Non periodic oscillations, bistability, coexistence of chaos and hyperchaos in the simplest resistorless Op-Amp based Colpitts oscillator.

In the framework of a project on simple circuits with unexpected high degrees of freedom, we report an autonomous microwave oscillator made of a CLC linear resonator of Colpitts type and a single general purpose operational amplifier (Op-Amp). The resonator is in a parallel coupling with the Op-Amp to build the necessary feedback loop of the oscillator. Unlike the general topology of Op-Amp-based oscillators found in the literature including almost always the presence of a negative resistance to justify the nonlinear oscillatory behavior of such circuits, our zero resistor circuit exhibits chaotic and hyperchaotic signals in GHz frequency domain, as well as many other features of complex dynamic systems, including bistability. This simplest form of Colpitts oscillator is adequate to be used as didactic model for the study of complex systems at undergraduate level. Analog and experimental results are proposed.

Didactic model of a simple driven microwave resonant T-L circuit for chaos, multistability and antimonotonicity.

A simple driven bipolar junction transistor (BJT) based two-component circuit is presented, to be used as didactic tool by Lecturers, seeking to introduce some elements of complex dynamics to undergraduate and graduate students, using familiar electronic components to avoid the traditional black-box consideration of active elements. Although the effect of the base-emitter (BE) junction is practically suppressed in the model, chaotic phenomena are detected in the circuit at high frequencies (HF), due to both the reactant behavior of the second component, a coil, and to the birth of parasitic capacitances as well as to the effect of the weak nonlinearity from the base-collector (BC) junction of the BJT, which is otherwise always neglected to the favor of the predominant but now suppressed base-emitter one. The behavior of the circuit is analyzed in terms of stability, phase space, time series and bifurcation diagrams, Lyapunov exponents, as well as frequency spectra and Poincaré map section. We find that a limit cycle attractor widens to chaotic attractors through the splitting and the inverse splitting of periods known as antimonotonicity. Coexisting bifurcations confirm the existence of multi-stability behaviors, marked by the simultaneous apparition of different attractors (periodic and chaotic ones) for the same values of system parameters and different initial conditions. This contribution provides an enriching complement in the dynamics of simple chaotic circuits functioning at high frequencies. Experimental lab results are completed with PSpice simulations and theoretical ones.