Improving efficiency of the largest Lyapunov exponent’s estimation by its determination from the vector field properties

Abstract

Controlling dynamics of nonlinear systems is one of the most important issues in science and engineering. Thus, there is continuous need to study and develop numerical algorithms of control methods. Among the most frequently applied invariants characterizing different aspects of a systems’ dynamics are Lyapunov exponents, fast Lyapunov index, angles of small deviations, fractal dimension or entropy. There exist many different methods of estimation of these indicators. In this paper, modification of our novel method is presented. We have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. Results of efficiency measurements for typical mechanical, electrical and random systems were discussed. We have proved that discussed modification introduced to our method makes the LLE estimation 17–53% faster than using classical algorithms. In addition, unlike the results presented in our previous publication, an improvement in performance was achieved for each of the analyzed cases. As such, the new approach lends to prospective application of LLE not only in dynamical systems' stability investigations, but also in real-time control of systems since the basic calculations and fast, effective method of LLE estimation can be applied even in simple microcontrollers. Our approach could be also applied in investigations of vector field properties, global stability or basins of attraction analyses, allowing for huge time savings.

Effect of a visual dual task on postural stability—A comparative study using linear and nonlinear methods

“…Higher values of LE indicate more divergence and instability. The stability of a dynamical system is inversely correlated with LE, so system stability increases as the LE decreases 32,33 . For this reason, it seems logical that patients suffering from stroke have a small Lyapunov spectrum and higher dynamic position stability compared with a healthy population.…”

Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method

“…Finally, method M5 has the best average efficiency equal to 0.780, so M5 saves on average about 22% of the computation time. The results for M1, M2, and M5 will be marginally inferior for more complex systems, as shown in [ 58 ]. However, they will be invariably superior to M3.…”

“…The last case is the application of our effective method presented in [ 58 ]. In this case, the value given by Equation (9) is calculated from the projection of the vector onto the direction of not normalized vector according to: …”

“…We investigated the stability of continuous systems [ 51 ] and systems with discontinuities [ 9 , 28 ], control system’s optimization [ 52 ], synchronization phenomena of energy flow [ 48 , 53 , 54 ] and chaos-based control of energy flow [ 55 , 56 , 57 ]. We have also investigated efficiency of our novel method of Lyapunov spectrum estimation in [ 58 ] and showed that it allows for significant computation time savings.…”

PI/PID controller stabilizing sets of uncertain nonlinear systems: an efficient surrogate model-based approach

“…Thus, the PID controller space can be explored and classified into stabilizing or non-stabilizing. Recent efforts have been made to speed up the computation of the LLE in nonlinear systems, e.g., methods based on non-logarithm computations [17] have been applied to the analysis of the regulation error of closed-loop systems [7] lead-…”