Stabilization of cycles for difference equations with a noisy PF control

TL;DR

This paper investigates stabilizing cycles in difference equations, like the Ricker map, using noisy proportional feedback control, and demonstrates both theoretically and numerically that stable blurred cycles can be achieved despite environmental and control variability.

Contribution

It extends stabilization methods to include both additive and multiplicative noise, providing theoretical justification and numerical evidence for stable blurred cycles in difference equations.

Findings

Stable blurred cycles can be achieved with noisy control.

Theoretical results confirm stabilization under multiplicative or additive noise.

Numerical simulations show stabilization with combined noise types.

Abstract

Difference equations, such as a Ricker map, for an increased value of the parameter, experience instability of the positive equilibrium and transition to deterministic chaos. To achieve stabilization, various methods can be applied. Proportional Feedback control suggests a proportional reduction of the state variable at every $k$th step. First, if $k \neq 1$, a cycle is stabilized rather than an equilibrium. Second, the equation can incorporate an additive noise term, describing the variability of the environment, as well as multiplicative noise corresponding to possible deviations in the control intensity. The present paper deals with both issues, it justifies a possibility of getting a stable blurred $k$-cycle. Presented examples include the Ricker model, as well as equations with unbounded $f$, such as the bobwhite quail population models. Though the theoretical results justify stabilization for either multiplicative or additive noise only, numerical simulations illustrate that a blurred cycle can be stabilized when both multiplicative and additive noises are involved.