Asymmetric Oscillator Research Articles

Asymmetric Duffing oscillator: the birth and build-up of a period-doubling cascade

Asymmetric Duffing oscillator: the birth and build-up of a period-doubling cascade

Harmonic Oscillator Staging Coordinates for Efficient Path Integral Simulations of Quantum Oscillators and Crystals.

Imaginary-time path integral (PI) is a rigorous quantum mechanical tool to compute static properties at finite temperatures. However, the stiff nature of the internal PI modes poses a sampling challenge. This is commonly tackled using staging coordinates, in which the free particle (FP) contribution of the PI action is diagonalized. We introduce novel and simple staging coordinates that diagonalize the entire action of the harmonic oscillator (HO) model, rendering it efficiently applicable to (exclusively) systems with harmonic character, such as quantum oscillators and crystals. The method is not applicable to fluids or systems with imaginary modes. Unlike FP staging, the HO staging provides a unique treatment of the centroid mode. We provide implementation schemes for PIMC and PIMD simulations in NVT ensemble. Sampling efficiency is assessed in terms of the precision and accuracy of estimating the energy and heat capacity of a one-dimensional HO and an asymmetric anharmonic oscillator (AO). In PIMC, the HO coordinates propose collective moves that perfectly sample the HO contribution, then (for AO) the residual anharmonic term is sampled using standard Metropolis method. This results in a high acceptance rate and, hence, high precision, in comparison to the FP staging. In PIMD, the HO coordinates naturally prescribe definitions for the fictitious masses, yielding equal frequencies of all modes when applied to the HO model. This allows for a substantially larger time step sizes relative to standard staging, without affecting accuracy or integrator stability. For completeness, we also present results using normal mode (NM) coordinates, based on both HO and FP models. While staging and NM coordinates show similar performance (for FP or HO), staging is computationally preferable due to its cheaper scaling with the number of beads. The simplicity and the enhanced sampling gained by the HO coordinates open avenues for efficient estimation of nuclear quantum effects in more complex systems with harmonic character, such as real molecular bonds and quantum crystals.

Response analysis of the Duffing oscillator in dielectric elastomer transducers

To elucidate the nonlinear oscillations inherent to dielectric elastomer transducers (DETs), this study delves into the dynamic behaviour exhibited by a one-dimensional asymmetric Duffing oscillator, which may exhibit either hardening or softening stiffness. The investigation utilizes hyperelastic and visco-hyperelastic constitutive models to capture the nonlinear stiffness properties of dielectric elastomers within compliant transducers. The canonical Duffing equation is employed to articulate the oscillator’s motion, complemented by both asymptotic and quantitative analyses of unforced vibrations and harmonic excitation. Our results reveal that the distinctive stiffness characteristics observed in DETs arise from the complex interplay among hyperelasticity, rheology and applied electric fields. Contrary to conventional oscillators, Maxwell stress substantially modifies the nonlinear dynamics of DETs, an aspect that has been quantitatively dissected in this study. Furthermore, the influence of additional system parameters on forced oscillation has also been meticulously evaluated.

Two-dimensional Coupled Asymmetric van der Pol Oscillator

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two dimensional coupled asymmetric van der Pol oscillator were obtained by Ms-DTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences.

An asymmetric model two-dimensional oscillator

We present a novel 2D oscillator with an asymmetric design and investigate its stable vibration utilizing the Ms-DTM method. Initially, we obtain the equations of motion for the proposed system. Subsequently, by employing Taylor expansion of [Formula: see text] and [Formula: see text], the derived nonlinear equations are transformed into linear ones, which we solve analytically using the eigenvalues-eigenfunctions technique. Additionally, we solve the obtained nonlinear system using the Ms-DTM method. Lastly, we examine the stability of the nonlinear system by visualizing the closed nature of the phase portrait.

