Damped Nonlinear Oscillator Research Articles

Vibrational resonance in a higher-order nonlinear damped oscillator with rough potential

We examine the vibrational resonance (VR) of particles moving in a strongly nonlinear damped medium with a harmonically perturbed potential consisting of a background smooth triple-well potential superimposed by a fast oscillating periodic function and subjected to weak and high-frequency (HF) driving forces. The combined effects of the nonlinear damping inhomogeneity and roughness induced by the harmonic perturbation on the phenomenon of VR were theoretically and numerically analysed. It was found that damping inhomogeneity contributed significantly to the enhancement of resonant states, while potential roughness can be optimised by the HF signal to assist resonance enhancement. Furthermore, the traditional smooth VR shapes occurring in the absence of roughness experienced significant distortion occasioned by potential roughness manifesting as spikes that could ultimately be optimised by large amplitudes of the fast signal to energetically facilitate the potential barrier crossing process, thereby enabling VR enhancement.

Transmission time delays organize the brain network synchronization.

The timing of activity across brain regions can be described by its phases for oscillatory processes, and is of crucial importance for brain functioning. The structure of the brain constrains its dynamics through the delays due to propagation and the strengths of the white matter tracts. We use self-sustained delay-coupled, non-isochronous, nonlinearly damped and chaotic oscillators to study how spatio-temporal organization of the brain governs phase lags between the coherent activity of its regions. In silico results for the brain network model demonstrate a robust switching from in- to anti-phase synchronization by increasing the frequency, with a consistent lagging of the stronger connected regions. Relative phases are well predicted by an earlier analysis of Kuramoto oscillators, confirming the spatial heterogeneity of time delays as a crucial mechanism in shaping the functional brain architecture. Increased frequency and coupling are also shown to distort the oscillators by decreasing their amplitude, and stronger regions have lower, but more synchronized activity. These results indicate specific features in the phase relationships within the brain that need to hold for a wide range of local oscillatory dynamics, given that the time delays of the connectome are proportional to the lengths of the structural pathways.This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

Practical examples of ISS systems

In this work we collect several practical examples that illustrate ISS-like properties of nonlinear finite dimensional systems. Depending on the nonlinearities involved we show under which conditions ISS, strong iISS, iISS or LISS property holds in these examples. Strict Lyapunov functions for these systems are provided. Supply rates or gains are calculated explicitly in most cases. Water tank, nonlinear damped elastic oscillator and nonlinear mathematical pendulum are considered as illustrative examples.

Application of AG method and its improvement to nonlinear damped oscillators

In this paper, a new and innovative semi analytical technique, namely Akbari-Ganji’s method (AGM), is employed for solving three nonlinear damped oscillatory systems. Applying this method to nonlinear problems is very simple because in solving process only a trial solution, the main differential equation and its derivatives are required. The analytical solutions obtained by the AGM are utilized to study the impact of amplitude on nonlinear frequency and damping ratio. It is found that the AGM leads to acceptable results for the problems considered in this paper. Also, in order to obtain a more accurate solution, instead of using a trial solution with higher-order terms which may result in complicated and time consuming mathematical calculations, the solution obtained by AGM is improved via variational iteration method (VIM). The usefulness and effectiveness of the approach is demonstrated through comparison of the obtained results with those achieved by the numerical method. Hence, the AGM can be applied to nonlinear problems consisting of significant nonlinear damping terms and, if necessary, can be easily improved.

Superposition principle for the simultaneous optimization of collective responses.

We study the dynamics of sets of independent systems, all of which are coupled to the same time-dependent external force. Using optimal control theory, we compute the most efficient temporal pulse shape for this force that can maximize simultaneously the collective response of these systems. This response can be a weighted sum of all amplitudes at the final interaction time. Remarkably, it turns out that for certain systems this optimal force for the collective response can be related to the individual forces that would optimize each system separately. We illustrate this superposition principle for the simultaneous optimization of collective responses with numerical and also analytical solutions for sets of damped linear and nonlinear oscillators. We also apply this principle to predict the optimal temporal profile of a laser pulse that can maximize the final macroscopic polarization (total dipole moment) of a set of quantum mechanical two-level atoms.

