Periodically forced delay limit cycle oscillator

This paper investigates the dynamics of a delay limit cycle oscillator under periodic external forcing. The system exhibits quasiperiodic motion outside of a resonance region where it has periodic motion at the frequency of the forcer for strong enough forcing. By perturbation methods and bifurcation theory, we show that this resonance region is asymmetric in the frequency detuning, and that there are regions where stable periodic and quasiperiodic motions coexist.     

Effect of parameter mismatch and time delay interaction on density-induced amplitude death in coupled nonlinear oscillators

Oscillations in the coupled systems can be suppressed by varying density of mean field. The presence of parameter mismatch or time delay interaction in such systems enhances the amplitude death (AD) region in parameter space. This behavior of stability of steady state is analyzed by analytical as well as numerical studies of specific cases of limit-cycle and Rössler oscillators. Experimental evidence of AD is also shown, using an electronic version of the Chua’s oscillator.     

Nonlinear dynamics and chaos in coupled shape memory oscillators

Shape memory and pseudoelastic effects are thermomechanical phenomena associated with martensitic phase transformations, presented by shape memory alloys. This contribution concerns with the dynamical response of coupled shape memory oscillators. Equations of motion are formulated by assuming a polynomial constitutive model to describe the restitution force of the oscillators and, since they are associated with a five-dimensional system, the analysis is performed by splitting the state space in subspaces. Free and forced vibrations are analyzed showing different kinds of responses. Periodic, quasi-periodic, chaos and hyperchaos are all possible in this system. Numerical investigations show interesting and complex behaviors. Dynamical jumps in free vibration and amplitude variation when temperature characteristics are changed are some examples. This article also shown some characteristics related to chaos–hyperchaos transition.     

Stable amplitude chimera states and chimera death in repulsively coupled chaotic oscillators

Amplitude chimera states, representing a spontaneous symmetry breaking of a population of coupled identical oscillators into two distinct clusters with one oscillating in spatial coherent amplitude, while the other displaying oscillations in a spatially incoherent manner, have been observed as a kind of transient dynamics in the process of transition to the in-phase synchronization in coupled limit-cycle oscillators. Here, we obtain a kind of stable amplitude chimera state in the chaotic regime of a system of repulsively coupled Lorenz oscillators. With the increment of the coupling strength, the coupled oscillators transit from spatiotemporal chaos to amplitude chimera states then to coherent oscillation death or chimera death states. Moreover, the number of clusters in amplitude chimera patterns has a power-law dependence on the number of coupled neighbors. The amplitude chimera and the chimera death states coexist at certain coupling strength. Moreover, the amplitude chimera and the amplitude death patterns are related to the initial condition for given coupling strength. Our findings of amplitude chimera states and chimera death states in coupled chaotic system may enrich the knowledge of the symmetry-breaking-induced pattern formation.     

Fractional derivative reconstruction of forced oscillators

Fractional derivatives are applied in the reconstruction from a single observable of the dynamics of a Duffing oscillator and a two-well experiment. The fractional derivatives of time series data are obtained in the frequency domain. The derivative fraction is evaluated using the average mutual information between the observable and its fractional derivative. The ability of this reconstruction method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and fractal dimensions. The reconstruction is compared to the true phase space and the delay reconstruction in order to assess the reconstruction parameters and the quality of results.     

Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities

We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes—NNMs), as well as, asynchronous periodic motions (elliptic orbits—EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.     

Hybrid dynamics of two coupled oscillators that can impact a fixed stop

We consider two linearly coupled masses, where one mass can have inelastic impacts with a fixed, rigid stop. This leads to the study of a two degree of freedom, piecewise linear, frictionless, unforced, constrained mechanical system. The system is governed by three types of dynamics: coupled harmonic oscillation, simple harmonic motion and discrete rebounds. Energy is dissipated discontinuously in discrete amounts, through impacts with the stop. We prove the existence of a non-zero measure set of orbits that lead to infinite impacts with the stop in a finite time. We show how to modify the mathematical model so that forward existence and uniqueness of solutions for all time is guaranteed. Existence of hybrid periodic orbits is shown. A geometrical interpretation of the dynamics based on action coordinates is used to visualize numerical simulation results for the asymptotic dynamics.     

