We consider a ring of n Rayleigh oscillators coupled hybridly. Using the symmetrical property of the system we demonstrate the degeneracy of the Hopf bifurcation of the equilibrium at the origin. The degeneracy implies the exstence and stability of the n-phase oscillation. We discuss some consequences of the perturbation of the symmetry. Then we study the case n = 3. We show the bifurcation diagram of the equilibria and of hte periodic solutions. Especially, we analyze the mechanism for the symmetry breaking bifurcation of the fully symmetric solution. We report and explain the occurrence of both chaotic attractors and repellors and show two types of symmetry recovering crisis they undergo.

In this paper we study the properties induced by the symmetrical properties of a system of hybridly coupled oscillators of the Rayleigh type on the bifurcations of its equilibria. We first discuss the symmetrical properties of the system. Then we classify the equilibria according to their symmetrical properties. Demonstrating the structural degeneracy of the system, we give the complete stability analysis of the equilibria.

In this paper we study the bifurcations of the periodic solutions induced by the symmetrical properties of a system of hybridly coupled oscillators of the Rayleigh type. By analogy with the results concerning with the equilibria, we classify the periodic solutions according to their spatial and temporal symmetries. We discuss the possible bifurcations of each type of periodic solution. Finally we analyze the phase portraits of the system when the parameters vary.

A radial line slot antenna (RLSA) is a high gain and high efficiency planar antenna proposed for DBS subscribers. Spirally arrayed slots are excited by a cylindrical wave with the rotational symmetry. In a small sized antenna where large slot coupling is adopted, aperture efficiency reduction due to rotational asymmetry associated with a spiral arrangement of the slots becomes notable. Authors proposed a RLSA with a concentric slot arrangement excited by a rotating mode in order to enhance the rotational symmetry. This is the first report of the normal operation of a rotating mode RLSA fed by a cavity resonator. The experiments confirm the basic operation of this novel antenna; the gain of 27.8dBi and the efficiency of 68% is measured at 11.85GHz for the RLSA with 0.24mφ.

Numerical simulations for the (3+1)-dimensional optical-field dynamics of nonstationary pulsed beams that propagate down Kerr-like nonlinear channel waveguides are carried out for what is to our knowledge the first time. Time-resolved intrapulse switching due to spontaneous symmetry breaking of optical fields from a quasilinear symmetric field to a nonlinear asymmetric field is analyzed. A novel instability phenomenon triggered by the symmetry breakdown is found.

A radial line slot antenna (RLSA) is a slotted waveguide planar array for the direct broadcast from satellite (DBS) subscriber antennas. A single-layered RLSA (SL-RLSA) is excited by a radially outward traveling wave. The antenna efficiency of more than 85% has already been realized. These antennas are designed on the assumption of perfectly rotationally symmetrical traveling wave excitation; the slot design is based upon the analysis of a slot pair on the rectangular waveguide model with periodic boundary walls. However, the slots perturb the inner field and the actual antenna operation is not perfectly symmetrical. This causes the efficiency reduction especially for very small size antenna. This paper presents a fundamental analysis of the inner field of the radial waveguide. It is impossible to analyze all the slot pairs in the aperture as it is and only the slots in the inner few turns are considered since these provide dominant perturbation. The calculated results are verified by the experiments and reasonable agreement is demonstrated. Some design policies are suggested for enhancing the rotational symmetry.