Third-order Resonance Research Articles

Stability of L 4 for radiated rigid primaries in resonant cases

The non-linear stability of the triangular libration point L 4 of the restricted three-body problem is studied under the presence of third- and fourth-order resonances, when the more massive primary is a triaxial rigid body and source of radiation. In this study, Markeev's theorems are applied with the help of Moser's theorem. It is found that the stability of the triangular libration point is unstable in the third-order resonance case and in the fourth-order resonance case, this is stable or unstable depending on A 1 and A 2, and a source of radiation parameter α, where A 1, A 2 depend upon the lengths of the semi-axes of the triaxial rigid body.

Wavelength selection for the far-infrared p-Ge laser using etched silicon lamellar gratings

A lamellar mirror made from Si wafer by anisotropic chemical etching and coated with gold has been demonstrated as an intracavity wavelength selector for the far-infrared p-Ge laser. The etching process produces rectangular grooves with precisely predetermined depth and 100 nm surface smoothness. This lamellar-grating structure defines the resonant laser wavelength within the broad tuning range of the p-Ge laser. Single wavelength laser operation with this mirror has been demonstrated on the third-order resonance with an active cavity finesse of at least 0.09.

Cycle in a system close to a resonance system

An autonomous system when the frequencies of the linear system satisfy a relation which is close, with a frequency detuning ε, to an exact two-frequency internal resonance is investigated. The fact that there is an isolated periodic solution in the general situation is established. The cycle occurs at a distance O(ε) from zero in the case of third-order resonance and, at a distance of O(√ε) in the case of second- and fourth-order resonances. Systems of a general from as well as Lyapunov systems are considered. The problem of the stability of the cycles is studied. It is shown that, when small periodic perturbations of the order of μ act on the system being investigated, periodic motions exist in the μ χ neighbourhood of zero for which ε = O( μ k ). In addition, χ= 1 2 ,μ ⩾ 1 2 in the case of third-order resonance and χ = 1 3 , k ⩾ 2 3 in the case of second- and fourth-orderresonances.

Three-dimensional instability of standing waves

We investigate the three-dimensional instability of finite-amplitude standing surface waves under the influence of gravity. The analysis employs the transition matrix (TM) approach and uses a new high-order spectral element (HOSE) method for computation of the nonlinear wave dynamics. HOSE is an extension of the original high-order spectral method (HOS) wherein nonlinear wave–wave and wave–body interactions are retained up to high order in wave steepness. Instead of global basis functions in HOS, however, HOSE employs spectral elements to allow for complex free-surface geometries and surface-piercing bodies. Exponential convergence of HOS with respect to the total number of spectral modes (for a fixed number of elements) and interaction order is retained in HOSE. In this study, we use TM-HOSE to obtain the stability of general three-dimensional perturbations (on a two-dimensional surface) on two classes of standing waves: plane standing waves in a rectangular tank; and radial/azimuthal standing waves in a circular basin. For plane standing waves, we confirm the known result of two-dimensional side-bandlike instability. In addition, we find a novel three-dimensional instability for base flow of any amplitude. The dominant component of the unstable disturbance is an oblique (standing) wave oriented at an arbitrary angle whose frequency is close to the (nonlinear) frequency of the original standing wave. This finding is confirmed by direct long-time simulations using HOSE which show that the nonlinear evolution leads to classical Fermi–Pasta–Ulam recurrence. For the circular basin, we find that, beyond a threshold wave steepness, a standing wave (of nonlinear frequency . From the cases we consider, the critical wave steepness is found to generally decrease/increase with increasing radial/azimuthal mode number of the base standing wave. Finally, we show that the instability we find for both two- and three-dimensional standing waves is a result of third-order (quartet) resonance.

A Pulse Stretcher by Slow Extraction Utilizing the Third-Order Resonance with the RF Knockout Method

A pulse stretcher operation of KSR using slow electron beam extraction by the third-order resonance in combination with an RF knockout method has been accomplished. The extracted beam energy, number of extracted electrons in a single pulse, maximum duration and duty factor are 60 MeV, 1.4×108, 40 s and 90%, respectively. The beam spill can be controlled from 0.2 s to 40 s by adjusting of the power for RF knockout.

