Qualitative analysis of periodically forced nonlinear oscillators responses and stability areas in the vicinity of bifurcation cascade
Bifurcation theory is the mathematical investigation of changes in the qualitative or topological structure of a studied family. In this paper, we numerically investigate the qualitative behavior of nonlinear RLC circuit excited by sinusoidal voltage source based on the bifurcation analysis. Poincare mapping and bifurcation methods are applied to study both dynamics and qualitative properties of the periodic responses of such oscillator. As numerically illustrated here, a small variation of amplitude or frequency of the driver sinusoidal voltage may involve qualitative changes for witch the system exhibits fold, period doubling and pitchfork bifurcations. In fact, the presence of these kinds of bifurcation necessitates an examination of the role of these singularities in the dynamical behavior of circuit. Particularly, we numerically study the qualitative changes may affect number and stability of the periodic solutions and the shapes of its basins of attraction associated while approaching the neighborhood of a particular bifurcation structure known as isoordinal lips cascade. Using a numerical scanning technique, higher harmonic domains which can prove the existence of such cascade of bifurcation are numerically computed. Furthermore, we report on some numerical simulations of bifurcation singularity and basins attractor which are useful tools for understanding and illustrating these effects.
Keywords: Qualitative behavior, bifurcations cascade, fold, flip, pitchfork, higher harmonic, Attraction basins
Download Full-Text
