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.     

Metamorphoses of resonance curves in systems of coupled oscillators:The case of degenerate singular points

We study dynamics of two coupled periodically driven oscillators. The internal motion is separated off exactly to yield a nonlinear fourth-order equation describing inner dynamics. Periodic steady-state solutions of the fourth-order equation are determined within the Krylov–Bogoliubov–Mitropolsky approach and we compute the corresponding amplitude profiles.Metamorphoses of these amplitude curves induced by changes of control parameters as well as the corresponding changes of dynamics are studied within the framework of theory of differential properties of algebraic curves. The major finding is that there is a very rich dynamics in neighborhoods of degenerate singular points.     

Free vibration of two coupled nonlinear oscillators

An analytical investigation is carried out on the free vibration of a two degree of freedom weakly nonlinear oscillator. Namely, the method of multiple time scales is first applied in deriving modulation equations for a van der Pol oscillator coupled with a Duffing oscillator. For the case of non-resonant oscillations, these equations are in standard normal form of a codimension two (Hopf-Hopf) bifurcation, which permits a complete analysis to be performed. Three different types of asymptotic states-corresponding to trivial, periodic and quasiperiodic motions of the original system-are obtained and their stability is analyzed. Transitions between these different solutions are also identified and analyzed in terms of two appropriate parameters. Then, effects of a coupling, a detuning, a nonlinear stiffness and a damping parameter are investigated numerically in a systematic manner. The results are interpreted in terms of classical engineering terminology and are related to some relatively new findings in the area of nonlinear dynamical systems.     

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.     

Dynamics and synchronization analysis of coupled fractional-order nonlinear electromechanical systems

A fractional-order (FO) nonlinear model is used to describe an electromechanical system. We make capital out of the fact that for realistic modeling, the electric characteristics of a capacitor include a fractional-order time derivative. The dynamics and synchronization of coupled fractional-order nonlinear electromechanical systems are analyzed. Detailed attention is granted to the bifurcations that can occur in the dynamics of a single uncoupled electromechanical system as the fractional-order varies. For example, the fractional-order counterparts of the chaotic 4-order system are periodic at orders less than 3.985. The effect of the fractional-order on the condition of occurrence of synchronization phenomena in the network of many mutually coupled fractional-order nonlinear electromechanical systems is analyzed, especially when they are chaotic. An insight on the overall dynamics of the network is provided. It is shown that the dynamics of both the uncoupled system and the network are very sensitive to changes in the order of the fractional derivative.     

Scaling behavior of coupled conservative weakly nonlinear oscillators

The scaling of the solution of coupled conservative weakly nonlinear oscillators is demonstrated and analyzed through evaluating the normal modes and their bifurcation with an equivalent linearization technique and calculating the general solutions with a method of multiple seales. The scaling law for coupled Duffing oscillators is that the coupling intensity should be proportional to the total energy of the system.Present address: Department of Chemistry and Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030, U.S.A.     

Dynamics of two delay coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol oscillators with delayed position and velocity coupling is studied by the method of averaging together with truncation of Taylor expansions. According to the slow-flow equations, the dynamics of 1:1 internal resonance is more complex than that of non-1:1 internal resonance. For 1:1 internal resonance, the stability and the number of periodic solutions vary with different time delay for given coupling coefficients. The condition necessary for saddle-node and Hopf bifurcations for symmetric modes, namely in-phase and out-of-phase modes, are determined. The numerical results, obtained from direct integration of the original equation, are found to be in good agreement with analytical predictions.     

Dynamics of two strongly coupled van der pol oscillators

A perturbation method is used to study the steady state behavior of two Van der Pol oscillators with strong linear diffusive coupling. It is shown that a bifurcation occurs which results in a transition from phase-locked periodic motions to quasi-periodic motions as the coupling is decreased or the detuning is increased. The analytical results are compared with a numerically generated solution.     

Transient mode localization in coupled strongly nonlinear exactly solvable oscillators

Coupled strongly nonlinear oscillators, whose characteristic is close to linear for low amplitudes but becomes infinitely growing as the amplitude approaches certain limit, are considered in this paper. Such a model may serve for understanding the dynamics of elastic structures within the restricted space bounded by stiff constraints. In particular, this study focuses on the evolution of vibration modes as the energy is gradually pumped into or dissipates out of the system. For instance, based on the two degrees of freedom system, it is shown that the in-phase and out-of-phase motions may follow qualitatively different scenarios as the system’ energy increases. So the in-phase mode appears to absorb the energy with equipartition between the masses. In contrast, the out-of-phase mode provides equal energy distribution only until certain critical energy level. Then, as a result of bifurcation of the 1:1 resonance path, one of the masses becomes a dominant energy receiver in such a way that it takes the energy not only from the main source but also from another mass.     

Dynamic instabilities in coupled oscillators induced by geometrically nonlinear damping

The dynamics of a system of coupled oscillators possessing strongly nonlinear stiffness and damping is examined. The system consists of a linear oscillator coupled to a strongly nonlinear, light attachment, where the nonlinear terms of the system are realized due to geometric effects. We show that the effects of nonlinear damping are far from being purely parasitic and introduce new dynamics when compared to the corresponding systems with linear damping. The dynamics is analyzed by performing a slow/fast decomposition leading to slow flows, which in turn are used to study transient instability caused by a bifurcation to 1:3 resonance capture. In addition, a new dynamical phenomenon of continuous resonance scattering is observed that is both persistent and prevalent for the case of the nonlinearly damped system: For certain moderate excitations, the transient dynamics “tracks” a manifold of impulsive orbits, in effect transitioning between multiple resonance captures over definitive frequency and energy ranges. Eventual bifurcation to 1:3 resonance capture generates the dynamic instability, which is manifested as a sudden burst of the response of the light attachment. Such instabilities that result in strong energy transfer indicate potential for various applications of nonlinear damping such as in vibration suppression and energy harvesting.     

On nonlinear energy flows in nonlinearly coupled oscillators with equal mass

Nonlinear Dynamics - Due to the near monopoly held by nonlinear energy sinks in the study of targeted energy transfer, little research has been done on the flow of mechanical energy between...     

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.     

Modeling phase synchronization in systems of two and three coupled oscillators

We determine regions of synchronization of two and three globally coupled oscillators and describe the main mechanisms and bifurcations through which the synchronization of systems is lost.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 311–327, July–September, 2004.     

The elliptic core of nonlinear oscillators

We devise a method which is apt for the approximation of periodic solutions of strongly nonlinear or statically unstable oscillating systems. To this purpose we consider firstly a system which is canonically associated with ours, the so called elliptic core, and solve it in closed form; this solution is then used as a basis for the construction of a suitable set of trial functions which enable us to apply the Galerkin method in order to obtain the sought approximation. The technique is employed in a number of vibrating systems previously considered in the literature; the results are supported by numerical evidence. Finally we present some further results concerning the relationship between amplitude and period of nonlinear oscillators.