Asymmetric Oscillator Research Articles

Oscillators with symmetric and asymmetric quadratic nonlinearity

In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Both oscillators are modeled as ordinary second-order differential equations with strong quadratic nonlinearities: one with positive quadratic term and the second with a quadratic term which changes the sign. Solutions for both equations are obtained in the form of Jacobi elliptic functions. For the asymmetric oscillator, conditions for the periodic motion are determined, while for the symmetric oscillator a new approximate solution procedure based on averaging is developed. Obtained results are tested on an optomechanical system where the motion of a cantilever in the intracavity field is oscillatory. Two types of quadratic nonlinearities in the system are investigated: symmetric and asymmetric. The advantage and disadvantage of both models is discussed. The analytical procedure suggested in the paper is applied. The obtained solution agrees well with a numerical one.

Entanglement evolution in the open quantum systems consisting of asymmetric oscillators

Using the theory of open quantum systems, we study the entanglement evolution in two and three-mode systems consisting of uncoupled harmonic oscillators which interact with a thermal bath as the environment. The evolution of the system is obtained with the use of the master equation in the Lindblad form with the Markovian approximation. The coherent states of a spinless charged particle in an anisotropic harmonic potential and a uniform magnetic field are considered as the initial states of two and three-mode systems. Then by the use of the positive partial transpose criterion for three-mode system and the logarithmic negativity for two-mode system, the entanglement evolution is obtained as a function of the temperature, dissipation coefficient, magnetic field and asymmetric parameter. In both two and three-mode systems, by increasing the dissipation coefficient and temperature, the entanglement sudden death occurs sooner. Also, for certain values of the magnetic field and asymmetric parameter which depend on the other parameters, the entanglement survives the most.

Periodic and unbounded motions in asymmetric oscillators at resonance

We consider the existence of periodic and unbounded solutions for the asymmetric oscillatorx″+ax+−bx−+g(x)=p(t), where x+=max⁡{x,0}, x−=max⁡{−x,0}, a and b are two positive constants, p(t) is a 2π-periodic smooth function and g(x) satisfies lim|x|→+∞⁡x−1g(x)=0. We have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions Φp(θ) and Λ(h). In this paper, we consider some critical cases and obtain some new sufficient conditions for the existence of 2π-periodic and unbounded solutions. In particular, the obtained periodic solutions are new and cannot be deduced from Landesman–Lazer's and Fabry–Fonda's existence conditions. In contrast with many existing results, the function g(x) may be unbounded or oscillatory without asymptotic limits.

M-shaped asymmetric nonlinear oscillator for broadband vibration energy harvesting: Harmonic balance analysis and experimental validation

Over the past few years, nonlinear oscillators have been given growing attention due to their ability to enhance the performance of energy harvesting devices by increasing the frequency bandwidth. Duffing oscillators are a type of nonlinear oscillator characterized by a symmetric hardening or softening cubic restoring force. In order to realize the cubic nonlinearity in a cantilever at reasonable excitation levels, often an external magnetic field or mechanical load is imposed, since the inherent geometric nonlinearity would otherwise require impractically high excitation levels to be pronounced. As an alternative to magnetoelastic structures and other complex forms of symmetric Duffing oscillators, an M-shaped nonlinear bent beam with clamped end conditions is presented and investigated for bandwidth enhancement under base excitation. The proposed M-shaped oscillator made of spring steel is very easy to fabricate as it does not require extra discrete components to assemble, and furthermore, its asymmetric nonlinear behavior can be pronounced yielding broadband behavior under low excitation levels. For a prototype configuration, linear and nonlinear system parameters extracted from experiments are used to develop a lumped-parameter mathematical model. Quadratic damping is included in the model to account for nonlinear dissipative effects. A multi-term harmonic balance solution is obtained to study the effects of higher harmonics and a constant term. A single-term closed-form frequency response equation is also extracted and compared with the multi-term harmonic balance solution. It is observed that the single-term solution overestimates the frequency of upper saddle-node bifurcation point and underestimates the response magnitude in the large response branch. Multi-term solutions can be as accurate as time-domain solutions, with the advantage of significantly reduced computation time. Overall, substantial bandwidth enhancement with increasing base excitation is validated experimentally, analytically, and numerically. As compared to the 3dB bandwidth of the corresponding linear system with the same linear damping ratio, the M-shaped oscillator offers 3200, 5600, and 8900 percent bandwidth enhancement at the root-mean-square base excitation levels of 0.03g, 0.05g, and 0.07g, respectively. The M-shaped configuration can easily be exploited in piezoelectric and electromagnetic energy harvesting as well as their hybrid combinations due to the existence of both large strain and kinetic energy regions. A demonstrative case study is given for electromagnetic energy harvesting, revealing the importance of higher harmonics and the need for multi-term harmonic balance analysis for predicting the electrical power output accurately.