Nonlinear dynamics of two harmonic oscillators coupled by Rayleigh type self-exciting force

The present paper reports some interesting phenomena observed in the nonlinear dynamics of two self-excitedly coupled harmonic oscillators. The system under consideration consists of two mechanical oscillators coupled by the Rayleigh type self-exciting force. Both autonomous and nonautonomous cases for weakly coupled systems are analyzed. When the natural frequencies of the two oscillators are close to each other, only one mode of oscillation exists. As two modes of oscillations get locked to a single mode, the system is said to be in a mode-locked condition. Under a mode-locked condition, the oscillators can oscillate with only a single frequency. However, when two oscillators are sufficiently detuned, the mode-locking condition does not persist and two distinct modes of oscillations emerge. Under these circumstances, particularly when detuning is large, one of the oscillators, depending on the initial conditions, oscillates with much larger amplitude as compared to the other oscillator, and hence mode localization is observed. When one of the oscillators is subject to a harmonic excitation, at two different frequencies, termed here as the decoupling frequencies, the coupling between the oscillators is almost lost, resulting in almost zero response of the unexcited oscillator. Analytical and numerical results are presented to analyze the above mentioned phenomena. Some potential applications of the aforesaid phenomena are also discussed.     

Aeroelastic inverse: Estimation of aerodynamic loads during large amplitude limit cycle oscillations

This paper presents an algorithm to compute the aerodynamic forces and moments of an aeroelastic wing undergoing large amplitude heave and pitch limit cycle oscillations. The technique is based on inverting the equations of motion to solve for the lift and moment experienced by the wing. Bayesian inferencing is used to estimate the structural parameters of the system and generate credible intervals on the lift and moment calculations. The inversion technique is applied to study the affect of mass coupling on limit cycle oscillation amplitude. Examining the force, power, and energy of the system, the reasons for amplitude growth with wind speed can be determined. The results demonstrate that the influence of mass coupling on the pitch–heave difference is the driving factor in amplitude variation. The pitch–heave phase difference not only controls how much aerodynamic energy is transferred into the system but also how the aerodynamic energy is distributed between the degrees of freedom.     

Bifurcation analysis and limit cycle oscillation amplitude prediction methods applied to the aeroelastic galloping problem

A global stability and bifurcation analysis of the transverse galloping of a square section beam in a normal steady flow has been implemented. The model is an ordinary differential equation with polynomial damping nonlinearity. Six methods are used to predict bifurcation, the amplitudes and periods of the ensuing Limit Cycle Oscillations: (i) Cell Mapping, (ii) Harmonic Balance, (iii) Higher Order Harmonic Balance, (iv) Centre Manifold linearization, (v) Normal Form and (vi) numerical continuation. The resulting stability predictions are compared with each other and with results obtained from numerical integration. The advantages and disadvantages of each technique are discussed. It is shown that, despite the simplicity of the system, only two of the methods succeed in predicting its full response spectrum. These are Higher Order Harmonic Balance and numerical continuation.     

On the transient response of forced nonlinear oscillators

We consider the transient response of a prototypical nonlinear oscillator modeled by the Duffing equation subjected to near resonant harmonic excitation. Of interest here is the overshoot problem that arises when the system is undergoing free motion and is suddenly subjected to harmonic excitation with a near resonant frequency, which leads to a beating type of transient response during the transition to steady state. In some design applications, it is valuable to know the peak value of this response and the manner in which it depends on system parameters, input parameters, and initial conditions. This nonlinear overshoot problem is addressed by considering the well-known averaged equations that describe the slowly varying amplitude and phase for both transient and steady state responses. For the undamped system, we show how the problem can be reduced to a single parameter χ that combines the frequency detuning, force amplitude, and strength of nonlinearity. We derive an explicit expression for the overshoot in terms of χ, describe how one can estimate corrections for light damping, and verify the results by simulations. For zero damping, the overshoot approximation is given by a root of a quartic equation that depends solely on χ, yielding a simple bound for the overshoot of lightly damped systems.     

Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments

We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency–energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers. PACS numbers: 05.45.Xt, 02.30.Hq     

Effects of gradient coupling on amplitude death in nonidentical oscillators

The effects of the gradient coupling on the amplitude death in an array and a ring of diffusively coupled nonidentical oscillators are explored, respectively. The gradient coupling plays a significant role on the amplitude death dynamics, however, it is strongly related to the boundary conditions of the coupled system. With the increment of the gradient coupling, the domain of the amplitude death is monotonically enlarged in an array of coupled oscillators. However, for a ring of coupled oscillators, it is firstly enlarged and then decreased as the gradient coupling increases. The domain of the amplitude death in parameter space is analytically predicted for a small number of gradiently coupled oscillators.     

Collective dynamics induced by diversity taken from two-point distribution in globally coupled chaotic oscillators

The effect of parameter mismatch (diversity) taken from two-point distribution is studied numerically and theoretically in globally coupled Rössler chaotic systems. Two cases including mixed populations consisting of elements with different timescales and attractors are considered. In these two cases, the probability p of two-point distribution, which acts as an asymmetrical coupling on the system, plays a crucial role in determining the evolution of systems, and the rich dynamical phenomena are observed, especially for amplitude death (AD). The relationships between various dynamics are also discussed.     

