Control of chaotic two-predator one-prey model with single state control signals
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Control of chaotic two-predator one-prey model with single state control signals Ugur ˘ Erkin Kocamaz1
· Alper Göksu2 · Harun Ta¸skın2 · Yılmaz Uyaroglu ˘ 3
Received: 10 February 2020 / Accepted: 20 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, the complex control dynamics of a predator–prey Lotka–Volterra chaotic system are studied. The main purpose is to control the chaotic trajectories of two-predator one-prey system which was introduced by Samardzija and Greller (Bull Math Biol 50(5):465–491. https://doi.org/10.1007/BF02458847, 1988). Lyapunov based nonlinear control and sliding mode control methods are used. The other purpose of this paper is to present the sliding mode control performances under different sliding surface choices. Based on the sliding mode control and Lyapunov stability theory, four alternative sliding surfaces are constructed to stabilize the chaotic two-predator one-prey model to its zero equilibrium point. The focused control signals realize the control from only one state which provides simplicity in implementation. Numerical simulations are demonstrated to validate the theoretical analyses and compare the effectiveness of proposed controllers for the chaotic Samardzija–Greller system. Keywords Two-predator one-prey model · Chaotic Samardzija–Greller system · Lotka–Volterra system · Lyapunov based nonlinear control · Sliding mode control · Chaos control
Introduction As a consequence of independently studies of Lotka (1925) and Volterra (1926), the simple model of two-species predator–prey system was introduced and called as Lotka–Volterra system. It consists of a pair of first-order nonlinear differential equations. Lotka–Volterra system is generally using for describing the dynamics of ecological systems in which
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U˘gur Erkin Kocamaz [email protected]; [email protected] Alper Göksu [email protected] Harun Ta¸skın [email protected] Yılmaz Uyaro˘glu [email protected]
1
Department of Computer Technologies, Vocational School of Karacabey, Bursa Uluda˘g University, 16700 Karacabey, Bursa, Turkey
2
Department of Industrial Engineering, Faculty of Engineering, Sakarya University, 54187 Serdivan, Sakarya, Turkey
3
Department of Electrical and Electronics Engineering, Faculty of Engineering, Sakarya University, 54187 Serdivan, Sakarya, Turkey
two species interact, one as a predator and the other as a prey. Chaos can be defined as the property of a complex system whose behaviour is so unpredictable as to appear random, owing to great sensitivity to small changes in parameter values and initial conditions. Chaotic systems are the systems which exhibit chaos. Since Lorenz (1963) discovered the first chaotic system in a simplified mathematical model of atmospheric convection, it has been shown that chaotic systems exist in many fields including physics, chemistry, biology, ecology and finance (Zhang et al. 2013; Bodale and Oancea 2015; Wei et al. 2017; Khajanchi et al. 2018), and can be useful in some
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