Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative dot product

Nowadays, the largest Lyapunov exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. The new method of the LLE estimation is presented in this paper. The method (LLEDP) applies the perturbation vector and its derivative dot product to calculate Lyapunov exponent in the direction of disturbance. Value of this exponent is the LLE. The theoretical improvement was introduced. Results of the numerical simulations were shown and compared with the Stefanski method (Stefanski in Chaos Solitons Fractals 11(15):2443–2451, 2000; Stefanski and Kapitaniak in Chaos Solitons Fractals 15:233–244, 2003; Stefanski et al. in Chaos Solitons Fractals 23:1651–1659, 2005; Stefanski in J. Theor. Appl. Mech. 46:665–678, 2008). Investigations of the Duffing oscillators with external excitation and three Duffing oscillators coupled in ring scheme without external excitation were made with use of the presented method. Fast time LLE estimation results convergence was shown.