Recurrence analysis of strange nonchaotic dynamics

We present methods to detect the transitions from quasiperiodic to chaotic motion via strange nonchaotic attractors (SNAs). These procedures are based on the time needed by the system to recur to a previously visited state and a quantification of the synchronization of trajectories on SNAs. The applicability of these techniques is demonstrated by detecting the transition to SNAs or the transition from SNAs to chaos in representative quasiperiodically forced discrete maps. The fractalization transition to SNAs—for which most existing diagnostics are inadequate—is clearly detected by recurrence analysis. These methods are robust to additive noise, and thus can be used in analyzing experimental time series.

Figure 1

(Color online) Phase diagram of the quasiperiodically forced cubic map [Eq. (9)], obtained using $N_{\mathit{MPRT}}$ [Eq. (3)]. $1\mathrm{T}$ and $2\mathrm{T}$ correspond to tori of period 1 and 2, respectively. GF corresponds to the region where the gradual fractalization of the torus occurs. HH represents the region where the SNA is created through the Heagy Hammel route. S1 and S3 denote regions where the SNA appears through type-I and type-III intermittencies, respectively. IC denotes the region where SNAs are created through crisis-induced intermittency. C1 and C2 correspond to chaotic regions. For the parameter values in the region marked D, the trajectories escape to infinity.

Figure 2

(Color online) Phase diagram of the quasiperiodically forced logistic map [Eq. (8)], obtained using $N_{\mathit{MPRT}}$ [Eq. (3)].

Figure 3

(Color online) Transition from a doubled torus to SNA through a HH mechanism in the cubic map [Eq. (9)]. (a) behavior of $T_{\mathit{MRT}}$; (b) behavior of $N_{\mathit{MPRT}}$; (c) variance of $T_{\mathit{MRT}}$; and (d) variance of $N_{\mathit{MPRT}}$. The critical value is $A\approx 1.88697$.

Figure 4

(Color online) Type-I intermittency route in the logistic map [Eq. (8)]. (a) Behavior of $T_{\mathit{MRT}}$; (b) behavior of $N_{\mathit{MPRT}}$; (c) variance of $T_{\mathit{MRT}}$; and (d) variance of $N_{\mathit{MPRT}}$. The critical value is ${\alpha }_c\approx 3.4058088$.

Figure 5

(Color online) Robustness of the recurrence time measures against noise for the HH route in the cubic map [Eq. (9)]. (a) and (c) $T_{\mathit{MRT}}$ and $N_{\mathit{MPRT}}$ for dynamical noise of amplitude ${10}^{-4}$; and (b) and (d) $T_{\mathit{MRT}}$ and $N_{\mathit{MPRT}}$ for observational noise of amplitude 0.01. The threshold used $\delta =0.001$ and $N=10000$ normalized data points.

Figure 6

(Color online) Fractalization route in the logistic map [Eq. (10)]. (a) Variance of $T_{\mathit{MRT}}$ and (b) Variance of $N_{\mathit{MPRT}}$; variances computed using $\delta =0.05$, $N=750000$ data points as the whole trajectory and $N=3000$ as the length of each segment. The critical value of the bifurcation parameter is ${\epsilon }_c\approx 0.1553$.

Figure 7

(Color online) Fractalization route in the Hénon map [Eq. (11)]. (a) Variance of $T_{\mathit{MRT}}$ and (b) variance of $N_{\mathit{MPRT}}$. Variances computed using $\delta =0.05,750000$ data points and 2500 as the length of each segment. The variances increase suddenly at the bifurcation parameter $b\approx 0.69$.

Figure 8

CRP of the forced logistic map [Eq. (8)]: (a) in the SNA regime with $\alpha =3.32$, ${\epsilon }^{\prime }=0.595$, and (c) in the chaotic regime with $\alpha =3.33$, ${\epsilon }^{\prime }=0.595$. Due to synchronization, the CRP and the ordinary RP are identical in the SNA regime, whereas this is not the case in the chaotic regime. In (b) and (d) only the main diagonals for the SNAs and chaotic attractors are shown. The threshold for the computation of the CRP is $\delta =0.2\sigma$ where $\sigma$ is the average of the individual standard deviations of the two time series. The embedding dimension is taken to be $m=3$.

Figure 9

(Color online) Comparison of the Lyapunov exponent $\left(\lambda \right)$ and $D_{\mathit{ET}}$ for the CRPs on the main diagonal line. (a) $\lambda$ for Eq. (8) with ${\epsilon }^{\prime }=0.595$, (b) the corresponding $D_{\mathit{ET}}$, (c) $\lambda$ for Eq. (8) with ${\epsilon }^{\prime }=1$, (d) the corresponding $D_{\mathit{ET}}$, (e) ${\Lambda }_T$, ${\Lambda }_x$ and ${\lambda }_x$ for Eq. (12) and (f) the corresponding $D_{\mathit{ET}}$. A $D_{\mathit{ET}}$ value of unity corresponds to SNAs dynamics, and a drop in the value of $D_{\mathit{ET}}$ to lower values indicates the SNAs to chaos transition. In all cases studied here, we use $\delta =0.2\sigma$ and embedding dimension is $m=3$.

Figure 10

(Color online) $D_{\mathit{ET}}$ (averaged over 200 stochastic runs) for Eq. (8) with increasing noise amplitude $\sigma$. The line with diamonds corresponds to the HH route at $\alpha =3.32$ and ${\epsilon }^{\prime }=0.595$. The line with filled circles corresponds to the intermittency route at $\alpha =3.385$ and ${\epsilon }^{\prime }=0.8$. We use $\delta =0.2\sigma$ and the embedding dimension is $m=3$.