Limit Cycles from Perturbed Center on the Invariant Algebraic Surface of Unified Lorenz-Type System

For a three-dimensional chaotic system, little seems to be known about the perturbation of invariant algebraic surface and the center on this surface. This question is very interesting and worth investigating. This paper is devoted to analyzing the limit cycles from perturbed center (trivial and nontrivial equilibria) on the invariant algebraic surface of the unified Lorenz-type system (ULTS), which contains some common chaotic systems as its particular cases. First, based on the parameter-dependent center manifold, we obtain the approximate two-dimensional center manifold from the perturbation of invariant algebraic surface, as well as the two-dimensional system on this center manifold. Second, by applying the averaging method of third order to the above two-dimensional system, we show that under suitable perturbation of parameters of the ULTS, there is one limit cycle bifurcating from the perturbed center on the invariant algebraic surface of the ULTS, and the stability of this limit cycle is determined as well. By using the averaging method of fourth order, we show the same results with the averaging method of third order. Finally, numerical simulation is used to verify the theoretical analyses.