On the forced vibrations of an oscillator with a periodically time-varying mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator with a time-varying mass. Not only solutions of the oscillator equation will be constructed, but also stability properties for the forced vibrations will be presented for various parameter values.     

On the stability properties of a damped oscillator with a periodically time-varying mass

In this paper the vibrations of a damped, linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be considered. Both the free and forced vibrations of the oscillator will be studied. For the free vibrations the minimal damping rates will be computed, for which the oscillator is always stable. The forced vibrations are partly due to small masses, which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator. Not only solutions of the oscillator equations will be constructed, but also stability properties for the free, and for the forced vibrations will be presented for various parameter values. For the external, harmonic forcing case an interesting resonance condition will be derived.     

Stability analysis of a nonlinear rotating blade with torsional vibrations

This paper discusses the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model. To obtain a reduced nonlinear model from nodal space, a standard modal reduction procedure based on matrix operation is developed with essential geometric stiffening nonlinearities retained in the equation of motion. For the linear model, the stability chart with various spinning parameters of the blade is studied via the Bolotin method, and an efficient boundary tracing algorithm is developed to trace the stability boundary of the linear model. For the geometric nonlinear model, the method of multiple time scale is employed to study the steady state solutions, and their stability and bifurcations for the periodically time-varying rotating blade. The backbone curves of steady-state motions are achieved, and the parameter map for stability and bifurcation is developed.     

An Instability Index Theory for Quadratic Pencils and Applications

Primarily motivated by the stability analysis of nonlinear waves in second-order in time Hamiltonian systems, in this paper we develop an instability index theory for quadratic operator pencils acting on a Hilbert space. In an extension of the known theory for linear pencils, explicit connections are made between the number of eigenvalues of a given quadratic operator pencil with positive real parts to spectral information about the individual operators comprising the coefficients of the spectral parameter in the pencil. As an application, we apply the general theory developed here to yield spectral and nonlinear stability/instability results for abstract second-order in time wave equations. More specifically, we consider the problem of the existence and stability of spatially periodic waves for the “good” Boussinesq equation. In the analysis our instability index theory provides an explicit, and somewhat surprising, connection between the stability of a given periodic traveling wave solution of the “good” Boussinesq equation and the stability of the same periodic profile, but with different wavespeed, in the nonlinear dynamics of a related generalized Korteweg–de Vries equation.     

Nonlinear evolutionary processes in a free-electron laser generator of diffracted radiation

Nonlinear evolutionary processes with two control parameters, one of which is related to the electrodynamic structure (positive feedback) and the other is related to the constant electric field applied to the electron flux, are studied in a free-electron laser, which is a diffracted-radiation oscillator. To this end, first, the linear spectral problem for an open-cavity resonator is investigated and the dispersion relation near the Morse critical point is established. Then the nonlinear evolutionary equation with two control parameters is constructed. Analysis of the latter makes it possible to determine the properties of the parametric dependence of the bifurcation and structural stability, which are determined by small variations of the control parameter (tuning of the cavity). This explains the operating efficiency of a diffracted-radiation oscillator in the millimeter and submillimeter wavelength ranges. Zh. éksp. Teor. Fiz. 111, 2243–2262 (June 1997)     

A Fractional-Order Sinusoidal Discrete Map

In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.     

High-codimensional static bifurcations of strongly nonlinear oscillator

The static bifurcation of the parametrically excited strongly nonlinear oscillator is studied. We consider the averaged equations of a system subject to Duffing--van der Pol and quintic strong nonlinearity by introducing the undetermined fundamental frequency into the computation in the complex normal form. To discuss the static bifurcation, the bifurcation problem is described as a 3-codimensional unfolding with $Z_{2}$ symmetry on the basis of singularity theory. The transition set and bifurcation diagrams for the singularity are presented, while the stability of the zero solution is studied by using the eigenvalues in various parameter regions.     

