Slow Manifolds of Lorenz-Haken System and its Application

In this paper, the slow manifolds of Lorenz-Haken system are discussed. By using three different methods, associated with the slow manifold equations of the L-H system are obtained. Firstly, the slow manifold equation of the nonlinear chaotic dynamic system is obtained by considering that the slow manifold is locally defined by a plane orthogonal to the tangent system’s left fast eigenvector. On the condition that zT λ1 (X) · Ẋ = 0, the slow manifold equation of the L-H system is built. And secondly, another method consists of defining the slow manifold as the surface generated by the two slow eigenvectors associated with the two eigenvalues λ2(X) and λ3(X) of J(X). Another slow manifold equation of the L-H system is obtained. Thirdly, by geometric singular perturbation theory, we give the new slow manifold equation that is concrete and terse of the L-H system. Finally, we apply our results to derive the slow manifold equations of some known classical chaotic systems, such as the chua’s system, the Lorenz system, the Chen’s system and Lü’s system, and analyze the dynamical behavior of the S-FADS.