Detection of the type of intermittency using characteristic patterns in recurrence plots

One of the common routes to chaos is intermittency. Identification of the intermittency type is usually made using the properties of the probability distribution of laminar phases and of the average length of the laminar phases. Both have a statistical character and to obtain them a long time series has to be examined. Here, we present a recurrence plot method applicable to the analysis of short time series and through which the type of intermittency may be identified. The three types of intermittency introduced by Pomeau and Manneville and a chaos-chaos intermittency induced by interior crisis were examined. The identification of the type of intermittency is equivalent to the identification of the bifurcation associated with it. Our result seems particularly interesting as our method allows the analysis of short time series. The effect of the measurement noise on the effectiveness of the method is also discussed. An application of the method to the detection of type I intermittency in measured heart rate variability data is discussed.

Figure 1

Examples of the recurrence plots (lower panels) for different signals together with the raw data (upper panels): (a) a periodic signal obtained from the logistic equation for the control parameter $r=3.829$; (b) a chaotic signal obtained from the logistic equation for $r=3.9999$; (c) white noise. Note that, in the RP for the deterministic chaotic signal, short lines parallel to the main diagonal are present. Such lines do not occur in the RP of the white noise.

Figure 2

The RP calculated for the time series obtained from the generic model of type I intermittency—Eq. (5). This model describes only the laminar phase of type I intermittency. For this type of intermittency, the image of the laminar phase is a black square adjacent to the main diagonal of the RP. The laminar phase ends abruptly—here at about $i=400$ (chaotic burst—not shown here).

Figure 3

RP calculated for a time series obtained from the generic model of type III intermittency—Eq. (6) (lower graph) shown with the original data (upper graph). A black square with rounded upper right corner is characteristic for the laminar phase of this type of intermittency. The elongated of the upper right corner of the pattern consists of lines parallel to the main diagonal. The distance between these lines is equal to 1—see the magnification inset.

Figure 4

RP calculated for a time series obtained from the generic model of type II intermittency—Eq. (7) (lower graphs) shown with the original data (upper graphs). The left column depicts results for the $x$ variable while the right—for the $y$ variable of the model. For the generic model, a black square with perforated upper and right edges is the characteristic shape for the laminar phase of this type of intermittency. The perforation structure is due to the periodic behavior at the end of the laminar phase and is similar to the pattern obtained for harmonic functions.

Figure 5

Recurrence plots and the time series calculated using the logistic equation for the control parameter $r$ smaller than $r_c$. (a) $r=3.82839$; (b) $r=3.828425$.

Figure 6

Enlargement of the RP shown in Fig. 5a. When a laminar phase occurs, lines parallel to the main diagonal, forming squares and rectangles, are present in the RP.

Figure 7

Recurrence plots of the signals obtained from the third iterate of the logistic equation for the same control parameter values as in Fig. 5: (a) $r=3.82839$; (b) $r=3.828425$.

Figure 8

Schematic illustrating the origin of the recurrence squares and rectangles for type I intermittency. Appearance of squares 8 and 9 is a consequence of the occurrence of the squares 1 and 3—images of the laminar phases of the same length. The rectangles 4 and 6 are a result of the occurrence of the squares 2 and 3 each with a different length of the side, similarly, rectangles 5 and 7.

Figure 9

The histograms of $\sqrt{F_a}$ for the time series with type I intermittency: part (a) $r=3.82839$, the number of laminar phases was 329; part (b) $r=3.828425$, the number of laminar phases—68. The maximum lengths of the laminar phases obtained analytically (solid curve) using Eq. (3.15) in Hirsch et al. [21] were equal to 165 and 686, respectively.

Figure 10

The distribution of $\sqrt{F_a}$ obtained from the RP for the simple model of the type I intermittency for $x_0=0.07$ and $a=0.001$. Black line: analytical result—according to Eq. (3.15) in Hirsch et al. [21].

Figure 11

The distribution of the parameter $F_b$ for type I intermittency: (a) $r=3.82839$, the number of laminar phases in the time series is 329; (b) $r=3.828425$, the number of laminar phases—68.

Figure 12

Trajectory during laminar phase of type II intermittency obtained from Eq. (10) with the following parameters: $b=0.14$, $\gamma =0.02$, $ϵ=0.001$, and $\mu =0.0001$. The data for the $x$ and $y$ variables are shown in the left panels. Each thin line and thick line, respectively, depict the data for separate laminar phases which are interleaved. The right panel depicts the shape of the trajectories with the arrows marking the direction of motion.

Figure 13

The RPs of a single laminar phase of type II intermittency obtained from Eq. (10). The parameters of the RP were set to $m=2$, $\tau =1$, and $ϵ=0.1$ in the left RP and $ϵ=0.01$ in the right RP. In the right RP, the white regions in the kitelike elongation of the upper right corner of the square—characteristic for type II intermittency is visible—for comparison see Fig. 4 and discussion in the text.

