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 .

Không thể tạo bản xem trước, hãy bấm tải xuống
TÀI LIỆU MỚI ĐĂNG
Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.