tailieunhanh - Degenerate Hopf bifurcations, hidden attractors, and control in the extended Sprott E system with only one stable equilibrium

In this paper, we introduce an extended Sprott E system by a general quadratic control scheme with 3 arbitrary parameters for the new system. The resulting system can exhibit codimension-one Hopf bifurcations as parameters vary. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 672 – 687 ¨ ITAK ˙ c TUB ⃝ doi: Degenerate Hopf bifurcations, hidden attractors, and control in the extended Sprott E system with only one stable equilibrium 1 Zhouchao WEI1,∗, Irene MOROZ2 , Anping LIU1 School of Mathematics and Physics, China University of Geosciences, Wuhan, . China 2 Mathematical Institute, Oxford University, Oxford, UK Received: • Accepted: • Published Online: • Printed: Abstract: In this paper, we introduce an extended Sprott E system by a general quadratic control scheme with 3 arbitrary parameters for the new system. The resulting system can exhibit codimension-one Hopf bifurcations as parameters vary. The control strategy used can be applied to create degenerate Hopf bifurcations at desired locations with preferred stability. A complex chaotic attractor with only one stable equilibrium is derived in the sense of having a positive largest Lyapunov exponent. The chaotic attractor with only one stable equilibrium can be generated via a period-doubling bifurcation. To further suppress chaos in the extended Sprott E system coexisting with only one stable equilibrium, adaptive control laws are designed to stabilize the extended Sprott E system based on adaptive control theory and Lyapunov stability theory. Numerical simulations are shown to validate and demonstrate the effectiveness of the proposed adaptive control. Key words: Chaotic attractor, stable equilibrium, Sil’nikov’s theorem, degenerate Hopf bifurcations, hidden attractor 1. Introduction Since chaotic attractors were found by Lorenz in 1963 [10], many chaotic systems have been constructed, such as the R¨ossler [16], the Chen [4], and the L¨ u [11] systems. Because of potential applications in engineering, the study of chaotic systems has attracted the interest of more and more researchers. By .