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Global dynamics of a state-dependent feedback control system

Global dynamics of a state-dependent feedback control system

Tang, Sanyi, Pang, Wenhong, Cheke, Robert ORCID: 0000-0002-7437-1934 and Wu, Jianhong (2015) Global dynamics of a state-dependent feedback control system. Advances in Difference Equations:322. pp. 1-70. ISSN 1687-1839 (Print), 1687-1847 (Online) (doi:

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The main purpose is to develop novel analytical techniques and provide a comprehensive qualitative analysis of global dynamics for a state-dependent feedback control system arising from biological applications including integrated pest management. The model considered consists of a planar system of differential equations with state-dependent impulsive control. We characterize the impulsive and phase sets, using the phase portraits of the planar system and the Lambert W function to define the Poincaré map for impulsive point series defined in the phase set. The existence, local and global stability of an order-1 limit cycle and sharp sufficient conditions for the global stability of the boundary order-1 limit cycle have been provided. We further examine the flip bifurcation related to the existence of an order-2 limit cycle. We show that the existence of an order-2 limit cycle implies the existence of an order-1 limit cycle. We derive sufficient conditions under which any trajectory initiating from a phase set will be free from impulsive effects after finite state-dependent feedback control actions, and we also prove that order-k (k ≥ 3) limit cycles do not exist, providing a solution to an open problem in the integrated pest management community. We then investigate multiple attractors and their basins of attraction, as well as the interior structure of a horseshoe-like attractor. We also discuss implications of the global dynamics for integrated pest management strategy. The analytical techniques and qualitative methods developed in the present paper could be widely used in many fields concerning state-dependent feedback control.

Item Type: Article
Additional Information: This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Uncontrolled Keywords: planar impulsive semi-dynamical system; integrated pest management; Poincaré map; impulsive set; phase set; global stability
Faculty / School / Research Centre / Research Group: Faculty of Engineering & Science > Natural Resources Institute > Agriculture, Health & Environment Department
Last Modified: 27 Apr 2016 16:09

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