Novel gradient calculation method for the largest Lyapunov exponent of chaotic systems

A novel method is presented to calculate the sensitivity gradients of the largest Lyapunov exponent (LLE) in dynamical systems. After the elimination of the discontinuity of state perturbation vector, the augmented system of differentiation equations is constructed to govern the time evolution of the LLE. To overcome the ill-posed property of the sensitivity problem associated with the augmented differentiation system, the improved least squares shadowing approach is developed. The simple algebraic formula depending on the final state value of the Lagrange multipliers is deduced from the discretization representation for the first-order optimal conditions of the improved least squares shadowing formulation. The LU factorization technique is introduced to solve the set of discretized linear equations, resulting in a better performance of the convergence problem and computational expense. The correctness and effectiveness of the present approaches are validated.