Random excitation of nonlinear coupled oscillators

This paper presents the experimental results of random excitation of a nonlinear two-degree-of-freedom system in the neighborhood of internal resonance. The random signals of the excitation and response coordinates are processed to estimate the mean squares, autocorrelation functions, power spectral densities, and probability density functions. The results are qualitatively compared with those predicted by the Fokker-Planck equation together with a non-Gaussian closure scheme. The effects of system damping ratios, nonlinear coupling parameter, internal detuning ratio, and excitation spectral density level are considered in both studies except the effect of damping ratios is not considered in the experimental investigation. Both studies reveal similar dynamic features such as autoparametric absorber effect and stochastic instability of the coupled system. The experimental results show that the autocorrelation function of the coupled system has the feature of ultra narrow band process and degenerates to a periodic one as the internal detuning departs from the exact internal resonance condition. The measured probability density functions of the response of the main system suggests that the Gaussian representation is sufticient as long as the excitation level is relatively low in the neighborhood of the system internal resonance condition.     

On the multi-scale analysis of strongly non-linear forced oscillators

We consider the resonant response of strongly non-linear oscillators of the form ü + 2ϵηu + mu + ϵƒ(u) = 2ϵpcosΩt, where ƒ(u) is an odd non-linearity, ϵ need not be small, and m = −1, 0, or + 1. Approximate solutions are obtained using a multiple-scale approach with two procedural steps which differ from the usual ones: (1) the detuning is introduced in the square of the excitation frequency Ω and as a deviation from the so called backbone curve and (2) a new expansion parameter α = α(ϵ) is defined, enabling accurate low order solutions to be obtained for the strongly non-linear case.     

Partially unstable attractors in networks of forced integrate-and-fire oscillators

The asymptotic attractors of a nonlinear dynamical system play a key role in the long-term physically observable behaviors of the system. The study of attractors and the search for distinct types of attractor have been a central task in nonlinear dynamics. In smooth dynamical systems, an attractor is often enclosed completely in its basin of attraction with a finite distance from the basin boundary. Recent works have uncovered that, in neuronal networks, unstable attractors with a remote basin can arise, where almost every point on the attractor is locally transversely repelling. Herewith we report our discovery of a class of attractors: partially unstable attractors, in pulse-coupled integrate-and-fire networks subject to a periodic forcing. The defining feature of such an attractor is that it can simultaneously possess locally stable and unstable sets, both of positive measure. Exploiting the structure of the key dynamical events in the network, we develop a symbolic analysis that can fully explain the emergence of the partially unstable attractors. To our knowledge, such exotic attractors have not been reported previously, and we expect them to arise commonly in biological networks whose dynamics are governed by pulse (or spike) generation.     

Periodic,almost periodic and chaotic motions of forced self-sustained oscillators

Steady motions of the Van der Pol oscillator and an oscillator with hysteresis are studied numerically in this paper. Some features of periodic, almost periodic and chaotic motions of forced self-sustained oscillators are investigated. This paper has been presented at the ICTAM XVI Lyngby.     

Parametric random excitation of nonlinear coupled oscillators

The stochastic bifurcation and response statistics of nonlinear modal interaction under parametric random excitation are studied analytically, numerically and experimentally. Two basic definitions of stochastic bifurcation are first introduced. These are bifurcation in distribution and bifurcation in moments. bifurcation in moments is examined for the case of a coupled oscillator subjected to parametric filtered white noise. The center frequency of the excitation is selected to be close to either twice the first mode or second mode natural frequencies or the sum of the two. The stochastic bifurcation in moments is predicted using the Fokker-Planck equation together with gaussian and non-Gaussian closures and numerically using the Monte Carlo simulation. When one mode is parametrically excited it transfers energy to the other mode due to nonlinear modal interaction. The Gaussian closure solution gives close results to those predicted numerically only in regions well remote from bifurcation points. However, bifurcation points predicted by the non-Gaussian closure are in good agreement with those estimated by numerical simulation. Depending on the excitation level, the probability density of the excited mode is strongly non-Gaussian and exhibits multi-maxima as predicted by Monte Carlo simulation. Experimental tests are carried out at relatively low excitation levels. In the neighborhood of stochastic bifurcation in mean square the measured results exhibit different regimes of response characteristics including zero motion and occasional small random motion regimes. These two regimes are characterized by the phenomenon of on-off intermittency. Both regimes overlap and thus it is difficult to locate experimentally the bifurcation point.