Synchronisation vs. resonance: Isolated resonances in damped nonlinear oscillators

We describe differences between synchronisation and resonance, and analyse different types of nonlinear resonances in a weakly damped Duffing oscillator using bifurcation theory techniques. In addition to previously reported (i) odd subharmonic resonances found on the primary branch of symmetric periodic solutions with the forcing frequency and (ii) even subharmonic resonances due to symmetry-broken periodic solutions that bifurcate off the primary branch and also oscillate at the forcing frequency, we uncover (iii) novel resonance type due to isolas of periodic solutions that are not connected to the primary branch. These occur between odd and even resonances, oscillate at a fraction of the forcing frequency, and give rise to a complicated resonance ‘curve’ with disconnected elements and high degree of multistability.We use bifurcation continuation to compute resonance tongues in the plane of the forcing frequency vs. the forcing amplitude for different but fixed values of the damping rate. Our analysis shows that identified here isolated resonances explain the intriguing “intermingled tongues” that were observed for weak damping and misinterpreted as (synchronisation) Arnold tongues in Paar and Pavin (1998). What is more, isolated resonances link “intermingled tongues” to a seemingly unrelated phenomenon of “bifurcation superstructure” described for moderate damping in Parlitz and Lauterborn (1985).

Externally excited undamped and damped linear and nonlinear oscillators: Exact solutions and tuning to a desired exact form of the response

This work presents a methodology on how to use exact closed-form solutions for the response of free undamped linear and nonlinear oscillators to design the external excitation of undamped or damped nonlinear oscillators to get such steady-state response. A variety of examples, including Duffing-type oscillators and purely nonlinear oscillators, are given to illustrate this methodology.

Analytical estimates of the effect of amplitude modulated signal in nonlinearly damped Duffing-vander Pol oscillator

The effect of amplitude modulated (AM) signal on horseshoe chaos is studied both analytically and numerically in nonlinearly damped Duffing-vander Pol (DVP) oscillator. We assume the nonlinear damping term is proportional to the power of velocity (x˙) in the form |x˙|p−1. Using the Melnikov analytical method, the analytical threshold condition for the prediction of onset of horseshoe chaos is obtained. Melnikov threshold curves are drawn in different external parameters space. Our results show that as the damping exponent p increases from a small value, the threshold value for the onset of horseshoe is decreased. The analytical results are verified by numerical simulations using the computation of stable and unstable manifolds of saddle, measuring the time elapsed between two successive transverse intersections, bifurcation diagram, maximal Lyapunov exponent, phase portrait and Poincare´ map.

Uniform magnetization dynamics of a submicron ferromagnetic disk driven by the spin–orbit coupled spin torque

A simple model of magnetization dynamics in a ferromagnet/doped semiconductor hybrid structure with Rashba spin–orbit interaction (SOI), driven by an applied pulse of the electric field, is proposed. The electric current excited by the applied field is spin-polarized, due to the SOI and therefore it induces the magnetization rotation in the ferromagnetic layer via s-d exchange coupling. Magnetization dynamics dependence on the electric pulse shape and magnitude is analyzed for realistic values of the parameters. We show that it is similar to the dynamics of a damped nonlinear oscillator with the time-dependent frequency proportional to the square root of the applied electric field. The magnetization switching properties of an elliptic magnetic element are examined as a function of the applied field magnitude and direction.

Research on the reliability of friction system under combined additive and multiplicative random excitations

In this paper, the reliability of a non-linearly damped friction oscillator under combined additive and multiplicative Gaussian white noise excitations is investigated. The stochastic averaging method, which is usually applied to the research of smooth system, has been extended to the study of the reliability of non-smooth friction system. The results indicate that the reliability of friction system can be improved by Coulomb friction and reduced by random excitations. In particular, the effect of the external random excitation on the reliability is larger than the effect of the parametric random excitation. The validity of the analytical results is verified by the numerical results.

Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation

Nonlinear oscillators are ubiquitous in sciences, being able to model the behavior of complex nonlinear phenomena, as well as in engineering, being able to generate repeating (i.e., periodic) or non-repeating (i.e., chaotic) reference signals. The state of the classical oscillators known from the literature evolves in the space Rn, typically with n=1 (e.g., the famous van der Pol vacuum-tube model), n=2 (e.g., the FitzHugh–Nagumo model of spiking neurons) or n=3 (e.g., the Lorenz simplified model of turbulence). The aim of the current paper is to present a general scheme for the numerical differential-geometry-based integration of a general second-order, nonlinear oscillator model on Riemannian manifolds and to present several instances of such model on manifolds of interest in sciences and engineering, such as the Stiefel manifold and the space of symmetric, positive-definite matrices.

High-frequency vibration energy harvesting from repeated impulsive forcing utilizing intentional dynamic instability caused by strong nonlinearity

The work in this study explores the excitation of high-frequency dynamic instabilities to enhance the performance of a strongly nonlinear vibration-based energy harvesting system subject to repeated impulsive excitations. These high-fraequency instabilities arise from transient resonance captures (TRCs) in the damped dynamics of the system, leading to large-amplitude oscillations in the mechanical system. Under proper forcing conditions, these high-frequency instabilities can be sustained. The primary system is composed of a grounded, weakly damped linear oscillator, which is directly subjected to impulsive forcing. A light-weight, damped nonlinear oscillator (nonlinear energy sink, NES) is coupled to the primary system using electromechanical coupling elements and strongly nonlinear stiffness elements. The essential (nonlinearizable) stiffness nonlinearity arises from geometric and kinematic effects resulting from the traverse deflection of a piano wire coupling the two oscillators. The electromechanical coupling is composed of a neodymium magnet and inductance coil, which harvests the energy in the mechanical system and transfers it to the electrical system which, in this present case, is composed of a simple resistive element. The energy dissipated in the circuit is inferred as a measure of energy harvesting capability. The large-amplitude TRCs result in strong, nearly irreversible energy transfer from the primary system to the NES, where the harvesting elements work to convert the mechanical energy to electrical energy. The primary goal of this work is to numerically and experimentally demonstrate the efficacy of inducing sustained high-frequency dynamic instability in a system of mechanical oscillators to achieve enhanced vibration energy harvesting performance. This work is a continuation of a companion paper (Remick K, Quinn D, McFarland D, et al. (2015) Journal of Sound and Vibration Final Publication) where vibration energy harvesting of the same system subject to single impulsive excitation is studied.

Statistical measures approximations for the Gaussian part of the stochastic nonlinear damped Duffing oscillator solution process under the application of Wiener Hermite expansion linked by the multi-step differential transformed method

Abstract In this paper, the stochastic Wiener Hermite expansion (WHE) is used to find the statistical measures (mean and variance) of the first order stochastic approximation (Gaussian part) of the stochastic solution processes related to the nonlinear damped Duffing oscillator model which is excited randomly by white noise process. Under the application of WHE, a deterministic model is generated to simulate the statistical measures. In next stages, smi-analytical treatments are performed under applying multi-step differential transformed method (Ms-DTM) and some cases study are illustrated related to the statistical properties using Mathematica10 software.

Linear and nonlinear resonance features of an erbium-doped fibre ring laser under cavity-loss modulation

The continuous-wave output of a single-mode erbium-doped fibre ring laser when subjected to cavity-loss modulation is found to exhibit linear as well as nonlinear resonances. At sufficiently low driving amplitude, the system resembles a linear damped oscillator. At higher amplitudes, the dynamical study of these resonances shows that the behaviour of the system exhibits features of a nonlinear damped oscillator under harmonic modulation. These nonlinear dynamical features, including harmonic and subharmonic resonances, have been studied experimentally and analysed with the help of a simple time-domain and frequency-domain information obtained from the output of the laser. All the studies are restricted to the modulation frequency lying in a regime near the relaxation oscillation frequency.

On the local analytic integrability for a generalized damped nonlinear oscillator equation

We consider the general damped nonlinear oscillator x′ = y, y′ = −a1x − a2xn+1 − a3x2n+1 − y(a4 + a5xn), where ai are real parameters for i = 1, 2, 3, 4, 5 and n is an integer. We note that these class of Liénard differential systems include a large number of physically important nonlinear oscillators. We provide a complete and rigorous characterization of these systems that admit a local analytic first integral in a neighborhood of the origin.

