Abstract:

The presence of homoclinic tangencies and homoclinic intersection makes it difficult if not impossible, to denoise or shadow the trajectory of a non-hyperbolic nonlinear system. By exploiting the properties of chaotic systems, a new highly-fast-but-stable algorithm is developed which improves both on the speed of convergence of the Gradient Descent algorithm and the inadequacy of stability of the Newton-Raphson algorithm. With this algorithm machine precision could be obtained for the noisy time series of non-hyperbolic nonlinear systems. Different from former methods, this method first computes the local stable and unstable manifolds of the noisy trajectory, and then determines the locations of the homoclinic tangencies. Thus the effects of the homoclinic tangencies on the algorithm can be reduced to a great extent. Different from those methods which take it for granted that the failure of denoising algorithms is related the homoclinic tangencies only, experiments in this article demonstrate a quantitative correlation between the minimal distance of homoclinic intersections and the standard variance of noise. Thus the probability is great that the algorithm converges to the true trajectory and this strategy could suggest a heuristic approach to similar methods.

Key words: non-hyperbolic nonlinear system, noise reduction, Newton-Raphson algorithm, gradient descent, homoclinic tangency, homoclinic intersection, parameter estimation