Fractional Oscillator Research Articles

Nonstationary response statistics of fractional oscillators to evolutionary stochastic excitation

While studying fractional oscillators to the excitation as an evolutionary stochastic process, this paper develops a novel pole-residue method to compute closed-form solutions for the nonstationary response statistics, including evolutionary power spectrum, correlation function and mean square value. The proposed method is not only suitable to the oscillators with an arbitrary fractional order, but also applies to oscillators with multiple fractional derivative terms. A necessary task of the proposed method is to obtain a pole-residue form of the oscillator’s transfer function (TF), which can be achieved through two steps: (1) obtaining a discrete impulse response function (IRF) from taking the inverse Fourier transform of the oscillator’s frequency response function (FRF), which is readily derived from its exact TF; (2) invoking the Prony-SS method to decompose this IRF signal for getting the poles and residues of the TF. Once the pole-residue TF is available, the remaining derivation can follow a similar pole-residue method that has been developed for ordinary linear oscillators. In the numerical studies, three fractional oscillators with one and two fractional derivative terms, respectively, are chosen to demonstrate the proposed method; considering the input to be a uniformly modulated random process modeled with either white noise or Kanai–Tajimi earthquake spectrum, the correctness of the proposed method is verified by Monte Carlo simulations.

Decay and revival of a transient trapped Fermi condensate

The authors investigate the response of a strongly-interacting superfluid Fermi gas to a spin-flip and corresponding change of the interaction strength. They observe decay and revival of the condensed fraction and shape oscillations of the gas.

Charge Oscillations in Quantum Spin Hall Constrictions

Constrictions between 1D helical edge states represent a useful tool for the realization of devices based on topological materials. Indeed, they complement superconductivity and magnetic barriers as gaps allowing to enlarge the functionalities of topological nanostructures. Recent experiments point to the direction that electronic interactions are not negligible in such systems. Herein, the interaction‐induced oscillations of the local electronic density at constrictions between helical edges are theoretically inspected. The system is in the quantum dot configuration thanks to magnetic barriers. By means of bosonization, it is shown that the fractional charge oscillations, predicted for single edges, can persist. On top of that, oscillations involving both edges and with different wave vectors can appear. It is argued that the dominant oscillation can be discriminated by varying the angle of magnetization of the barriers, thus providing information about the dominant interaction mechanisms.

Collective behaviors of two coupled harmonic oscillators driven by different frequency fluctuations with fractional damping

In a complex environment, the long-time memory and coupling effect are two important system characteristics that can be described by the fractional damping force and coupling force. This paper investigates the collective behaviors of two coupled fractional harmonic oscillators driven by different frequency fluctuations, including stability, synchronization and stochastic resonance (SR). Theoretically, the ‘synchronization condition’ and ‘stability condition’ of the system are derived. Comparative analysis shows that the latter is stricter than the former. Based on this, an analytical expression of the output amplitude gain is obtained. The numerical results show that when the stability condition is met, the average trajectories of two particles are both bounded and synchronous. Otherwise, they will diverge to infinity. Increasing ɛ (coupling strength) and decreasing α (fractional order) can both accelerate the synchronization speed. SR mainly occurs in the high-α or high-σ (noise amplitude) region, which means that SR emergence can be controlled by adjusting α or σ. The damping force, coupling force and frequency fluctuations compete with each other; thus, the SR intensity should be maximized by adjusting α, ɛ and σ simultaneously.

Solving Coupled Fractional Differential Equations Using Differential Transform Method.

This paper presents the solution of coupled equations which are of fractional order using differential transform method. In this paper we extend the scope of differential transform method to system of fractional differential equations so that we get the analytical solutions. The coupled fractional differential equations of a physical system, namely, coupled fractional oscillator with some applications is given via differential transform method. Here we introduce the solution of coupled oscillation of equal fractional order which can be enhanced to unequal fractional order.

A Practical Fractional-Order Sinusoidal Oscillator Design and Implementation

In this study, a new sinusoidal fractional oscillator circuit in which the fractional low-pass filter and fractional differentiator form a closed loop has been introduced. The single series [Formula: see text] pair and Valsa method have been used to imitate the fractional-order capacitor. Two different sinusoidal signals with different phases are available at output ports. An integer version of the proposed oscillator is not possible, only the fractional version is valid, that is a noticeable feature. The introduced circuit has been simulated by employing SPICE software. The proposed fractional oscillator is also verified by implementing circuit with AD817ANs and passive components, experimentally.

