Research On Complicated Dynamics Of Three-dimensional Systems Based On A Family Of Lorenz-type Systems

In the early 1960s, the first chaotic system was proposed by a meteorologist namedEdward Lorenz. The Lorenz system is the first mathematical and physical model of chaos,thereby becoming a touchstone for the modern chaos theory. As the first mathematicalmodel of chaos-Lorenz system, it led to further think about the complex phenomena in thereal world. The study on a family of Lorenz-type systems have played in the developmentof chaos theory and several disciplines.Based on the family of Lorenz-type systems, this dissertation will mainly considerthe stability, bifurcation and chaos of three dimensional Lorenz-like chaotic systems withonly two stable equilibria, and complex dynamics of the di?usionless Lorenz equations(DLE) with periodic parametric perturbation. The main research work are as follows.In Chapter 1, the research background and the significance of this paper are pre-sented. The research developments, main methods and achievements of chaos theory arebriefed. The advanced studies for the family of Lorenz-type systems, their homoclinicorbits and heteroclinic orbits are also introduced.In Chapter 2, an unusual three-dimensional autonomous quadratic Lorenz-like chaoticsystem with only two stable equilibria is considered. The system contains the di?usion-less Lorenz system and the Burke-Shaw system, and some others, as special cases. Thealgebraic form of the new chaotic system is similar to the other Lorenz-type systems,but they are topologically nonequivalent. Three kinds of chaotic attractors (Shilnikov'ssaddle-focus, non-hyperbolic and only two stable equilibria) are investigated. To furtheranalyze the new system, some dynamical behaviors such as Hopf bifurcation and singu-larly degenerate heteroclinic, are rigorously proved with simulation verification. In orderto further understand the complex 3-D dynamical system, codimension one, two, andthree Hopf bifurcations of the controlled system depending on five parameters are alsostudied.In Chapter 3, generalized Sprott C system is proposed and its chaos mechanismis analyzed. This system can display a double-scroll chaotic attractor with only twostable equilibria. With the help of rigorous maths analysis and symbol computation, twocomplete mathematical characterizations for Hopf bifurcation are derived and studied, bymeans of the high Hopf bifurcation theorem. Some chaotic behaviors of novel attractorare illustrated by bifurcation diagram, Lyapunov-exponent spectrum, phase portraits, etc.In particular, when all equilibria are asymptotically stable or non-hyperblic, system has chaotic attractors, periodic solutions and fixed points coexisting, and has four di?erentattractor evolution process.In Chapter 4, a chaotic system of three-dimensional quadratic autonomous ordinarydi?erential equations by introducing an exponential quadratic term is presented. Thissystem can display a double-scroll chaotic attractor with only two stable equilibria. It istopologically non-equivalent to the original Lorenz and all Lorenz-like systems. Of par-ticular interest is that the chaotic system can generate double-scroll chaotic attractors ina very wide parameter domain with only two stable equilibria. The existence of singu-larly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.Periodic solutions and chaotic attractors can be found when these cycles disappear. Fi-nally, the complicated dynamics are studied by virtue of theoretical analysis, numericalsimulation and Lyapunov exponents spectrum. The obtained results clearly show thatthe chaotic system deserves further detailed investigation.In Chapter 5, a simple one-parameter version of the well-known Lorenz model- Dif-fusionless Lorenz equations (DLE) is discussed, which was obtained in the limit of highRayleigh and Prandtl numbers, physically corresponding to di?usionless convection. Asimple control method is presented to control chaos by using periodic parameter perturba-tion in DLE. Using the generalized Melnikov method, the parameter conditions could beobtained to guide the controlled DLE to a low-periodic motion. Moreover, the existenceconditions of periodic orbits and homoclinic orbits in the system are given. Some resultsof the numerical simulation are also explained clearly by a rigorous analysis.