On the nonlocal stability of periodic motion of a Hamiltonian system at the third-order resonance

On the nonlocal stability of periodic motion of a Hamiltonian system at the third-order resonance

Time-Trimming Tricks for Dynamic Simulations: Splitting Force Updates to Reduce Computational Work

Time-Trimming Tricks for Dynamic Simulations: Splitting Force Updates to Reduce Computational Work

Phase space structure for matched intense charged-particle beams in periodic focusing transport systems

Test particle motion is analyzed analytically and numerically in the field configuration consisting of the equilibrium self-electric and self-magnetic fields of a well-matched, thin, continuous, intense charged-particle beam and an applied periodic focusing solenoidal magnetic field. The self fields are determined self-consistently, assuming the beam to have a uniform-density, rigid-rotor Vlasov equilibrium distribution. Using the Hamilton–Jacobi method, the betatron oscillations of test particles in the average self fields and applied focusing field are analyzed, and the nonlinear resonances induced by periodic modulations in the self fields and applied field are determined. The Poincaré surface-of-section method is used to analyze numerically the phase-space structure for test particle motion outside the outermost envelope of the beam over a wide range of system parameters. For vacuum phase advance σv=80°, it is found that the phase-space structure is almost entirely regular at low beam intensity (phase advance σ≳70°, say), whereas at moderate beam intensity (30°≲σ≲70°), nonlinear resonances appear, the most pronounced of which is the third-order primary nonlinear resonance. As the beam intensity is further increased (σ≲30°), the widths of the higher-order nonlinear resonances increase, and the chaotic region of phase space increases in size. Furthermore, the many chaotic layers associated with the separatrices of the primary and secondary nonlinear resonances are still divided by the remaining invariant Kolmogorov–Arnold–Moser surfaces, even at very high beam intensities. The implications of the rich nonlinear resonance structure and chaotic particle motion found in the present test-particle studies are discussed in the context of halo formation.

The non-linear oscillations of a satellite with third-order resonance

Plane non-linear oscillations of an artificial satellite—a rigid body—about its centre of mass in an elliptical orbit of small eccentricity are considered. It is assumed that three times the frequency of small oscillations of the satellite in a circular orbit is close to the frequency of revolution of its centre of mass. Methods of classical perturbation theory are used to reduce the problem to that of a model system, described by a Hamiltonian which is characteristic for problems involving the motion of Hamiltonian systems with one degree of freedom in the case of third-order resonance. A detailed analysis of such systems is carried out. The theory of periodic Poincare motions and KAM-theory are used to transfer the results for the model system to the complete system and to apply them to the problem of satellite motion. The question of the existence, number and stability of periodic motions with period equal to three times the period of revolution of the centre of mass of the satellite in orbit is considered, depending on the inertial parameter of the satellite and the eccentricity of the orbit. It is shown that motions of the satellite beginning in a certain neighbourhood of its eccentricity oscillations are bounded, and an estimate is given for the size of that neighbourhood.

Stability and Geometry of Third-Order Resonances in four-Dimensional Symplectic Mappings

We analyze four-dimensional symplectic mappings in the neighbourhood of an elliptic fixed point whose eigenvalues are close to satisfy a third-order resonance. Using the perturbative tools of resonant normal forms, the geometry of the orbits and the existence of elliptic or hyperbolic one-dimensional tori (fixed lines) is worked out. This allows one to give an analytical estimate of the stability domain when the resonance is unstable. A comparison with numerical results for the four-dimensional Henon mapping is given.

Steady-state Vibration Responses of a Beam with a Nonlinear Boundary Condition

An analysis is presented for the response of a beam constrained by a nonlinear spring to a harmonic excitation. The system is governed by a linear partial differential equation with a nonlinear boundary condition. The method of multiple scales is used to reduce the nonlinear boundary value problem to a system of autonomous ordinary differential equations of the amplitudes and phases. The case of the third-order subharmonic resonance is considered in this study. The autonomous system is used to determine the steady-state responses and their stability.

The Integrable cases of the general and restricted elliptic spatial three-body problems at both third-order orbital and secondary resonances under the oblateness of the central body

For the general and restricted elliptic planetary spatial three-bodyproblems under the oblateness of the central body and at both third-orderorbital and secondary resonances, the analytical solutions are obtained bythe use of the Weierstrass functions with an accuracy of the fourth-degreeterms in the eccentricities and inclinations. The behaviour of solutions isstudied on the phase plane.

Distributed Bragg reflection in integrated waveguides of polymethylmethacrylate

Distributed Bragg reflection from integrated polymethylmethacrylate waveguides fabricated on corrugated spin-on-glass substrate is reported. Utilizing third-order Bragg resonance, a maximum reflection of 92% from the grating filter is measured at a wavelength of 423.6 nm. A maximum bandwidth of the Bragg reflection is found to be ∼0.5 nm for a 6 mm long filter. The broad bandwidth and the significant transmission loss (∼90%) away from resonance are believed to be related to the radiation coupling by the grating. It is shown that the radiation loss can be minimized without sacrificing high reflectivity by limiting the waveguide length to 0.5 mm, the saturation length of the Bragg reflector.