Horseshoe Dynamics in an Asymmetric Duffing-Van der Pol Oscillator Driven by a Narrow-Band Frequency Modulated Force

We investigate both analytically and numerically the occurrence of horseshoe chaos in three different asymmetric Duffing-Van der Pol oscillators driven by a narrow-band frequency modulated force f(cos ωt-g sin Ωt sin ωt). We identify the regions of horseshoe chaos in (ω = Ω; f) parameter space and bring out the effect of the parameter α. We present the results for α = 0.9 and 1.2 in the three systems. When g = 0 and for fixed values of the other parameters there are regions in the (α, f) plane where (i) the horseshoe chaos can occur in both the x ＜ 0 and x ＞ 0 regions, and (ii) no horseshoe dynamics occurs in both the x ＜ 0 and x ＞ 0 regions. We illustrate that by varying the parameter g one can suppress chaotic dynamics, say, in the x ＞ 0 region or induce chaos in the region x ＜ 0. This is realized in the three asymmetrical systems.

Amplification and generation of terahertz plasmons in gated two-dimensional channels: Modal analysis

A theoretical analysis is presented of plasmons in gated two-dimensional electron channels supporting dc currents. In contrast to previous treatments, the model takes into account complete mode spectra and transverse field distributions. Conditions for plasmon amplification are determined, and asymmetric plasmonic oscillators are analysed. The results are compared with the traditional treatment of the Dyakonov-Shur instability. The limitations of the traditional treatment are thus revealed. The present electrodynamically rigorous model can be used to design and analyse terahertz plasmonic oscillators.

The influence of anharmonic and solvent effects on the theoretical vibrational spectra of the guanine–cytosine base pairs in Watson–Crick and Hoogsteen configurations

The theoretical IR and Raman spectra of the guanine-cytosine DNA base pairs in Watson-Crick and Hoogsteen configurations were computed using DFT method with M06-2X meta-hybrid GGA exchange-correlation functional, including the anharmonic corrections and solvent effects. The results for harmonic frequencies and their anharmonic corrections were compared with our previously calculated values obtained with the B3PW91 hybrid GGA functional. Significant differences were obtained for the anharmonic corrections calculated with the two different DFT functionals, especially for the stretching modes, while the corresponding harmonic frequencies did not differ considerable. For the Hoogtseen case the H⁺ vibration between the G-C base pair can be characterized as an asymmetric Duffing oscillator and therefore unrealistic anharmonic corrections for normal modes where this proton vibration is involved have been obtained. The spectral modification due to the anharmonic corrections, solvent effects and the influence of sugar-phosphate group for the Watson-Crick and Hoogsteen base pair configurations, respectively, were also discussed. For the Watson-Crick case also the influence of the stacking interaction on the theoretical IR and Raman spectra was analyzed. Including the anharmonic correction in our normal mode analysis is essential if one wants to obtain correct assignments of the theoretical frequency values as compared with the experimental spectra.

A Note on Forced Oscillations in Differential Equations with Jumping Nonlinearities

The goal of this paper is to study bifurcations of asymptotically stable \(2\pi \)-periodic solutions in the forced asymmetric oscillator \(\ddot{u}+\varepsilon c \dot{u}+u+\varepsilon a u^+=1+\varepsilon \lambda \cos t\) by means of a Lipschitz generalization of the second Bogolubov’s theorem due to the authors. The small parameter \(\varepsilon >0\) is introduced in such a way that any solution of the system corresponding to \(\varepsilon =0\) is \(2\pi \)-periodic. We show that exactly one of these solutions whose amplitude is \(\frac{\lambda }{\sqrt{a^2+c^2}}\) generates a branch of \(2\pi \)-periodic solutions when \(\varepsilon >0\) increases. The solutions of this branch are asymptotically stable provided that \(c>0\).

Kinematics of a 2D Asymmetric Nonlinear Oscillator

Motion of a point-like massive particle under the influence of two nonidentical linear springs conducive to an irregular planar oscillation is analyzed. For a two dimensional oscillations the equation of motion is a coupled highly nonlinear differential equation. The set of equations cannot be solved analytically. Utilizing a Computer Algebra System (CAS) such as Mathematica [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000005F005200650066003300390031003500360035003100380030000000 we solve the equations numerically. Kinematics of the particle is presented. For a comprehensive visual understanding the oscillations are simulated. We also include an extended atlas of useful two-dimensional time-folded diagrams.

