Continuous Adaptive Stabilization of the Unstable Period-1 Orbit of the Fractional Difference Logistic Map

A continuous adaptive stabilization technique for the unstable period-1 orbit of the fractional difference logistic map is presented in this paper. An impulse-based control technique without short oscillatory transients right after the control impulse is designed for the fractional map with a long memory horizon. However, it appears that the coordinate of the unstable period-1 orbit may drift due to the continuous application of the impulse-based control scheme. An adaptive scheme capable of tracking the drifting coordinate of the unstable period-1 orbit is designed and validated by a number of computational experiments. The proposed control scheme is minimally invasive compared to the continuous feedback control as it preserves the model of the system while requiring only a series of sparse, small, instantaneous control impulses to achieve continuous adaptive stabilization of the unstable period-1 orbit of the fractional difference logistic map.