Figure 14

(a) RP $\left(m=2,\tau =1,ϵ=0.1\right)$ and the time series for type II intermittency obtained from Eq. (10) (parameters: $b=0.14$, $\gamma =0.02$, $ϵ=0.001$, and $\mu =0.0001$). (b) Histogram of $\sqrt{F_a}$, the number of laminar phases—213 [black line—the analytical distribution of the laminar phases for the type II intermittency—Eq. (11)]; (c) the probability distribution of $F_b$.

Figure 15

(a) RP and data for type III intermittency (parameters: $b=0.14$, $\gamma =0.02$, and $\mu =0.005$); RP parameters: $m=2$, $\tau =1$, and $ϵ=0.1$. (b) Histogram of the $\sqrt{F_a}$; the number of laminar phases—55 [black line—the analytical distribution of the laminar phases for the type III intermittency—Eq. (13)]; (c) the probability distribution of $F_b$.

Figure 16

The RP for chaos-chaos intermittency induced by an interior crisis. The time series was obtained from the third iterate of the logistic equation for control parameter $r=3.857$.

Figure 17

Magnification of the RP obtained from the first iterate of the logistic equation for control parameter $r=3.857$—an example of a laminar phase.

Figure 18

(a) The distribution of $\sqrt{F_a}$ obtained from chaos-chaos intermittency (logistic equation; $r=3.857$). Solid line—distribution of the laminar phases for type II intermittency; dashed line—distribution of the laminar phases for type III intermittency. (b) The probability distribution of $F_b$.

Figure 19

The characteristic shape of the trajectory for type II intermittency is visible only in the stroboscopic section—(b) of the raw data obtained from Eq. (14)—(a) (see [23] for details).

Figure 20

RP of systems with continuous time: part (a)—type II intermittency [Eq. (14)] and part (b)—type III intermittency [Eq. (15)]. For type II intermittency, the characteristic kitelike shape is visible. The distances between the lines forming the elongated upper right corner of the pattern are due to the sampling rate (integration time step). For type III intermittency, the shape characteristic for it is also visible—a square with a rounded upper right corner and perforated outer edges.

Figure 21

The recurrence plots of four different types of intermittencies with measurement noise added to the time series. The noise level was 10%. (a) Type I intermittency $r=3.828425$; (b) type II intermittency; (c) type III intermittency; (d) chaos-chaos intermittency $r=3.857$.

Figure 22

The distribution of $\sqrt{F_a}$ for different types of intermittencies obtained from the RPs presented in Fig. 21. Noise level—10%. (a) Type I intermittency $r=3.828425$ [solid curve—the analytical distribution of the laminar phases for the simple model of type I intermittency—Eq. (9)]; (b) type II intermittency [solid curve—the analytical distribution of the laminar phases for the type II intermittency—Eq. (11)]; (c) type III intermittency [solid curve—the analytical distribution of the laminar phases for the type III intermittency—Eq. (13)]; (d) chaos-chaos intermittency $r=3.857$ (solid line—distribution of the laminar phases for type II intermittency; dashed line—distribution of the laminar phases for type III intermittency).

Figure 23

The distribution of $F_b$ for different types of intermittencies with measurement noise added to the time series. Noise level—10%. (a) Type I intermittency $r=3.828425$; (b) type II intermittency; (c) type III intermittency; (d) chaos-chaos intermittency $r=3.857$.

Figure 24

The comparison between type I intermittency without (a), and with (b) measurement noise, with chaos-chaos intermittency without (c) and with (d) noise for the first iterate of the logistic equation. For type I intermittency, lines parallel to the main diagonal are visible. In case of noise these lines have gaps. For chaos-chaos intermittency the lines parallel to the main diagonal are also visible but even without noise these lines have gaps. For chaos-chaos intermittency parts of the lines are shifted with respect to each other. The gray rectangles mark corresponding patterns in the two cases.

Figure 25

Application of the method to heart rate variability data. Part (a) upper panel: the RP of a time series extracted from a 24 h recording of the heart rate (morning hours). For legibility, only a 1000 data-point long fragment of the RP is presented; the complete data set analyzed by the extended recurrence plot analysis was $20000$ data points long. Lower panel: the recurrence plot of the data in the upper panel obtained with the parameters $m=6$, $\tau =1$, and $ϵ=300$; part (b): the distribution of the parameter $\sqrt{F_a}$ for the complete data set; part (c): the distribution of $F_b$ for the complete data set. The total number of laminar phases was 1038.

Figure 26

A comparison of the results presented in Fig. 25b with results presented in Żebrowski [25]. Light gray—results from Żebrowski [25] (for this part of the figure the $y$ axis is on the right side of the plot and the $x$ axis—at the top); dark gray—results presented in Fig. 25b; black line—distribution of the laminar phases for the simple model of type I intermittency [Eq. (3.15) in Hirsch et al. [21]].

Figure 27

(a) A fragment of the RP of the HRV data in Fig. 18—black squares are the signature of type I intermittency. (b) The raw data used to obtain (a). The detected laminar phases are marked by dashed lines. (c) The first return map for the selected parts of the HRV data in part (b)—two passages through a single intermittency channel are visible. The arrows depict the direction of motion.