Dynamical transition and chaos for a five‐dimensional Lorenz model

In this paper, we study the dynamical transition and chaos for a five‐dimensional Lorenz system. Based on the eigenvalue analysis, the principle of exchange of stabilities conditions is obtained. By using the dynamical transition theory, three different types of dynamical transition for the five‐dimensional Lorenz system are derived. More precisely, when the control parameter , the system has a continuous transition and bifurcates to two stable steady states. As r further increases, the system undergoes two successive transitions. That is, under some condition, the transition is continuous and a stable limit cycle is bifurcated; if not, the system undergoes a jump transition and an unstable periodic orbit occurs. Especially, the chaotic orbits occur when . Finally, numerical results are given to illustrate our theoretical analysis.