Resonance in the dynamics of chemical systems simulated by the implicit midpoint scheme

The numerical behavior of the symplectic, implicit midpoint method with a wide range of integration timesteps is examined through an application to a diatomic molecule governed by a Morse potential. Our oscillator with a 12.6 fs period exhibits notable, integrator induced, timestep- ( Δt) dependent resonances and we predict approximate values of Δt where they will occur. The particular case of a third-order resonance ( Δt ≈ 7 fs here) leads to instability, and higher-order resonances ( n = 4, 5) to large energetic fluctuations and/or corrupted phase diagrams. Significantly, for Δt > 10 fs the energy errors remain bound.

The integrable cases of the general spatial three-body problem at third-order resonance

For the general three-body problem at third-order resonance an analytical solution is obtained by the use of the Weierstrass functions.

Thin-Film Bulk Acoustic Resonators Employing ZnO/Pyrex-Glass Composite Diaphragm Structure

This paper discusses the application of ZnO/Pyrex-glass composite diaphragm structure to thin-film bulk acoustic resonators (FBARs). Analysis of ZnO/Pyrex-glass composite diaphragm structure showed for thickness-longitudinal vibration that zero temperature coefficient of frequency with relatively large electromechanical coupling constant of about 20% is realised for the second- and third-order resonances as well as for the fundamental resonance. An FBAR employing the ZnO/Pyrex-glass composite diaphragm structure was fabricated, and Q values of 970 and 858 were obtained, respectively, for the fundamental and second-order resonances at 298.7 MHz and 497.5 MHz.

Small-signal resonance distortion of semiconductor lasers

Small-signal expressions for the second- and third-order resonance distortion of a semiconductor injection laser are reported. Results are valid even if the frequencies involved are not closely spaced, and are applicable to high-power lasers for which previously published results may not be valid. The new results are more accurate than prior results when the product of the gain compression factor and the photon density is relatively large, or when low frequencies are of interest. The second-order distortion-to-carrier ratio depends on the resultant frequency but not on the individual frequencies involved in nonlinear mixing. The third-order distortion-to-carrier ratio in general depends on individual mixing frequencies, but if all frequencies are small compared to the resonance frequency, then this result also depends primarily on the resultant frequency.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Third-order resonance in a Hamiltonian system with one degree of freedom

Non-linear oscillations of a nearly integrable Hamiltonian system with one degree of freedom, which is 2π-periodic in t, are investigated in a small finite neighbourhood of equilibrium. The Hamiltonian is assumed to be analytic, the linearized system is stable, and its characteristic exponents ± iv are purely imaginary, where 3 v is an integer. The equilibrium position of such a system is generally unstable and six trajectories exist that asymptotically approach the equilibrium point as t →±∞ [1,2]. It is shown that for most initial data the motion is quasi-periodic in the neighbourhood of the equilibrium. The existence of stable 6π-periodic motions near an unstable equilibrium position is established. It is shown that, irrespective of instability, trajectories beginning sufficiently close to an equilibrium point will always remain at a finite distance from it. An estimate is obtained for this distance. The stochastic nature of the motion near trajectories asymptotic to the equilibrium point is discussed.

Polyaniline sol-gels and their third-order nonlinear optical effects

Sol-gel matrices can provide an inert host for optically active organic compounds. Here we describe the room temperature synthesis of high quality silica glasses containing the conjugated polymers polyaniline and 2-ethyl-polyaniline. Phase conjugate reflectivity (R pc) values for both solutions and sol-gels containing the polymers were obtained via degenerate four-wave mixing experiments (DFWM). The third-order resonance enhanced molecular susceptibility value for emeraldine base in solution was calculated to be 1.6 × 10 −11 esu, which is greater than that of a carbon disulfide reference (1.7 × 10 −12 esu). Sol-gel films containing 5 mg/mL of the ring substituted 2-Et-PANi were synthesized and found to have third-order susceptibilities of 4.8 × 10 −13 esu.

The stability of a class of reversible systems

The problem of the stability of the point of rest of an autonomous system of ordinary differential equations from a class of reversible systems [1] characterized by the critical case of m zero roots and n pairs of pure imaginary roots is considered. When there are no internal resonances [2, 3], the point of rest always has Birkhoff complete stability [2]. Internal resonances may lead to Lyapunov instability. The conditions of stability and instability of the model system when there are third-order resonances may be obtained from a criterion previously developed [4] for the case of pure imaginary roots. The results are used to analyse the stability of the translational-rotational motion of an active artificial satellite in a non-Keplerian circular orbit, including a geostationary satellite in any latitude [4, 5]. The region of stability of relative equilibria and regular precession of the satellite is constructed assuming a central gravitational field and the resonance modes are analysed.