Self-sustained and chaotic oscillations with excursions and asymmetry in oscillators with nonlinear damping singularities and exponential terms

We study, theoretically and experimentally with microcontrollers, the dynamics of second-order nonlinear differential equations with rational damping terms having singularities and exponential damping terms. Because of the singularities, one finds a series of excursions of the dynamical variable from a permanent self-sustained or forced oscillatory state to instantaneous high values before returning back to oscillations with almost constant amplitude. Chaotic behaviors with excursions are observed even in the autonomous case. The occurrence of chaos in the autonomous system indicates that one way to generate chaos is the use of singularities in the damping term. In the case of the oscillator with the exponential damping term, one finds asymmetric self-sustained oscillations as well as asymmetric chaotic dynamics. The theoretical results are confirmed by the experimental investigation in a mixed circuit containing both discrete electrical components and a microcontroller. To the best of our knowledge, this work offers a place to three interesting novelties in nonlinear dynamics. The first one is the analysis of oscillators with singularities and exponential terms in their damping coefficient. The second is the use of a mixed circuit consisting of discrete electric components in series with a microcontroller generating nonlinear characteristics. The third one is the fact that chaos can be generated because of the presence of the singularities in the damping term.