Attractor selection in nonlinear oscillators by temporary dual-frequency driving

AbstractThis paper presents a control technique capable of driving a harmonically driven nonlinear system between two distinct periodic orbits. A vital component of the method is a temporary dual-frequency driving with tunable driving amplitudes. Theoretical considerations revealed two necessary conditions: one for the frequency ratio of the dual-frequency driving and another one for torsion numbers of the two orbits connected by bifurcation curves in the extended dual-frequency driving parameter space. Although the initial and the final states of the control strategy are single-frequency driven systems with distinct parameter sets (frequencies and driving amplitudes), control of multistability is also possible via additional parameter tuning. The technique is demonstrated on the symmetric Duffing oscillator and the asymmetric Toda oscillator.

Nonreciprocity of energy transfer in a nonlinear asymmetric oscillator system with various vibration states

Nonreciprocity of energy transfer in a nonlinear asymmetric oscillator system with various vibration states

Exact solution and coherent states of an asymmetric oscillator with position-dependent mass

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.

More on Superintegrable Models on Spaces of Constant Curvature

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalised Kepler potential. The angular components, on the contrary, are given implicitly by a transcendental, in general, equation. In the present note, devoted to the previously less studied models with the radial potential of the generalised Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator.

Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance

We deal with the following system of coupled asymmetric oscillators x¨1+a1x1+-b1x1-+ϕ1(x2)=p1(t)x¨2+a2x2+-b2x2-+ϕ2(x1)=p2(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\ddot{x}_1+a_1x_1^+-b_1x^-_1+\\phi _1(x_2)=p_1(t) \\\\ \\ddot{x}_2+a_2\\,x_2^+-b_2\\,x^-_2+\\phi _2(x_1)=p_2(t), \\end{array} \\right. \\end{aligned}$$\\end{document}where ϕi:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi _i: \\mathbb {R} \\rightarrow \\mathbb {R}$$\\end{document} is locally Lipschitz continuous and bounded, pi:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_i: \\mathbb {R} \\rightarrow \\mathbb {R}$$\\end{document} is continuous and 2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\pi $$\\end{document}-periodic and the positive real numbers ai,bi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_i, b_i$$\\end{document} satisfy 1ai+1bi=2n,forsomen∈N.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\dfrac{1}{\\sqrt{a_i}}+\\dfrac{1}{\\sqrt{b_i}}=\\dfrac{2}{n}, \\quad \ ext{ for } \ ext{ some } n \\in \\mathbb {N}. \\end{aligned}$$\\end{document}We define a suitable function L:T2→R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L: \\mathbb {T}^2 \\rightarrow \\mathbb {R}^2$$\\end{document}, appearing as the higher-dimensional generalization of the well known resonance function used in the scalar setting, and we show how unbounded solutions to the system can be constructed whenever L has zeros with a special structure. The proof relies on a careful investigation of the dynamics of the associated (four-dimensional) Poincaré map, in action-angle coordinates.

The effect of fractional damping and time-delayed feedback on the stochastic resonance of asymmetric SD oscillator

This paper proposes the stiffness nonlinearities and asymmetric SD (smooth and discontinuous) oscillator under time-delayed feedback control with the fractional derivative damping. With the effect of displacement and velocity feedback, the oscillator can be adjusted to the desired vibration state, and then the stochastic resonance (SR) is achieved. This article discusses the contribution of various system parameters and time-delayed feedback to SR, especially which induced by fractional order. It should be noted that this paper provides effective guidance for fault diagnosis and weak signal detection, energy harvesting, vibration isolation and vibration reduction.

Discontinuous dynamics of an asymmetric 2-DOF friction oscillator with elastic and rigid impacts

