Revisiting linear augmentation for stabilizing stationary solutions: potential pitfalls and their application

TL;DR

This paper critically examines linear augmentation for stabilizing stationary solutions, revealing its limitations and potential pitfalls through examples in conservative and dissipative systems, and discusses implications for predator-prey models.

Contribution

It provides a detailed analysis of the conditions under which linear augmentation fails to stabilize solutions, highlighting scenarios that challenge its general applicability.

Findings

Linear augmentation can fail to stabilize desired solutions in certain systems.

Examples include conservative and dissipative systems where the scheme leads to complex dynamics.

Potential applications in predator-prey systems are discussed.

Abstract

Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the highlight of this scheme but at the same time questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Appropriate examples from conservative as well as dissipative systems are presented in this regard and potential applications for relevant observations in dissipative predator--prey systems are also discussed.