[en] The time-dependent oscillator describing parametric oscillation, the concept of invariant and Noether's theorem are important issues in physics education. Here, it is shown how they can be interconnected in a simple and unified manner.

[en] Highlights: • The hybrid Van der Pol–Rayleigh oscillator exhibits new dynamics. • There exist two limit cycles and a heteroclinic cycle. • Complete bifurcation diagram and the corresponding phase portraits are sketched. • The locations of these limit cycles are characterized. • These results are helpful to understand the motion of human or bipedal robot. We study the dynamics of a hybrid van der Pol–Rayleigh oscillator that has been used to model the self-sustained walking behaviors of humans or bipedal robots in literature. By using qualitative and bifurcation analysis, we discover new dynamics that are different from van der Pol and Rayleigh oscillators. This hybrid oscillator can exhibit two limit cycles and rich bifurcations phenomena. The global bifurcation diagram is shown in the parameter space, and the corresponding global phase portraits of this hybrid oscillator are sketched in the phase space. Furthermore, the locations or amplitudes of the periodic oscillations (limit cycles) are also characterized.

[en] A system of two magnets hung from two vertical springs and oscillating in the hollows of a pair of coils connected in series is a new, interesting and useful example of coupled oscillators. The electromagnetically coupled oscillations of these oscillators are experimentally and theoretically studied. Its coupling is electromagnetic instead of mechanical, and easily adjustable by the experimenter. The coupling of this new coupled oscillator system is determined by the currents that the magnets induce in two coils connected in series, one to each magnet. It is an interesting case of mechanical oscillators with field-driven coupling, instead of mechanical coupling. Moreover, it is both a coupled and a damped oscillating system that lends itself to a detailed study and presentation of many properties and phenomena of such a system of oscillators. A set of experiments that validates the theoretical model of the oscillators is presented and discussed.

[en] The vibrational transition probability expressions for the forced Morse oscillator have been derived using the commutation relations of the anharmonic Boson operators. The formulation is based on the collinear collision model with the exponential repulsive potential in the framework of semiclassical collision dynamics. The sample calculation results for H₂ + He collision system, where the anharmonicity is large, are in excellent agreement with those from an exact, numerical quantum mechanical study by Clark and Dickinson, using the reactance matrix. Our results,however, are markedly different from those of Ree, Kim, and Shin's in which they approximate the commutation operator I_o as unity, the harmonic oscillator limit. We have concluded that the quantum number dependence in I_o must be retained to get accurate vibrational transition probabilities for the Morse oscillator

[en] Amongst all stars observed to pulsate, the Sun has by far the largest number and variety of modes of oscillation. This presents a unique opportunity to apply and test stellar oscillation theory. To match the observational accuracy, very precise calculations of oscillation frequencies are needed. Asymptotic methods have proved useful in the analysis and interpretation of the frequencies. The results provide tight constraints on solar models; they may also enable a direct determination of properties of the solar interior. There are difficulties in reconciling the amplitudes obtained in Doppler velocity with those observed in the apparent position of the solar limb. The excitation of the oscillations is so far not well understood, although it is probable that the interaction between pulsation and convection plays an important role. (orig.)

[en] The issue of asymmetric modes output of a Ka-band super overmoded coaxial Cerenkov oscillator is analyzed in this paper. Due to serious passband overlapping in a super overmoded coaxial slow wave structure (SWS), the asymmetric competition mode EH₁₁ can hardly be suppressed thoroughly by the methods adopted in moderately overmoded devices, especially in the startup of oscillation. If the output structures reflect the asymmetric modes, the asymmetric mode competition in SWS will be aggravated and the normal operation state will be destroyed. In order to solve this problem, a taper waveguide is inserted at a specific position to achieve the destructive interference of the reflected TM₁₁, and a special support structure is designed to avoid reflection of TE₁₁. With these methods, asymmetric mode competition can be successfully eliminated, and the oscillator is capable of achieving a steady fundamental mode operation performance. (paper)

[en] High-l nonradial nonadiabatic fundamental oscillations of a star are considered. It is found that the f-mode 5-minute oscillations may be exited only by turbulent convection

[en] The delayed response of a damped harmonic oscillator (RLC circuit) to a slow periodic disturbance is presented. This communication is supplementary to the paper published recently (Krupska et al 2001 Eur. J. Phys. 22 133-8)

[en] Highlights: • Bursting oscillations of the vector field with a Double Hopf bifurcation are observed. • Bifurcation mechanism of bursting oscillations is presented. • Different forms of bursting oscillations with both single mode and mixed modes are obtained. Up to now, most of the work related to the slow-fast dynamical systems are based on the low dimensional systems with only codimension-one bifurcations at the transitions between the quiescent states and the spiking states, while there usually only exists one slow variable in the system. Since the coupling effect of two scales on the behaviors of high dimensional systems with two or more slow variables as well as high co-dimensional bifurcations lacks exploration, in this paper, we investigate the slow-fast effect on a 6-dimension system, which can be regarded as the coupling of two subsystems. The fast subsystem is expressed as the normal form of the vector field with a double Hopf bifurcation at the origin, while the slow-varying parameter in the fast subsystem is controlled by the slow subsystem. For the fast subsystem, the trajectory is projected onto two independent sub-planes for better observation, and the movement of bursting oscillators can be synthesized with information from each dimension. In the sub-plane, the mechanism of bursting oscillations is derived by employing the overlap of the transformed phase portrait and the equilibrium branches as well as the bifurcations. It is found that, Hopf bifurcation sets may cause the fast subsystem to transform between the single mode and the mixed-mode or between different single modes, leading to bursting oscillations in certain sub-planes or the whole four dimensions of the fast subsystem. Furthermore, interesting phenomena may be caught, such as the trajectory jumping in the process of oscillation as well as transforming between different oscillation modes of the fast subsystem by the way of jump as a result of the fold bifurcation set.

[en] Observation of global oscillations of the Sun constitutes a primitive seismology of the solar interior. The frequencies, if correctly identified with definite normal modes of vibration. Provide a measure of the average velocity of sound in the interior and thereby of its composition and temperature. Fine structure in the frequencies of nonradial modes may provide information on their character (multiplicity) and on the rotation of the solar interior. Study of the amplitudes and phase fluctuations of the vibrations may clarify the excitation and damping of the vibrations. After a brief historical review emphasizing global velocity spectroscopy an account is given of the present status of the observations of global oscillations in the range of periods of 3 to 160 min. Finally the future capabilities of the observational techniques and their resultant potential is discussed. (orig.)