Rendering phase portraits and bifurcation diagrams in multidimensional oscillating systems

Abstract

Numerical simulation and visualization of nonlinear dynamical systems behavior can be easily produced, but these graphical results are of little aid if there are parameters in the system, perhaps causing it to exhibit some kind of qualitative change in its phase portrait bifurcations, or oscillations, for instance. For this reason, in this work we propose some methods to graphically detect, track and represent bifurcations and oscillatory behavior in the phase portrait. We propose an algorithm to advect the stationary behavior of the dynamic system and the representation of several periodic orbits in a higher dimension space. We consider two examples: a tunnel diode oscillator and the Colpitts oscillator. In both cases, our algorithm detects the different limit cycles that occur at different parameter values. These limit cycles are used to render the overall oscillatory behavior of the system as a 3D solid in parameter space.