Nonlinear Functional Analysis of Boundary Value Problems 2013

and Applied Analysis 3 solve a class of boundary value problems for second-order singular differential equations. (iii) In the paper titled “Existence and uniqueness of solution to nonlinear boundary value problems with sign-changing Green’s function,” by using the cone theory and the Banach contraction mapping principle, some existence and uniqueness results are established for a nonlinear higher-order differential equation boundary value problem with signchanging Green’s function. (iv) In the paper titled “Onboundedness and attractiveness of nonlinear switched delay systems,” the authors study the boundedness and attractiveness problems for a class of nonlinear switched delay systems. Some sufficient conditions are established to guarantee the system’s boundedness. The authors also work out the region where the solution will remain and the relationship between the initial function and the bounded region. (v) In the paper titled “Unbounded solutions of asymmetric oscillator,” the authors establish sufficient conditions for the existence of unbounded solutions of a nonlinear differential equation with a continuous, bounded, and periodic source function. (6) Stochastic Differential Equation Models (i) In the paper titled “A stochastic string with a compound poisson process,” the authors introduce a compound Poisson process with a constant jump intensity and random jump-size to capture information burst and the resulting discontinuous path and derive the no-arbitrage condition on the drift of instantaneous forward rates in the compound model. With the proposedmodel, the impact of random jump on the price of the zero coupon bond is shown in the paper. (ii) In the paper titled “Equilibrium asset and option pricing under jump diffusion model with stochastic volatility,” the authors study the equity premium and option pricing under a jump-diffusion model with stochastic volatility. The pricing kernel is established and the exact expression for the option price is derived by using the Fourier transformation method. (7) Fixed-Point Theory (i) In the paper titled “Generalizations of fixed-point theorems of Altman and Rothe types,” the authors present some extensions of the Altman and Rothe type fixed-point theorems. Some new fixed-point theorems for continuous operators are established by use of the theory of topological degree. (ii) In the paper titled “Commonfixed-points for weak contractive mappings in ordered metric space with applications,” the authors establish some new common fixed-point theorems satisfying a weak contractive condition in the framework of partially ordered metric spaces. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

Bounded and Unbounded Motions in Asymmetric Oscillators at Resonance

We consider the boundedness and unboundedness of solutions for the asymmetric oscillator $$\begin{aligned} x''+ax^+-bx^-+g(x)=p(t), \end{aligned}$$ where $$x^+=\max \{x,0\},x^-=\max \{-x,0\}, a$$ and $$b$$ are two positive constants, $$ p(t)$$ is a $$2\pi $$ -periodic smooth function and $$g(x)$$ satisfies $$\lim _{|x|\rightarrow +\infty }x^{-1}g(x)=0$$ . We establish some sharp sufficient conditions concerning the boundedness of all the solutions and the existence of unbounded solutions. It turns out that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions $$\Phi _p(\theta )$$ and $$\Lambda (h)$$ . Some explicit conditions are given for the boundedness of all the solutions and the existence of unbounded solutions. Unlike many existing results in the literature where the function $$g(x)$$ is required to be a bounded function with asymptotic limits, here we allow $$g(x)$$ be unbounded or oscillatory without asymptotic limits.

Generalized Analysis of Symmetric and Asymmetric Memristive Two-Gate Relaxation Oscillators

Memristive oscillators are a novel topic in nonlinear circuit theory, where the behavior of the reactive elements is emulated by the memristor. This paper presents symmetric and asymmetric memristive two-gate relaxation oscillators. First, the analysis of the two series memristors is introduced to study the effect of changing their polarities, as well as the mobility factor to be used in the two-gate relaxation oscillator instead of the RC circuit. The generalized analysis for the proposed memristive two-gate oscillator is introduced, where the generalized expressions for the oscillation frequency and conditions for oscillation are derived then four special cases for different mismatching of the memristors are introduced; showing a perfect matching with the PSPICE simulations. Finally, the discussion and comparison are proposed to discuss the four special cases and MATLAB simulations are also provided to study the effect of the memristance and the mobility ratio between the memristors on the oscillation frequency.

Quantum dynamics in the partial Wigner picture

Recently we have shown how the partial Wigner representation of quantum mechanics can be used to study hybrid quantum models where a system with a finite number of energy levels is coupled to linear or nonlinear oscillators (Beck and Sergi 2013 Phys. Lett. A 377 1047). The purpose of this work is to provide a detailed derivation of the partially Wigner-transformed quantum equations of motion for nonlinear oscillator subsystems under the action of general polynomial potentials. Such equations can be written in terms of a propagator, which can then be expanded in a power series. The linear terms of the series describe quantum-classical dynamics while the nonlinear terms provide the corrections needed to restore the fully quantum character of the evolution. In the case of polynomial potentials and position dependent couplings, the number of nonlinear terms is finite and the corrections can be calculated explicitly. In this work we show how to implement numerically the above scheme where, in principle, no assumption about the strength of the coupling must be taken. We illustrate the formalism by studying a two-level system interacting with an asymmetric quartic oscillator. We integrate the quantum dynamics of the total system and provide a comparison with the case of the quantum-classical dynamics of the quartic oscillator. The approach presented here is expected to be effective for studying hybrid quantum circuits in quantum information theory and for witnessing the quantum-to-classical transition in nano-oscillators coupled to pseudo-spins.

