Testing for chaos in deterministic systems with noise

Abstract

Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent.In this paper, we investigate the capability of the test to cope with moderate amounts of noisy data. Comparisons are made between an improved version of our test and both the "tangent space" and "direct method" for computing the maximal Lyapunov exponent. The evidence of numerical experiments, ranging from the logistic map to an eight-dimensional Lorenz system of differential equations (the Lorenz 96 system), suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods.

Regular and chaotic vibration in a piezoelectric energy harvester

“…This can be done by means of the 0-1 test [30,31] which is especially simple in systems with many degrees-of-freedom. The corresponding analysis and the estimation results will be presented in the next section.…”

“…The '0-1 test', invented by Gottwald and Melbourne [30,31], can be applied for any system of finite dimension to identify chaotic dynamics, but it is based on the statistical properties of a single coordinate only.…”

New Approach and Microsystem Technology of Advanced Non-Linear Analysis and Modelling Chaotic Environmental Radioactivity Dynamics

“…The first step of the whole methodology begins with the use of a known test for the presence of chaos in the system noise, which was proposed Gottwald and Melbourne [7]. Its main idea boils down to the choice of some constant c, rather, several values of c, which further define the value (with accounting for Eq.…”

“…Certainly, these models can't principally used in treating the chaotic nature systems (see [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). Although the use of methods of a chaos theory establishes certain fundamental limitation on the long-term predictions, however, as has been shown in a series of the papers (see, for example, [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]), these methods can be successfully applied to a shortor medium-term forecasting. In Ref.…”

Efficient time-series detection of the strong stochasticity threshold in Fermi-Pasta-Ulam oscillator lattices

“…It has been successfully applied to both numerically obtained [4][5][6][7][8] as well as to simple experimental data [9]. However, in a previous work [10] we have presented evidence that, when applied to time series stemming from nonhomogeneous, i.e dissimilar mass, many-degrees-offreedom systems, the test has serious limitations as an efficient chaos detector for a large system size N .…”