Exact Bloch States of a Spin-orbit Coupled Bose-Einstein Condensate in an Optical Lattice

We study the exact Bloch states of a spin-orbit (SO) coupled Bose-Einstein condensate (BEC) held in an optical lattice. Under a natural condition of the symmetry between the two species, we obtain two different forms of exact solutions corresponding to different existing conditions. Then, we analytically demonstrate that (a) the average atomic number per well can enlarge the region area (consisting of instability and stability parameter regions) existing exact solutions; (b) the sizes of the instability and stability parameter regions exhibit opposite variation trend with the increase in Rabi coupling strength, and the results of different solutions are just opposite. Besides, we find that spin-orbit coupling (SOC) results in the generation of spin-motion entanglement for the Bloch states, the SOC strength and lattice depth can influence the population transfer between two BEC components, and varying the SOC strength and lattice depth can also reveal the dynamical superfluid-insulator transition from the superfluid state to the critical insulating state. These results present a feasible scheme to manipulate the stable superfluid currents, which will be useful to control quantum transport of BEC.

     

A memristor oscillator based on a twin-T network

A novel inductance-free nonlinear oscillator circuit with a single bifurcation parameter is presented in this paper. This circuit is composed of a twin-T oscillator, a passive RC network, and a flux-controlled memristor. With an increase in the control parameter, the circuit exhibits complicated chaotic behaviors from double periodicity. The dynamic properties of the circuit are demonstrated by means of equilibrium stability, Lyapunov exponent spectra, and bifurcation diagrams. In order to confirm the occurrence of chaotic behavior in the circuit, an analog realization of the piecewise-linear flux-controlled memristor is proposed, and Pspice simulation is conducted on the resulting circuit.     

Control of instability in a semiconductor laser using a functional pump current generator with a dynamical parameter

In this paper, an electric circuit to control the dynamic output of a semiconductor laser is introduced. The circuit controls chaos and instability of the laser output by changing its pumping current. The change of the current is also introduced by a nonlinear map. The most important element of this nonlinear map is a dynamical variable parameter. We have studied the dynamic behavior of the laser before and after applying the control using bifurcation curves and time series. We have shown that the laser output, in the intervals of the feedback phase and strength where it is chaotic, can be totally inverted to the quasi periodic (QP) and period one (P1) oscillation modes, by control method.     

Bifurcations of a parametrically excited oscillator with strong nonlinearity

A parametrically excited oscillator with strong nonlinearity, including van der Pol and Duffing types, is studied for static bifurcations. The applicable range of the modified Lindstedt－Poincaré method is extended to 1/2 subharmonic resonance systems. The bifurcation equation of a strongly nonlinear oscillator, which is transformed into a small parameter system, is determined by the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analysed.     

Order parameter equation and model equation for high Prandtl number. Rayleigh-Bénard convection in a rotating large aspect ratio system

We derive the order parameter equation which describes the evolution of spatio-temporal patterns close to the Bénard instability in a rotating large aspect ratio system for high Prandtl number fluids. Since this order parameter equation contains rather complicated nonlinear terms we present a model equation which can be obtained from the order parameter equation by suitable simplification of the nonlinearity. For this model equation we calculate the family of roll solutions and investigate their stability with respect to long scale instabilities and examine the onset of the Küppers-Lortz instability. Then we present spatiotemporal patterns which are obtained from a numerical evaluation of the model equation.     

Dynamics and stability of a new class of periodic solutions of the optical parametric oscillator

The stability and dynamics of a new class of periodic solutions is investigated when a degenerate optical parametric oscillator system is forced by an external pumping field with a periodic spatial profile modeled by Jacobi elliptic functions. Both sinusoidal behavior as well as localized hyperbolic (front and pulse) behavior can be considered in this model. The stability and bifurcation behaviors of these transverse electromagnetic structures are studied numerically. The periodic solutions are shown to be stabilized by the nonlinear parametric interaction between the pump and signal fields interacting with the cavity diffraction, attenuation, and periodic external pumping. Specifically, sinusoidal solutions result in robust and stable configurations while well-separated and more localized field structures often undergo bifurcation to new steady-state solutions having the same period as the external forcing. Extensive numerical simulations and studies of the solutions are provided.     