Fractional quadrature oscillator using VDTAs with grounded capacitors

In this proposed work, the fractional order quadrature oscillator using voltage differencing trans-conductance amplifier (VDTA) as an active building block is designed. The circuit configuration of VDTA with its characteristics is shown. The expression for the frequency of oscillation is derived. The non-ideal effects and sensitivity of the proposed design for the parasitic components are determined. The stability of the proposed work is analyzed with the help of value and the location of roots of the proposed circuit. The approximation of the fractional order capacitor is done with the help of a single parallel circuit of passive components and General impedance converter. The comparison of the proposed work is done with the existing works. The results of the proposed work are verified using PSPICE with TSMC CMOS 180 nm technology.

Fractional order biological snap oscillator: Analysis and control

The manuscript studies the four dimensional fractional order biological snap oscillator. Its thorough dynamical analysis is performed using dynamical tools such as phase portraits, Lyapunov and bifurcation analysis, system’s solution, stagnation points analysis etc. The impact of fractional order on the chaotic system is also discussed by plotting phase diagrams, Lyapunov and bifurcation diagrams for variable order. Considering uncertainties and disturbances trajectories are stabilized about its unique stagnation point using adaptive SMC technique. The controllers are constructed to compensate undesirable chaotic/ hyper chaotic behavior of the biological system which may give rise to fatal health pathological conditions. Also SMC technique show immediate effect in controlling of chaos in the system. Numerical simulations are performed using MATLAB.

Fractional-order harmonic resonance in a multi-frequency excited fractional Duffing oscillator with distributed time delay

The general investigation on the vibration mechanism at arbitrary fraction-order harmonic resonance of the fractional Mathieu-Duffing system driven by multiple periodic excitations with distributed delay is reported theoretically and numerically. In terms of numerical scheme, a precise numerical algorithm based on the definition of Caputo derivative is applied. The unified analytical results of the steady-state response amplitude at each resonant frequency are obtained by utilizing the method of direct partition of motions, the self-consistent technique and the Krylov-Bogolubov-Mitropolsky asymptotic method. It relies on two kinds of parametric-forced pattern. It is found that for some parametric setting, both the amplitude of the external high-frequency excitation and the frequency of the external low-frequency excitation can give rise to saddle-node bifurcation of the steady-state amplitude in the analytical results. It is equivalent to amplitude jump phenomenon in the numerical observation. While comparing to the integer-order system, it is more likely to happen in the fractional-order situation. With the variant of the fractional order of the derivative, the steady-state amplitude solutions can be converted from monostability to bistability, then to monostability, and finally to the bistability again. This new multi-saddle-node bifurcation has not yet been reported before. The novel transcritical bifurcation induced by the strength of the distributed delay is discussed and verified in detail for the first time.

Regularity estimates for the Cauchy problem to a parabolic equation associated to fractional harmonic oscillators

Let H=−Δ+|x|2 be the harmonic oscillator on Rn. In this paper, we prove estimates on Besov spaces associated to the operator H and the end-point maximal regularity estimates for the fractional harmonic oscillator Hα, 0<α≤1, on the Besov spaces associated to the harmonic oscillator H. These spaces are the appropriate function spaces for the study of estimates on Besov type spaces and the end-point maximal regularity estimates for the fractional power Hα in the sense that similar estimates might fail with the classical Besov spaces.

A new approach on the modelling, chaos control and synchronization of a fractional biological oscillator

This research aims to discuss and control the chaotic behaviour of an autonomous fractional biological oscillator. Indeed, the concept of fractional calculus is used to include memory in the modelling formulation. In addition, we take into account a new auxiliary parameter in order to keep away from dimensional mismatching. Further, we explore the chaotic attractors of the considered model through its corresponding phase-portraits. Additionally, the stability and equilibrium point of the system are studied and investigated. Next, we design a feedback control scheme for the purpose of chaos control and stabilization. Afterwards, we introduce an efficient active control method to achieve synchronization between two chaotic fractional biological oscillators. The efficiency of the proposed stabilizing and synchronizing controllers is verified via theoretical analysis as well as simulations and numerical experiments.

Statistical and spectral analysis of wind power: Fractional oscillation dynamics

Statistical and spectral analysis of wind power: Fractional oscillation dynamics

Излучение дробного осциллятора

The spectral energy density of the oscillator radiation is calculated in the dipole approximation. The oscillator motion is described by an equation with fractional integro-differentiation. The fractional oscillator model can describe various types of radiation, including those with a nonexponential relaxation law. Found the shape of the spectral line of radiation. The obtained result is compared with the classical Lorentzian spectrum and experimental emission spectra of monochromatic and phosphor LEDs. The order of fractional integro-differentiation in the model sets the magnitude of the broadening of the radiation spectrum.

