motion of the inner surface of the shell. Motion of the shell is nonlinear periodic or quasi-periodic oscillation when the average load of the periodic load or the constant load is less than its critical value. However, the shell will be destroyed when the load exceeds the critical value. Solution to the static equilibrium problem is a fixed point for the dynamical response of the corresponding system under a suddenly applied constant load. The property of fixed point is related to the property of the dynamical solution and motion of the shell. The effects of thickness and load parameters on the critical value and oscillation of the shell are discussed.

ON THE EXISTENCE AND STABILITY OF PERIODIC SOLUTIONS FOR HOPFIELD NEURAL NETWORK EQUATIONS WITH DELAY

Sufficient conditions are obtained for the existence, uniqueness and stability of T-periodic solutions far the Hopfield neural network equations with delay [GRAPHICS]

Analysis of the periodic solutions of a smooth and discontinuous oscillator

In this paper, the periodic solutions of the smooth and discontinuous （SD） oscillator, which is a strongly irra- tional nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the un- perturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equa- tion using Lyapunov method. The results presented herein this paper are valid for both smooth （e 〉 0） and discontin- uous （ce = 0） stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.

Oscillation properties of matter–wave bright solitons in harmonic potentials

We investigate the oscillation periods of bright soliton pair or vector bright soliton pair in harmonic potentials. We demonstrate that periods of low-speed solitons are greatly affected by the position shift during their collisions. The modified oscillation periods are described by defining a characterized speed, with the aid of asymptotic analysis on related exact analytic soliton solutions in integrable cases. The oscillation period can be used to distinguish the inter-and intra-species interactions between solitons. However, a bright soliton cannot oscillate in a harmonic trap, when it is coupled with a dark soliton(without any trapping potentials). Interestingly, it can oscillate in an anti-harmonic potential, and the oscillation behavior is explained by a quasi-particle theory. The modified period of two dark-bright solitons can be also described well by the characterized speed. These results address well the effects of position shift during soliton collision, which provides an important supplement for previous studies without considering phase shift effects.

Sub-Harmonic Resonances of Periodic Parameter Excited Oscillators with Discontinuities

It is difficult to obtain analytic approximations of nonlinear problems such as parameter excited system with strong nonlinearity. An analytic approach based on the homotopy analysis method( HAM) is proposed to study the sub-harmonic resonances of highly nonlinear parameter excited oscillating systems with absolute value terms. The non-smoothness of absolute value terms is handled by means of an iteration approach with Fourier expansion. Two typical examples are employed to illustrate the validity and flexibility of this approach. The square residuals of the homotopy-approximations of the two examples decrease to 10-6and 10-5,respectively. Thus,the HAM combining with other methods gives hope to solve complex singular oscillating systems analytically.

On the Relationship between the Pure Delay and the Natural Period of Oscillation

This paper provides a proof of the well-known relationship between the pure delay and the natural period of oscillation in industrial systems.

Numerical Modelling of Wash Waves Generated by Ships Moving over An Uneven Bottom

Unsteady wash waves generated by a ship with constant speed moving across an uneven bottom topography are investigated by numerical simulations based on a Mixed Euler–Lagrange(MEL) method. The transition is accomplished by the ship traveling from the depth h1 into the depth h2 via a step bottom. A small tsunami would be created after this transition. However, the unsteady wave-making resistance induced by this new phenomenon has not been well documented by literature. Therefore, the main purpose of the present study is to quantify the effects of an uneven bottom on the unsteady wash waves and wave-making resistance acting on the ship. An upwind differential scheme is commonly used in the Euler method to deal with the convection terms under free-surface condition to prevent waves in the upstream. Evidently, it cannot be applied to the present problem due to upstream waves generated by the ship would be dampened by the upwind scheme. The central differential scheme provides more accurate results,but it is not unconditionally stable. An MEL method is therefore employed to investigate the upstream wave generated by the ship moving over the uneven bottom. Simulation results show that the hydrodynamic interaction between the ship and the uneven bottom could initiate an upstream tsunami, as well as unsteady wave-making resistance on ships.The unsteady wave-making resistance oscillates periodically, and the amplitude and period of the oscillations are highly dependent on speed and water depth.

QUALITATIVE ANALYSIS OF DYNAMICAL BEHAVIOR FOR AN IMPERFECT INCOMPRESSIBLE NEO-HOOKEAN SPHERICAL SHELL

The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. A second-order nonlinear ordinary differential equation that describes the radial motion of the inner surface of the shell was obtained. And the first integral of the equation was then carded out. Via analyzing the dynamical properties of the solution of the differential equation, the effects of the prescribed imperfection parameter of the material and the ratio of the inner and the outer radii of the underformed shell on the motion of the inner surface of the shell were discussed, and the corresponding numerical examples were carded out simultaneously. In particular, for some given parameters, it was proved that, there exists a positive critical value, and the motion of the inner surface with respect to time will present a nonlinear periodic oscillation as the difference between the inner and the outer presses does not exceed the critical value. However, as the difference exceeds the critical value, the motion of the inner surface with respect to time will increase infinitely. That is to say, the shell will be destroyed ultimately.