• An asymmetric 2 DOF oscillator with elastic and rigid impacts is investigated. • Negative feedback forces and nonlinear spring of cubic term are considered. • Switching conditions of motion at separation boundaries are explored. • Several typical motions are simulated numerically by MATLAB software. In this paper, by using flow switching theory of discontinuous dynamical systems , the discontinuous dynamic behavior of a 2-DOF (two-degree-of-freedom) friction oscillator with elastic impact and rigid impact on different sides is investigated. For the 2-DOF friction oscillator, when the direction of object’s velocity changes, the negative feedback works and the inequality of maximum static friction force and kinetic friction force leads to the existence of flow barriers. In addition, the elastic impact and rigid impact can change motion states of object, resulting in the discontinuity of this 2-DOF system. Considering the multiple motion states of object and better analyzing the equation of object’s motion, the phase space of each object is divided into different motion domains and boundaries in relative and absolute coordinates by means of the discontinuities resulted from the friction and impact. Moreover, because the elastic impact exists in a time period, there will be a variety of motion behaviors coexisting, such as elastic and rigid impacts occur simultaneously. With that in mind, in absolute coordinate system, the phase space of the system is divided in six cases according to whether there are stick motion and stuck motion. Based on flow switching theory, the switching conditions at discontinuous boundaries for all possible motions are given by using G-functions, and it should be noted that the vanishing conditions of the sliding motion become more complicated due to the existence of flow barriers at speed boundaries. Through defining the switching sets on separation boundaries, the mapping structures are further introduced to demonstrate the global dynamic behavior and periodic motions. For a better understanding of the object’s motion and the switching criteria at separation boundaries, the numerical simulation method is used to demonstrate several typical motions such as passable motion, sliding motion, grazing motion, stick motion (elastic impact), rigid impact and periodic motions etc. The results obtained have reference value for the optimization design and noise control of mechanical systems with gap couplings and negative feedbacks.

Ghost stochastic resonance in an asymmetric Duffing oscillator

We report the occurrence of ghost stochastic resonance (GSR) for a Brownian particle in three types of asymmetric potentials. The asymmetries are controlled by the asymmetry parameter and introduced in the potential by varying the left well. These variations are namely (i) the depth alone, (ii) the width alone and (iii) both depth and width. The properties of GSR in the asymmetric systems are different from the symmetric systems. It is demonstrated that the resonance observed at the ghost frequency is due to the existence of this frequency during the dynamics of the particle. We also examine numerically the effect of the asymmetry parameter on GSR phenomenon. It is found that the asymmetry parameter has a significant influence on the GSR and allows a precise control of the appearance of GSR.

A Generalized Accurate Approximate Solution to the Symmetric and Asymmetric Oscillators with Polynomial Restoring Forces

A Generalized Accurate Approximate Solution to the Symmetric and Asymmetric Oscillators with Polynomial Restoring Forces

Coupled liquid crystalline oscillators in Huygens' synchrony.

In the flourishing field of soft robotics, strategies to embody communication and collective motion are scarce. Here we report the synchronized oscillations of thin plastic actuators by an approach reminiscent of the synchronized motion of pendula and metronomes. Two liquid crystalline network oscillators fueled by light influence the movement of one another and display synchronized oscillations in-phase and anti-phase in a steady state. By observing entrainment between the asymmetric oscillators we demonstrate the existence of coupling between the two actuators. We qualitatively explain the origin of the synchronized motion using a theoretical model and numerical simulations, which suggest that the motion can be tuned by the mechanical properties of the coupling joint. We thus anticipate that the complex synchronization phenomena usually observed in rigid systems can also exist in soft polymeric materials. This enables the use of new stimuli, featuring a first example of collective motion by photo-actuation.

An Optimized-Parameter Spectral Clustering Approach to Coherent Structure Detection in Geophysical Flows

In Lagrangian dynamics, the detection of coherent clusters can help understand the organization of transport by identifying regions with coherent trajectory patterns. Many clustering algorithms, however, rely on user-input parameters, requiring a priori knowledge about the flow and making the outcome subjective. Building on the conventional spectral clustering method of Hadjighasem et al. (2016), a new optimized-parameter spectral clustering approach is developed that automatically identifies optimal parameters within pre-defined ranges. A noise-based metric for quantifying the coherence of the resulting coherent clusters is also introduced. The optimized-parameter spectral clustering is applied to two benchmark analytical flows, the Bickley Jet and the asymmetric Duffing oscillator, and to a realistic, numerically generated oceanic coastal flow. In the latter case, the identified model-based clusters are tested using observed trajectories of real drifters. In all examples, our approach succeeded in performing the partition of the domain into coherent clusters with minimal inter-cluster similarity and maximum intra-cluster similarity. For the coastal flow, the resulting coherent clusters are qualitatively similar over the same phase of the tide on different days and even different years, whereas coherent clusters for the opposite tidal phase are qualitatively different.

