Continual Impulse-Based Finite-Time Fractional Difference Logistic Map Stabilization Scheme

This research presents an impulse-based control scheme capable of stabilizing the unstable period-1 orbit of the fractional-difference logistic map. In 1876, a biologist named Robert May popularized the first known simple mathematical model with chaotic solutions. This system utilizes ordinary differences and has a memory horizon of one step. A few decades later, the fractional-difference logistic map was introduced, exhibiting a memory horizon that reaches the initial condition by using Caputo fractional differences. Due to its valuable properties, the fractional difference logistic map has already been adapted in economics, steganography, epidemiology, and engineering. In recent years, we have explored some impulse-based control schemes to stabilize this map's unstable period-1 orbit. In our recent research, it was shown that a naive control scheme does not achieve finite-time stabilization of the unstable period-1 orbit. Instead, an H-rank-based approach is used to achieve finite-time stabilization. However, this version of the control scheme exhibited violent oscillatory transient processes following the control impulse. To improve the stabilization scheme, several findings were made. Firstly, at the cost of some stabilization duration, it is possible to stabilize the unstable period-1 orbit by finding a suitable initial condition within the tolerance corridor δ. Secondly, the coordinate for the unstable period-1 orbit drifts each time a control impulse is applied. And finally, this drift is caused not only by perturbations arising from numerous stabilization impulses but also by computational effects. In conclusion, this proposed control scheme is minimally invasive compared to continuous feedback control because it preserves the system model and requires only a series of small, sparse, and instantaneous control impulses to achieve continuous adaptive stabilization of the unstable period-1 orbit of the fractional difference logistic map.