Complex dynamics in a simple model of pulsations for super-asymptotic giant branch stars

When intermediate mass stars reach their last stages of evolution they show pronounced oscillations. This phenomenon happens when these stars reach the so-called asymptotic giant branch (AGB), which is a region of the Hertzsprung-Russell diagram located at about the same region of effective temperatures but at larger luminosities than those of regular giant stars. The period of these oscillations depends on the mass of the star. There is growing evidence that these oscillations are highly correlated with mass loss and that, as the mass loss increases, the pulsations become more chaotic. In this paper we study a simple oscillator which accounts for the observed properties of this kind of stars. This oscillator was first proposed and studied in Icke et al. [Astron. Astrophys. 258, 341 (1992)] and we extend their study to the region of more massive and luminous stars -the region of super-AGB stars. The oscillator consists of a periodic nonlinear perturbation of a linear Hamiltonian system. The formalism of dynamical systems theory has been used to explore the associated Poincare map for the range of parameters typical of those stars. We have studied and characterized the dynamical behavior of the oscillator as the parameters of the model are varied, leading us to explore a sequence of local and global bifurcations. Among these, a tripling bifurcation is remarkable, which allows us to show that the Poincare map is a nontwist area preserving map. Meandering curves, hierarchical-islands traps and sticky orbits also show up. We discuss the implications of the stickiness phenomenon in the evolution and stability of the super-AGB stars. (c) 2002 American Institute of Physics.     

Nonlinear Waves in an Inhomogeneous Fluid Filled Elastic Tube

In a thin-walled, homogeneous, straight, long, circular, and incompressible fluid filled elastic tube, small but finite long wavelength nonlinear waves can be describe by a KdV (Korteweg de Vries) equation, while the carrier wave modulations are described by a nonlinear Schrödinger equation (NLSE). However if the elastic tube is slowly inhomogeneous, then it is found, in this paper, that the carrier wave modulations are described by an NLSE-like equation. There are soliton-like solutions for them, but the stability and instability regions for this soliton-like waves will change, depending on what kind of inhomogeneity the tube has.     

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameter-dependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green–Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.     

分数阶时滞反馈对Duffing振子动力学特性的影响

研究了含分数阶时滞耦合反馈的Duffing自治系统, 通过平均法得到了系统周期解的一阶近似解析形式, 定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数, 发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用. 比较了三种参数条件下近似解析解与数值积分的结果, 二者的吻合精度都很高, 证明了近似解析解的正确性和准确性. 分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.     

Phase synchronization in tilted deterministic ratchets

We study phase synchronization for a ratchet system. We consider the deterministic dynamics of a particle in a tilted ratchet potential with an external periodic forcing, in the overdamped case. The ratchet potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of external periodic forcing. This oscillator has an intrinsic frequency that can be entrained with the frequency of the external driving. We introduced a linear phase through a set of discrete time events and the associated average frequency, and show that this frequency can be synchronized with the frequency of the external driving. In this way, we can properly characterize the phenomenon of synchronization through Arnold tongues, which represent regions of synchronization in parameter space, and discuss their implications for transport in ratchets.     

Rogue waves: New forms enabled by GPU computing

A method to determine parameters governing periodic Riemann theta function rogue-wave solutions to the nonlinear Schrödinger equation is presented. A map of parameter values leading to candidate solutions is developed. In addition to candidate solutions, an overview of qualitative aspects of the solution space can be gained from this map. Based on these findings, several new extreme wave solutions are presented. Although the computations required to determine the map are quite demanding, it is shown that these computations can be efficiently accelerated with a parallel computing architecture. A general purpose computing on a graphics processor unit (GPGPU) implementation yielded a 400× acceleration over a single threaded high level implementation. This acceleration enabled exploration and examination of the solution space, which otherwise would not have been possible. In addition, the solution methodology presented here can be extended to explore other classes of solutions.