Research On Complex Dynamics Of A New 5D Hyperchaotic System Based On Conjugate Lorenz System

The theory of chaos has obtained great development in various fields since Lorenz first found a chaotic attractor in 1963.In the recent half century,hyperchaos which is based on chaos has been extensively studied.Compared to chaotic system,hyperchaotic system has at least two positive Lyapunov exponents and attractors can extend in several different directions simultaneously,then system’s randomness and indeterminacy has enhanced a lot.Thus,dynamical behavior of hyperchaotic system is much more complex.Due to its theoretical and practical applications in technological fields,hyperchaos has recently become a central topic in nonlinear sciences research.This paper reports a new five-dimensional(5D)hyperchaotic system with three pos-itive Lyapunov exponents,which is generated by feedback controller based on conjugate Lorenz system.This hyperchaotic system has four quadratic nonlinearities and exhibits complex dynamical behaviors.Of particular interest are the observations that the hy-perchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under a unique equilibrium,two or three equilibria,and there are seven types of co-existing attractors about this new 5D hyperchaotic system.Then dynamic behaviors including the stability of hyperbolic or nonhyperbolic equilibria and Hopf bifurcation are further analyzed.Moreover,the corresponding hyperchaotic and chaotic attractors are numerically verified through Lyapunov exponents,bifurcation,Poincare projections and powerspectrum.The specific research works of this paper are as follows:The first chapter mainly presents the research background and the meaning of this paper.Basic concepts as well as required methods about chaos are also introduced,then the current research status in typical hyperchaotic systems is stated.In the second chapter,a new 5D hyperchaotic system with four quadratic nonlin-earities is generated by employing feedback control method based on conjugate Lorenz system.Of particular note are the facts that this new hyperchaotic system has a hyper-chaotic attractor with three positive Lyapunov exponents under a unique equilibrium,two or three equilibria.Meanwhile,Lyapunov exponents and hyperchaotic behaviors are further numerically analyzed.The third chapter studies the local dynamical behaviors of the new 5D hyperchaotic system in theory.The stability of hyperbolic or nonhyperbolic equilibria is detailed ana-lyzed by using the center manifold theory,normal form and geometric theory of differen-tial equation.Furthermore,the Hopf bifurcation is investigated through high-dimensional Hopf bifurcation theory and strict symbolic inference,the stability and direction of Hopf bifurcation is analyzed,then the period and characteristic index of bifurcating periodic solution are rigorously given.In the forth chapter,the global dynamic behavior of the new 5D hyperchaotic system is discussed.Through observing Lyapunov exponents,bifurcation diagram and Poincare map,the corresponding hyperchaotic,chaotic,quasi-periodic and periodic attractors are numerically verified.Moreover,seven types of coexisting attractors including chaotic with quasi-periodic attractor and quasi-periodic with periodic attractor are discovered in the new 5D hyperchaotic system.