The effect of an external electric field on the optical properties of a quantum-dot molecule with a resonant state of the D 2 − center

The effect of an external electric field on the probability of the electron radiative transition from a resonant u-state to the localized g-state of the D2− center in the presence of dissipative tunneling has been investigated in the model of zero-radius potential. It is shown that the probability of radiative transition increases by about two orders of magnitude in the case of the external electric field intensity for which the initially asymmetric double-well oscillator potential simulating a quantum-dot molecule becomes symmetrical.

Transient and Steady-State Responses of an Asymmetric Nonlinear Oscillator

We study the dynamical response of an asymmetric forced, damped Helmholtz-Duffing oscillator by using Jacobi elliptic functions, the method of elliptic balance, and Fourier series. By assuming that the modulus of the elliptic functions is slowly varying as a function of time and by considering the primary resonance response of the Helmholtz-Duffing oscillator, we derived an approximate solution that provides the time-dependent amplitude-frequency response curves. The accuracy of the derived approximate solution is evaluated by studying the evolution of the response curves of an asymmetric Duffing oscillator that describes the motion of a damped, forced system supported symmetrically by simple shear springs on a smooth inclined bearing surface. We also use the percentage overshoot value to study the influence of damping and nonlinearity on the transient and steady-state oscillatory amplitudes.

Boundary Value Problems for a Super-Sublinear Asymmetric Oscillator: The Exact Number of Solutions

Properties of asymmetric oscillator described by the equation (i), where and , are studied. A set of such that the problem (i), (ii), and (iii) have a nontrivial solution, is called α-spectrum. We give full description of α-spectra in terms of solution sets and solution surfaces. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (i), and (ii) is given.

Unbounded Solutions of Asymmetric Oscillator

We obtain sufficient conditions for the existence of unbounded solutions of the following nonlinear differential equation(φp(x′))+(p−1)[αφp(x+)−βφp(x−)]=(p−1)f(t,x,x′), whereφp(u)=|u|p−2u, p>1, x+=max{x,0}, x−=max{−x,0},α,βare positive constants, andfis continuous, bounded, andT-periodic intfor someT>0.

Exact Solutions of the Klein–Gordon Equation for Spherically Asymmetrical Singular Oscillator

The relativistic Klein–Gordon equation with equal scalar and vector spherically asymmetrical singular oscillators is solved using the asymptotic iteration method. The energy eigenvalues equation and the corresponding wave functions are obtain explicitly. It was found that the asymptotic iteration method provides the closed-forms for the energy eigenvalues as well as the eigenfunctions. The non-relativistic limit \({c \rightarrow \infty}\) of the energy spectrum, where c is the speed of light, have also been discussed.

Flow Control Mono and Bi-stable Fluidic Device for Micromixer-injection System

Fluidics is a new technology arising from a re-appraisal of a very old technology, namely fluid power and its control. Fluidic technology based on natural oscillation phenomena is a relevant topic in several strategic areas. This technology of using the flow characteristics of liquids or gases to operate a control system is fairly old; it was in the 1960s that researchers started to use fluidics.This paper is focused on the modeling of a microinjection systems composed of passive amplifier without mechanical part. The micro-system modeling is based on geometrical oscillators form. An asymmetric micro oscillator design based on a monostable fluidic amplifier. Several configuration of associate device are simulating to found adequate logical signal form. The characteristic size of the channels is generally about a few hundred of microns.Click here and insert your abstract text.

Application of a Modified Homotopy Perturbation Method for Calculation of Secular Axial Frequencies in a Nonlinear Ion Trap with Hexapole, Octopole and Decapole Superpositions

In this paper we have used a modified homotopy perturbation method used previously by A. Belendez and his coworkers, for calculation of axial secular frequencies of a nonlinear ion trap with hexapole, octopole and decapole superpositions. We transform the motion of the ion in a rapidly oscillating field to the motion in an effective potential and obtain a nonlinear differential equation in the form of a Duffing-like equation. With only octopole superposition the resulted nonlinear equations are symmetric; however, in the presence of hexapole and decapole superpositions, they are asymmetric. For asymmetric oscillators, it has been pointed out that the angular frequency for positive amplitudes is different from the angular frequency for negative amplitudes. Considering this problem, the modified homotopy perturbation method is used for solving the resulted nonlinear equations. As a result, the ion axial secular frequencies as a function of nonlinear field parameters are obtained. The calculated secular frequencies are compared with the results of modified Lindstedt-Poincare approximation and the exact results. There is an excellent agreement between the results of this paper and the exact results.