Results

In Figures 1a-c the results for the baker map (4) are presented: The LE as the analytic function (7) of the parameter (Fig. 1a), the GPER estimated from time series plotted against (Fig. 1b), and the GPER plotted against the LE (Fig. 1c). The latter plot demonstrates that in the case of the chaotic baker map (4) the LE/KSE and the GPER are related by a nonlinear one-to-one function. Considering precision of the GPER estimates, the same conclusion can be drawn for the Lorenz system (6) for the parameter r varying from 34 to 65 (Figs. 1d-f).

The situation is different in the case of the logistic map (5) (Fig. 2): the basic trends in the dependences of the LE and the GPER on the parameter a (Figs. 2a, 2b, respectively) agree, however, there are clear discrepancies larger than the estimation errors and the functional relation between the GPER and the LE/KSE is lost (Fig. 2c).

Comparing the plots in Fig. 1 and Fig. 2, one can see that in Fig. 1 the LE (KSE) varies smoothly with variations of a system parameter, i.e., the systems change only quantitatively remaining in the chaotic regime, while in the case of the logistic map in Fig. 2 bifurcations into periodic states interrupt the regime of chaotic states. Similarly, the Lorenz system (6) with r> 65 enters the bifurcation region (Figs. 3a,b and 3d,e) and deviations from the bijective functional dependence between the KSE/LE and the GPER occur in the LE values related to the bifurcation region (Fig. 3c and 3f).

When a system parameter exactly fits a periodic-state value, the periodic state with zero KSE and negative LE occurs, which is indicated also by a very low (but positive) GPER value (cf. plot a with plot b, or plot d with plot e in Fig. 2 and Fig. 3). The functional relation between the KSE/LE and the GPER, however, is broken not only in periodic states, but apparently also at any point near a bifurcation. Only two bifurcations appeared in Fig. 2a, when the plot was obtained by increasing the parameter a from 3.857 to 4 by step . Using smaller step ( ), seven periodic states were ``hit'' (Fig. 2d). In fact, it is impossible to find any ``bifurcation free'' sub-interval of chaotic states of the logistic map. Note, that there are no bifurcations in the case of the baker map studied in Fig. 1a-c. In the case of the Lorenz system, both situations were observed: A chaotic region with smooth (``bifurcation free'') dependence of the KSE/LE on the parameter r for in which a one-to-one relation between the KSE/LE and the GPER exists (Fig. 1); and for r>65 a regime of chaotic states suddenly interrupted by bifurcations into periodic states, where digressions from the one-to-one functional dependence of the GPER on the KSE/LE occur (Fig. 3).