Dynamics of a nonlinear parametrically excited partial differential equation

We investigate a parametrically excited nonlinear Mathieu equation with damping and limited spatial dependence, using both perturbation theory and numerical integration. The perturbation results predict that, for parameters which lie near the 2:1 resonance tongue of instability corresponding to a single mode of shape cos nx, the resonant mode achieves a stable periodic motion, while all the other modes are predicted to decay to zero. By numerically integrating the p.d.e. as well as a 3-mode o.d.e. truncation, the predictions of perturbation theory are shown to represent an oversimplified picture of the dynamics. In particular it is shown that steady states exist which involve many modes. The dependence of steady state behavior on parameter values and initial conditions is investigated numerically. (c) 1999 American Institute of Physics.     

Experiments on periodic and chaotic motions of a parametrically forced pendulum

An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic route to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.     

Stability analysis of a nonlinear rotating blade with torsional vibrations

This paper discusses the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model. To obtain a reduced nonlinear model from nodal space, a standard modal reduction procedure based on matrix operation is developed with essential geometric stiffening nonlinearities retained in the equation of motion. For the linear model, the stability chart with various spinning parameters of the blade is studied via the Bolotin method, and an efficient boundary tracing algorithm is developed to trace the stability boundary of the linear model. For the geometric nonlinear model, the method of multiple time scale is employed to study the steady state solutions, and their stability and bifurcations for the periodically time-varying rotating blade. The backbone curves of steady-state motions are achieved, and the parameter map for stability and bifurcation is developed.     

High-order resonances in a parametrically excited magneto-optical trap

We present analytical and numerical study of high-order parametric resonance in a driven magneto-optical trap of cold atoms. We have obtained the general solutions for parametric resonance of arbitrary order. In particular, the amplitude and phase of atomic limit-cycle motion is expressed as a function of the modulation amplitude and frequency. Moreover, the atomic dynamics for high-order parametric resonance is investigated in terms of the Hamiltonian approach, which is useful in studying transitions between attractors. We find that the analytical results are in good agreement with the numerical calculations.     

Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: A dynamical integrity perspective

The objective of this paper is showing how global safety arguments can be fruitfully used to interpret experimental results of a pendulum parametrically excited by wave motion. In fact, the results of an experimental campaign developed with the aim of simulating sea-waves energy production by a parametric pendulum show that rotations exist in a region which is smaller than the theoretical one. This discrepancy can be partially attributed to the experimental approximations and constraints, but it has a deeper theoretical motivation. By comparing the experimental results with the dynamical integrity profiles we have found that experimental rotations exist only where a measure of dynamical integrity accounting for both attractor robustness and basin compactness is large enough, so that they can support experimental imperfections leading to changes in initial conditions.     

Existence and analytical predictions of periodic motions in a periodically forced,nonlinear friction oscillator

The grazing bifurcation, stick phenomena and periodic motions in a periodically forced, nonlinear friction oscillator are investigated. The nonlinear friction force is approximated by a piecewise linear, kinetic friction model with the static force. The total forces for the input and output flows to the separation boundary are introduced, and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces. The periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure. Illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction. The force responses are presented, which agreed very well with the force criteria. If the fully nonlinear friction force is modeled by several portions of piecewise linear functions, the periodically forced, nonlinear friction oscillator can be predicted more accurately. However, for the fully nonlinear friction force model, only the numerical investigation can be carried out.     

Multi-pulse homoclinic orbits and chaotic dynamics of a parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities

In this paper,the complicated dynamics and multi-pulse homoclinic orbits of a two-degree-of-freedom parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities are studied.The damping,parametrical excitation and the nonlinearities are regarded as weak.The averaged equation depicting the fast and slow dynamics is derived through the method of multiple scales.The dynamics near the resonance band is revealed by doing a singular perturbation analysis and combining the extended Melnikov method.We are able to determine the criterion for the existence of the multi-pulse homoclinic orbits which can form the Shilnikov orbits and give rise to chaos.At last,numerical results are also given to illustrate the nonlinear behaviors and chaotic motions in the nonlinear nano-oscillator.     

