DUAL VARIABLE VARIATIONAL METHOD FOR NONLINEAR LEAST SQUARES SHADOWING PROBLEM

The least squares shadowing (LSS) method, to compute the shadowing trajectories of dynamical systems, has been presented in recent years. The ill-posed initial value problem within the conventional sensitivity analysis algorithm for nonlinear systems can be effectively avoided based on the LSS method, which therefore has significant applications in the sensitivity analysis of chaotic systems. To achieve nonlinear LSS problem, it will be firstly represented as a nonlinear optimal control problem subject to constraints. By introducing the costate variables, the Hamiltonian function is represented depending on state and costate variables. The integral time of objective function is then discretized, and the state variables at ends of time interval are taken as independent variables. Then approximate the state and costate variables in the time interval using the Lagrange polynomials. This problem is finally transformed into solving nonlinear equations via dual variable variation principle. The linearizing process is avoided for the proposed algorithm, and then the errors, caused by the complex linearizing process, are also reduced, which provides affinew solution for solving the shadowing problem. Two state trajectories with designing parameters slightly changed can be obtained by LSS problem, and then the sensitivity of the nonlinear system is calculated from the two trajectories. Van der Pol oscillator as a numerical example shows that this method is effective for the LSS problem and sensitivity analysis of nonlinear systems.