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This paper presents a method of calculating an interval including a bifurcation point. Turning points, simple bifurcation points, symmetry breaking bifurcation points and hysteresis points are calculated with guaranteed accuracy by the extended systems for them and by the Krawczyk-based interval validation method. Taking several examples, the results of validation are also presented.
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.
Kazuya HAYATA Masanori KOSHIBA
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.