The Effect of Slow Flow Dynamics on the Oscillations of a Singular Damped System with an Essentially Nonlinear Attachment

We study a three degree of freedom autonomous system with damping, composed of two linear coupled oscillators with an essentially nonlinear lightweight attachment. In particular, we are interested in strongly nonlinear interactions between the linear oscillators and the essentially nonlinear attachment. First, we reduce our system to a non-autonomous second order nonlinear damped oscillator. Then, we introduce a slow-fast partition of the dynamics and average out the main frequency components in order to obtain a reduced system that is studied through the Slow Invariant Manifold (SIM) approach. Depending on the pa- rameters of the system we find different interesting nonlinear dynamical phenomena. With the help of the SIM approach we can study how the parameters of the original problem influence the asymptotic behavior of the orbits of the system. This is accomplished with the application of Tikhonov’s theorem. We classify the different cases of the dynamics according to the values of the parameters and the theoretically predicted asymptotic behavior of the orbits. Interesting phenomena are reported such as orbit capture, relaxation oscillations and complex structure of the basins of attraction.

Relaxation function for the non-Debye relaxation spectra description

This study presents the new relaxation function describing the non-Debye relaxation phenomena. The relaxation function is based on a new theoretical model of the relaxation polarization. The non-Debye relaxation is explained with the model of nonlinear damped oscillator. It is shown that the relaxation function describes the relaxation spectra of the Davidson-Cole and Havriliak–Negami types as well as spectra with the left-skewed loss peak.

Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations

A stochastic averaging method for quasi-integrable and resonant Hamiltonian systems subject to combined Gaussian and Poisson white noise excitations is proposed. The case of resonance with α resonant relations is considered. An (n + α)-dimensional averaged Generalized Fokker–Plank–Kolmogorov (GFPK) equation for the transition probability density of n action variables and α combinations of phase angles is derived from the stochastic integrodifferential equations (SIDEs) of original quasi-integrable and resonant Hamiltonian systems by using the jump-diffusion chain rule. The reduced GFPK equation is solved by using finite difference method and the successive over relaxation method to obtain the stationary probability density of the system. An example of two nonlinearly damped oscillators under combined Gaussian and Poisson white noise excitations is given to illustrate the proposed method. The good agreement between the analytical results and those from digital simulation shows the validity of the proposed method.

Analytical formulas for the energy, velocity and displacement decays of purely nonlinear damped oscillators

Accurate formulas for obtaining the energy, velocity and displacement decay envelopes in purely nonlinear damped oscillators are presented here. These purely nonlinear oscillators have no linear stiffness components, and the new formulas derived in this paper are found to be highly accurate in determining the decay envelopes, regardless of the system’s physical parameters and its initial condition. The mechanism of these new formulas for obtaining the decay envelopes of the nonlinear oscillators is found to be exactly the same as the mechanism of the well-known amplitude decay formulas used for the linear oscillator. Finding such formulas is significant to many scientific applications of these nonlinear oscillators, such as the passive targeted energy transfer through the stiffness-based nonlinear energy sinks (NESs). In these types of NESs, these formulas can be easily applied for system identification and dynamic analysis. In addition, they are expected to have a significant application in different scientific fields for such oscillators.

Keldysh approach for nonequilibrium phase transitions in quantum optics: Beyond the Dicke model in optical cavities

We investigate nonequilibrium phase transitions for driven atomic ensembles interacting with a cavity mode and coupled to a Markovian dissipative bath. In the thermodynamic limit and at low frequencies, we show that the distribution function of the photonic mode is thermal, with an effective temperature set by the atom-photon interaction strength. This behavior characterizes the static and dynamic critical exponents of the associated superradiance transition. Motivated by these considerations, we develop a general Keldysh path-integral approach that allows us to study physically relevant nonlinearities beyond the idealized Dicke model. Using standard diagrammatic techniques, we take into account the leading-order corrections due to the finite number $N$ of atoms. For finite $N$, the photon mode behaves as a damped classical nonlinear oscillator at finite temperature. For the atoms, we propose a Dicke action that can be solved for any $N$ and correctly captures the atoms' depolarization due to dissipative dephasing.