Analysis Of Dynamics And Numerical Simulations Of Two Lorenz-like Systems

Chaos is an interesting phenomenon in the nature world. It reflects the irregular andstochastic complex dynamics in time and in space following deterministic laws andequations. As an important part of Nonlinear Analysis, the extensive research of pastthree decades indicates that it has the necessary properties as follows:(1) stemmingfrom nonlinear systems,(2) the determining,(3) unpredictability of future behaviorsfor a long-term perspective,(4) extremely sensitive dependence on initial conditions,(5) widely distributing in nature,(6) the positive maximum Lyapunov exponent,(7)the infinite broadband frequency spectrum,(8) the ergodicity， etc. Furthermore, ithas played widely important role in such the real world applications as real-time dataencryption, fluid mixing, forecast and other areas. All of these make it the frontierbranch of Nonlinear Science.Since1963, American meteorologists E. N. Lorenz discovered the first chaoticsystem, which is the famous Lorenz system[1]called later. Motivated by thetopological and algebraic structure of it, most researchers and scientists from variousfields investigate chaos in the framework of autonomous differential equations. Up tonow, many chaotic systems has been discovered, such as R ssler system[2], Chensystem[3], Liu system[4], the Lorenz system family[5-7], the conjugate Lorenz-typesystem[8], the hyperchaotic system[9], the complex chaos system[10], the chaoticsystem with time delay[11]，fractional-order chaotic system[12]and other chaoticsystems[13-14]and so on. The study of these systems not only enrich the chaos theorybut also develop some new research method.Along this line, we study two3D chaotic systems in this article as soon possible.The rich dynamical behaviors of them, such as, the local one consisting of distributionof equilibrium, the stability of the isolated and non-isolated equilibria, the Hopfbifurcation and the global one on the the existence of singular orbits: infinitely manysingularly degenerate heteroclinic cycles, homoclinic and heteroclinic orbits anddynamics at infinity and so on, are studied respectively. Further, numericalsimulations not only verify the theoretical analysis results but also illustrate other rich dynamics. The main tools used in this paper are the Center Mainfold Theorem, theHopf bifurcation, the Routh-Hurwitz criterion, Poincaré compactification, symboliccomputation and the MATLAB program of ode45solver based Runge-kutta formula.The main innovations in this paper lie in:1. Present a new chaotic system with trumpet-shaped chaotic attractor;2. Combine the theoretical analysis and numerical simulations, the twodifferent topological and algebraic chaotic system have similar dynamicssuch as the Hopf bifurcation and the existence of non-isolated equilibria,infinitely many singularly degenerate heteroclinic cycles, homoclinic andheteroclinic orbits and so on.Our work has enriched the research contents of the Lorenz-like system and willlay the foundation for the further work.