Structural and functional synthesis of the continuous wave NQR temperature sensor with increased conversion linearity

The paper describes development of the detailed structure and circuit diagrams of the continuous wave NQR temperature sensor with increased conversion linearity. It is experimentally established that at amplitude modulation of 40% and change of input voltage in the range of 20–1000 ​mV, the circuit of a symmetric marginal oscillator with a linear active demodulator provides better linearity of transfer characteristic than the circuits of asymmetric marginal oscillators with JFET or diode detectors. As a thermometric substance of the proposed NQR sensor, copper oxide Cu2O was used, which is characterized by a strong temperature dependence of the resonance frequency of 63Cu NQR. In contrast to 35Cl NQR in KClO3, for cuprous oxide the temperature dependence of 63Сu NQR frequency in the frequency range 26.621–25.658 ​MHz is linear in the temperature range 100–390 ​K. It is experimentally confirmed that the use of a low mass sample (less than 200 ​mg) as a thermometric substance of the proposed NQR sensor is quite sufficient for successfully observation of the resonance line at the SNR equal to 9.1 ​dB.

Two New Asymmetric Boolean Chaos Oscillators with No Dependence on Incommensurate Time-Delays and Their Circuit Implementation

This manuscript introduces two new chaotic oscillators based on autonomous Boolean networks (ABN), preserving asymmetrical logic functions. That means that the ABNs require a combination of XOR-XNOR logic functions. We demonstrate analytically that the two ABNs do not have fixed points, and therefore, can evolve to Boolean chaos. Using the Lyapunov exponent’s method, we also prove the chaotic behavior, generated by the proposed chaotic oscillators, is insensitive to incommensurate time-delays paths. As a result, they can be implemented using distinct electronic circuits. More specifically, logic-gates–, GAL–, and FPGA–based implementations verify the theoretical findings. An integrated circuit using a CMOS 180nm fabrication technology is also presented to get a compact chaos oscillator with relatively high-frequency. Dynamical behaviors of those implementations are analyzed using time-series, time-lag embedded attractors, frequency spectra, Poincaré maps, and Lyapunov exponents.

Local control of an array of the globally coupled oscillators

Two control methods for stabilization of the steady states of the globally coupled second-order oscillators via local feedback are described. The first technique is based on the proportional feedback with a constant adjustable reference. The second technique is an adaptive one, employing the first-order stable filter. Three different types of oscillators have been considered: (1) strongly asymmetric version of the FitzHugh–Nagumo oscillators, (2) the classical FitzHugh–Nagumo oscillators, and (3) the van der Pol oscillators. Numerical simulations have been performed for all three types of oscillators. Detailed mathematical analysis and hardware experiments have been carried out with the asymmetric FitzHugh–Nagumo oscillators.

Dynamics of a Modified Leslie-Gower Model with Crowley-Martin Functional Response and Prey Harvesting

This paper considers a mariner under PDμ controller and analyses the effects of controller parameters on the yaw rate by using the Nomoto model. The Nomoto model describing the time evolution of the yaw rate of the steering dynamics of a mariner is reduced to an asymmetric Duffing oscillator with fractional order derivative. Under the approximation of calm water, the steady behavior of the mariner shows an “imperfect” supercritical pitchfork bifurcation. Region of safe behavior is identified and strategy to reduce the yaw rate by an appropriated selection of controller parameters are discussed. The frequency analysis of the mariner shows the prominence of hysteresis is reduced for small order of the fractional derivative as well as the amplitude of the yaw rate. Evidence of chaotic response is illustrated using robust chaotic indicators such as the Lyapunov exponent and the fast Fourier transform.