Stability analysis of a parametrically excited functionally graded piezoelectric, MEM system

In this paper the mechanical behavior of a parametrically actuated functionally graded piezoelectric (FGP) clamped-clamped micro-beam is investigated. The micro-beam is supposed to be a composite material with silicon and piezoelectric base. The mechanical properties of the structure, including elasticity modulus, density, and piezoelectricity coefficient are supposed to vary along the height of the micro-beam with an exponential functionality. It is supposed that the FGP clamped-clamped micro-beam is actuated with a combination of direct and alternative electric potential difference. Application of DC and AC actuation voltage leads in a constant and a time-varying axial force. The governing differential equation of the motion is derived using Hamiltonian principle and discretized using expansion theorem with the corresponding shape functions of a clamped-clamped beam. The discretized system is governed by Mathieu equation which’s stability is investigated using Floquet theory for single degree of freedom systems and verified using multiple time scales of perturbation technique.     

Bifurcations of a parametrically excited oscillator with strong nonlinearity

A parametrically excited oscillator with strong nonlinearity, including van der Pol and Duffing types, is studied for static bifurcations. The applicable range of the modified Lindstedt－Poincaré method is extended to 1/2 subharmonic resonance systems. The bifurcation equation of a strongly nonlinear oscillator, which is transformed into a small parameter system, is determined by the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analysed.     

Field dispersion in parametrically excited oscillator upon relaxation

A complete system of solutions of the equation describing the parametric excitation of an oscillator is presented. This system makes it possible to take into account losses in a parametrically excited oscillator in terms of the perturbation theory. It is shown that, although the oscillator losses are small (the relative spectral width of a typical laser cavity is 10⁻⁸), they limit the achievable squeezing coefficients by values on the order of ten. At present, the progress in resonator technique allows one to achieve relative spectral widths on the order of 10⁻¹¹–10⁻¹². Therefore, it can be expected to achieve squeezing coefficients on the order of 10³.     

Control of spatiotemporal disorder in parametrically excited surface waves

Interacting surface waves, parametrically excited by two commensurate frequencies, yield a number of nonlinear states. Near the system's bicritical point, a state, highly disordered in space and time, results from competition between nonlinear states. Experimentally, this disordered state can be rapidly stabilized to a variety of nonlinear states via open-loop control with a small-amplitude third frequency excitation, whose temporal symmetry governs the temporal and the spatial symmetry of the selected nonlinear state. This technique also excites rapid switching between nonlinear states.     

Necessary conditions for mode interactions in parametrically excited waves

We study the spatial and temporal structure of nonlinear states formed by parametrically excited waves on a fluid surface (Faraday instability), in a highly dissipative regime. Short-time dynamics reveal that 3-wave interactions between different spatial modes are only observed when the modes' peak values occur simultaneously. The temporal structure of each mode is functionally described by the Hill's equation and is unaffected by which nonlinear interaction is dominant.     

Defects and drift of structures in parametrically excited capillary ripples

The onset of defects is experimentally studied in a system of capillary waves parametrically excited in a thin layer of liquid with different bottom topography. It is shown that the onset of defects occurs through the excitation of bounded wave-field regions corresponding to the areas with higher depths. The drift of defects and structures is associated with the near-bottom mass flow generated by rapidly decaying surface waves. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, 1998) Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp. 1537–1542, December, 1998.     

High-intensity brillouin scattering from parametrically excited phonons in FeBO3

The saturation of the ferromagnetic resonance in FeBO₃ is due to the parametric decay into short wavelength phonons of half the exciting frequency. These phonons are observed using the technique of inelastic light scattering. Their intensity is so high (10⁵ above thermal level) that the scattered light can be observed visually. Light scattering and parametric decay selection rules result in well defined scattering angles. The analysis of these directions allows the evaluation of anisotropic sound velocities.