The optimal displacement control for fractional incommensurate mass-spring oscillators by Shifted Legendre polynomials

Based on the Riemann–Liouville fractional derivative, an optimal displacement control strategy for fractional incommensurate mass-spring oscillators has been presented. According to the calculus of variations and the Lagrange multiplier technique, the optimality conditions for the given problem are obtained. Using Shifted Legendre polynomials, the problem of solving differential equations is transformed into the problem of solving a set of algebraic equations. The validity, high efficiency and applicability of the proposed method are demonstrated through the simulation results.

Anthropogenic climate change on a non-linear arctic sea-ice model of fractional Duffing oscillator

Abstract In this contribution, a non-linear arctic sea-ice model of fractional Duffing oscillator is given. The solution of the model was obtained using a new proposed analytical method, which is an elegant combination of asymptotic and Laplace methods. The result obtained showed that this method is a very powerful and efficient technique for finding the analytical solution of nonlinear fractional differential equation. From the analysis of the result, we observed that the impact of anthropogenic climate change on arctic sea-ice could lead to flooding in many coastal areas and low-lying island nations.

The periodic response of a fractional oscillator with a spring-pot and an inerter-pot

Abstract The fractional oscillation system with two Weyl-type fractional derivative terms $_{ - \infty }D_t^\beta x$ (0 &amp;lt; β &amp;lt; 1) and $_{ - \infty }D_t^\alpha x$ (1 &amp;lt; α &amp;lt; 2), which portray a “spring-pot” and an “inerter-pot” and contribute to viscoelasticity and viscous inertia, respectively, was considered. At first, it was proved that the fractional system with constant coefficients under harmonic excitation is equivalent to a second-order differential system with frequency-dependent coefficients by applying the Fourier transform. The effect of the fractional orders β (0 &amp;lt; β &amp;lt; 1) and α (1 &amp;lt; α &amp;lt; 2) on inertia, stiffness and damping was investigated. Then, the harmonic response of the fractional oscillation system and the corresponding amplitude–frequency and phase–frequency characteristics were deduced. Finally, the steady-state response to a general periodic incentive was obtained by utilizing the Fourier series and the principle of superposition, and the numerical examples were exhibited to verify the method. The results show that the Weyl fractional operator is extremely applicable for researching the steady-state problem, and the fractional derivative is capable of describing viscoelasticity and portraying a “spring-pot”, and also describing viscous inertia and serving as an “inerter-pot”.

Ordinary Differential Equation with Left and Right Fractional Derivatives and Modeling of Oscillatory Systems

We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fractional oscillator equation containing the left and right Gerasimov–Caputo fractional derivatives with origins at plus and minus infinity. We derive a representation for the composition of the left and right derivatives with origins at plus and minus infinity in terms of the Riesz potential, and introduce special functions with which we give solutions to the model fractional oscillator equation with a complex coefficient. This approach can be useful for describing dissipative dynamical systems with the property of heredity.

Mathematical Model of Fractional Duffing Oscillator with Variable Memory

The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann–Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann–Liouville type by a discrete analog—the fractional derivative of Grunwald–Letnikov. The adequacy of the numerical scheme is verified using specific examples. Using a numerical algorithm, oscillograms and phase trajectories are constructed depending on the values of the model parameters. Chaotic regimes of the Duffing fractional oscillator are investigated using the Wolf–Bennetin algorithm. The forced oscillations of the Duffing fractional oscillator are investigated using the harmonic balance method. Analytical formulas for the amplitude-frequency, phase-frequency characteristics, and also the quality factor are obtained. It is shown that the fractional Duffing oscillator possesses different modes: regular, chaotic, multi-periodic. The relationship between the order of the fractional derivative and the quality factor of the oscillatory system is established.

Design and realisation of a fractional‐order sinusoidal oscillator

In this work, a new fractional-order sinusoidal oscillator is proposed. The proposed oscillator consists of one fractional-order all-pass filter and one fractional-order lossless integrator blocks. In order to emulate fractional-order capacitors, three different approximation methods the R–C pair, Matsuda and Oustaloup are employed and the results are compared. Three sinusoidal voltage signals with different phases that are controlled by fractional orders are provided by the proposed oscillator topology that is not possible for the classical integer-order case. Grounded passive components are used in the introduced oscillator circuit. Additionally, the output voltage signals are available at the low impedance terminal W of AD844ANs. Moreover, the time constant of the fractional-order all-pass filter can be adjusted accurately by means of second resistor Rx in the all-pass filter section. In addition to the simulation results, the proposed fractional oscillator is also implemented to verify the oscillator circuit experimentally. For the case of fractional orders α = 1 and β = 0.5, the measurement results are 10.3 kHz, 43.22° and 67.99°, respectively, while the desired values are 10 kHz, 45° and 67.36°, respectively.

Mathematical Modeling of Linear Fractional Oscillators

In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out.