Stability analysis of radial inflation of incompressible compositerubber tubes

The inflation mechanism is examined for a composite cylindrical tube composed of two incompressible rubber materials, and the inner surface of the tube is subjected to a suddenly applied radial pressure. The mathematical model of the problem is formulated, and the corresponding governing equation is reduced to a second-order ordinary differential equation by means of the incompressible condition of the material, the boundary conditions, and the continuity conditions of the radial displacement and the radial stress of the cylindrical tube. Moreover, the first integral of the equation is obtained. The qualitative analyses of static inflation and dynamic inflation of the tube are presented. Particularly, the effects of material parameters, structure parameters, and the radial pressure on radial inflation and nonlinearly periodic oscillation of the tube are discussed by combining numerical examples.

EXACT SOLUTIONS OF A DIPOLAR FLUID FLOW

Exact solutions for three canonical flow problems of a dipolar fluid are obtained: (i) The flow of a dipolar fluid due to a suddenly accelerated plate, (ii) The flow generated by periodic oscillation of a plate, (iii) The flow due to plate oscillation in the presence of a transverse magnetic field. The solutions of some interesting flows caused by an arbitrary velocity of the plate and of certain special oscillations are also obtained.

MHD UNSTEADY FLOWS DUE TO NON—COAXIAL ROTATIONS OF A DISK AND A FLUID AT INFINITY

Exact analytical solution for flows of an electrically conducting fluid over an infinite oscillatory disk in the presence of a uniform transverse magnetic field is constructed. Both the disk and the fluid are in a state of non-coaxial rotation. Such a flow model has a great significance not only due to its own theoretical interest, but also due to applications to geophysics and engineering. The resulting initial value problem has been solved analytically by applying the Laplace transform technique and the explicit expressions for the velocity for steady and unsteady cases have been established. The analysis of the obtained results shows that the flow field is appreciably influenced by the applied magnetic field, the frequency and rotation parameters.

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

Some nonlinear dynamic properties of axisymmetric deformation are ex- amined for a spherical membrane composed of a transversely isotropic incompressible Rivlin-Saunders material. The membrane is subjected to periodic step loads at its inner and outer surfaces. A second-order nonlinear ordinary differential equation approximately describing radially symmetric motion of the membrane is obtained by setting the thick- ness of the spherical structure close to one. The qualitative properties of the solutions are discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane are proposed. In certain cases, it is proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type ＂∞＂, and the amplitude growth of the periodic oscillation is discontinuous. Numerical results are provided.

DYNAMICAL FORMATION OF CAVITY IN A COMPOSED HYPER-ELASTIC SPHERE

The dynamical formation of cavity in a hyper_elastic sphere composed of two materials with the incompressible strain energy function, subjected to a suddenly applied uniform radial tensile boundary dead_load, was studied following the theory of finite deformation dynamics. Besides a trivial solution corresponding to the homogeneous static state, a cavity forms at the center of the sphere when the tensile load is larger than its critical value. An exact differential relation between the cavity radius and the tensile land was obtained. It is proved that the evolution of cavity radius with time displays nonlinear periodic oscillations. The phase diagram for oscillation, the maximum amplitude, the approximate period and the critical load were all discussed.

Experimental Study of Nonlinear Behaviors of A Free-Floating Body in Waves

Nonlinear behaviors of a free-floating body in waves were experimentally investigated in the present study. The experiments were carried out for 6 different wave heights and 6 different wave periods to cover a relatively wide range of wave nonlinearities. A charge-coupled device （CCD） camera was used to capture the real-time motion of the floating body. The measurement data show that the sway, heave and roll motions of the floating body are all harmonic oscillations while the equilibrium position of the sway motion drifts in the wave direction. The drift speed is proportional to wave steepness when the size of the floating body is comparable to the wavelength, while it is proportional to the square of wave steepness when the floating body is relatively small. In addition, the drift motion leads to a slightly longer oscillation period of the floating body than the wave period of nonlinear wave and the discrepancy increases with the increment of wave steepness.

A Kinetic Evidence for the Nitroxyl Radicals Recycling Mechanism in the Photostabilizing Process of HALS

The photoinduced bulk polymerization of a reactive-hindered amine light stabilizers (r-HALS), 4-acryloyl-2, 2, 6,6-tetramethylpiperidinyl (ATMP), was performed at 80 C by using a DPC technique. An unique periodic exponential attenuation-type oscillating curve was found when the polymerization was carried out in air, but this phenomenon was not found in nitrogen. It is supposed that this unique kinetic performance may be attributed to nitroxyl radicals that are produced in situ from the oxidation of ATMP. ATMP polymer with narrow polydispersity (d = 1.03) can be obtained by photoinduced solution polymerization of ATMP. The signal detected in ESR may be assigned to the nitroxyl radicals in the matrix of ATMP polymer. Since this kind of recycling of nitroxyl radicals is well documented for the photostabilizing mechanism of HALS, the present results may serve as a kinetic evidence for this mechanism.