Ghost attractors in blinking Lorenz and Hindmarsh–Rose systems

In this paper, we consider blinking systems, i.e., non-autonomous systems generated by randomly switching between several autonomous continuous time subsystems in each sequential fixed period of time. We study cases where a non-stationary attractor of a blinking system with fast switching unexpectedly differs from the attractors of composing subsystems. Such a non-stationary attractor is associated with an attractor of the averaged system being a ghost attractor of the blinking system [Belykh et al., Phys. D: Nonlinear Phenom. 195, 188 (2004); Hasler et al., SIAM J. Appl. Dyn. Syst. 12, 1031 (2013); Belykh et al., Eur. Phys. J. Spec. Top. 222, 2497 (2013)]. Validating the theory of stochastically blinking systems [Hasler et al., SIAM J. Appl. Dyn. Syst. 12, 1031 (2013); Hasler et al., SIAM J. Appl. Dyn. Syst. 12, 1007 (2013)], we demonstrate that fast switching between two Lorenz systems yields a ghost chaotic attractor, even though the dynamics of both systems are trivial and defined by stable equilibria. We also study a blinking Hindmarsh–Rose system obtained from the original model of neuron activity by using randomly switching sequence as an external stimulus. Despite the fact that the values of the external stimulus are selected from a set corresponding to the tonic spiking mode, the blinking model exhibits bursting activity. For both systems, we analyze changes in the dynamical behavior as the period of stochastic switching increases. Using a numerical approximation of the invariant measures of the blinking and averaged systems, we give estimates of a non-stationary and ghost attractors’ proximity.