Estimation of shadowing trajectory of the nonlinear system from a noisy scalar series

The presence of homoclinic tangencies and homoclinic intersection makes it very difficult, sometimes even impossible, to estimate the shadowing trajectory of the non-hyperbolic nonlinear system. A new algorithm for shadowing the non-hyperbolic nonlinear system is presented in this paper, which is geometrical in nature and tries to exploit the properties of the chaotic systems. Different from former methods, this method computes the stable and unstable manifolds of the noisy trajectory firstly, and then the locations of the homoclinic tangencies are determined. Thus the effects of the homoclinic tangencies on the algorithm can be decreased to a great extent, and the length of the shadowing trajectories are estimated by the locations of these homoclinic tangencies. Also different from those methods which take it as granted that the mechanism of failure of shadowing algorithms is related with the homoclinic tangencies only, experiments in this paper demonstrate a quantitative relation between the minimal distance of homoclinic intersections and the amplitude of noise. Thus the probability that the algorithm converges to the true trajectory can be boosted efficiently, and without any doubts, this strategy can be as a heuristic approach to other methods.