How to test for partially predictable chaos

Abstract

For a chaotic system pairs of initially close-by trajectories become eventually fully uncorrelated on the attracting set. This process of decorrelation can split into an initial exponential decrease and a subsequent diffusive process on the chaotic attractor causing the final loss of predictability. Both processes can be either of the same or of very different time scales. In the latter case the two trajectories linger within a finite but small distance (with respect to the overall extent of the attractor) for exceedingly long times and remain partially predictable. Standard tests for chaos widely use inter-orbital correlations as an indicator. However, testing partially predictable chaos yields mostly ambiguous results, as this type of chaos is characterized by attractors of fractally broadened braids. For a resolution we introduce a novel 0–1 indicator for chaos based on the cross-distance scaling of pairs of initially close trajectories. This test robustly discriminates chaos, including partially predictable chaos, from laminar flow. Additionally using the finite time cross-correlation of pairs of initially close trajectories, we are able to identify laminar flow as well as strong and partially predictable chaos in a 0–1 manner solely from the properties of pairs of trajectories.

Determining the chaotic behaviour of copper prices in the long-term using annual price data

“…3). Sensitivity to the initial conditions is asserted by the rapid exponential divergence of initially close trajectories through time (Becks et al, 2005;Chen and Aihara, 1995;Chen et al, 2016;Cvitanović et al, 2013;Graham et al, 2007;Kodba et al, 2005;Navarro-Urrios et al, 2017;Panas and Ninni, 2000;Perc, 2006;Pincus, 1995;Raffalt et al, 2017;Reynolds et al, 2016;Rosenstein et al, 1993;Savi, 2005;Wernecke et al, 2017). Thus, if at least one positive λ exists, the chaotic behaviour of the system is presumed (Supplementary Table 2).…”

“…The Lyapunov exponent λ has been widely used to examine the sensitivity to initial conditions and detect the presence of chaotic behaviour (Becks et al, 2005;Chen et 2017; Panas, 2001;Panas and Ninni, 2000;Perc, 2006;Pincus, 1995;Raffalt et al, 2017;Reynolds et al, 2016;Rosenstein et al, 1993;Savi, 2005;Wernecke et al, 2017;Zhong et al, 2017) as it quantifies the exponential divergence of nearby trajectories between initial close space-states ( Supplementary Fig. 3).…”

“…Sensitivity to changes in the initial conditions is the most important characteristic of chaotic behaviour in complex dynamic systems (Showalter and Hamilton, 2015). It is demonstrated by the fast exponential growth of the divergence between two near initial trajectories of the system across time known as strange attractors (Azar and Vaidyanathan, 2015, p 10;Becks et al, 2005;Guegan, 2009;Hegger et al, 1999;Kodba et al, 2005;Navarro-Urrios et al, 2017;Panas, 2001;Panas and Ninni, 2000;Reynolds et al, 2016;Rosenstein et al, 1993;Savi, 2005;Vlad et al, 2010;Wernecke et al, 2017;Yamamoto, 1999). Strange attractors contain the state of the system and plot its motion in the form of infinite non-periodic unique orbits that never close themselves.…”

“…They occur in parallel sets with a gap between any two members of the set. The most useful tool for examining the sensitivity to initial conditions and detecting the presence of chaotic behaviour has been the widely documented largest Lyapunov exponent (λ) method (Becks et al, 2005;Chen et al, 2016;Gaspard et al, 1998;Gottwald, 2009;Kodba et al, 2005;Navarro-Urrios et al, 2017;Panas, 2001;Panas and Ninni, 2000;Perc, 2006;Reynolds et al, 2016;Rosenstein et al, 1993;Savi, 2005;Showalter and Hamilton, 2015;Wernecke et al, 2017;Zhong et al, 2017). According to a large body of literature,, large datasets are required to assess chaotic behaviour, but this is often not achievable due to cost, technical or temporal limitations (Abraham et al, 1986;Kantz, 1994;Rosenstein et al, 1993).…”

Entrenched time delays versus accelerating opinion dynamics: are advanced democracies inherently unstable?

“…It is of interest to examine, for subcritical β < β c , the time scale T λ needed to close in to the equilibrium state (D, V ) = (1, 1), which is given by the inverse of the largest Lyapunov exponent of the fixed point [26]. In Figure 4 we present alternatively the results of a numerical experiment simulating the recovery from an external shock.…”