Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect

In this paper, the dynamical behavior of a three-dimensional (3D) autonomous oscillator (Pehlivan and Uyarogˇlu, 2012) is further investigated in some detail (including local stability and bifurcation structures analysis) in order to reveal bursting dynamics. This system has the unusual feature of involving three equilibria asymmetrically distributed. We find that for specific parameters, the system exhibits periodic and chaotic bursting oscillations. By rescaling its dynamical variables, we show that this system evolves on two time scales, corresponding respectively to fast and slow dynamics that lie at the basis of bursting dynamics. The mechanism underlying this phenomenon is drawn via the bifurcation analysis of the fast subsystem with respect to the slow subsystem variable. This reveals that the bursting oscillations found there result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. This class of bursting has been referred to as “source/sink bursting of node/focus type”. In addition, an analog circuit is designed and implemented to realize experimentally the periodic and chaotic bursting oscillations. Good agreement is shown between the numerical and the experimental results. Further, the commensurate fractional-order version of the chaotic slow–fast system is studied using stability theorem of fractional-order systems and numerical simulations. By tuning the fractional-order, the slow–fast system displays a wide variety of dynamical behaviors ranging from chaotic bursting to fixed point dynamics via continuous chaotic spiking. Because of the asymmetric distribution of the system equilibrium points, this transition includes peculiar phenomena such as transient chaos and the coexistence of point and chaotic attractors.