Estimating the dimension of the attractor for the complex Ginzburg-Landau system

In this paper, the upper bound of the dimension of the global attractor associated with the four-dimensional cubic complex Ginzburg-Landau system is derived using the Lyapunov exponent method. First, by employing the Faedo-Galerkin method and energy estimation, the existence and uniqueness of solutions under the variational form of the system are established. Subsequently, a specific quantity is constructed and transformed into an inequality, from which the upper bound of the fractal dimension corresponding to the global attractor of the dynamical system is obtained.