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Fig. 4 | Progress in Earth and Planetary Science

Fig. 4

From: Nonlinear dynamical analysis of GNSS data: quantification, precursors and synchronisation

Fig. 4

Cross and joint recurrence plots. a Cross recurrence plot between sin(x) (upper trace) and cos(x) (lower trace). High recurrence is represented by the dark blue (straight) lines of synchronisation at 45o to the horizontal axis. b Cross recurrence plot between sin(x) (upper trace) and sin(2x) (lower trace). High recurrence is represented by the dark blue (straight) lines of synchronisation at tan−1(1/2) = 26.6° to the horizontal axis. The ratio ½ is the ratio of the frequencies of the two signals. c Cross recurrence plot between sin(x) (upper trace) and sin(x) + sin(5x2) (lower trace). High recurrence is represented by the (curved) dark blue lines of synchronisation. The local slope of the line of synchronisation is the arctangent of the ratios of the local frequencies of the signals. d Cross recurrence between two quasi-periodic signals: \( y=\sin (x)+\sin \left(\sqrt{2}x\right) \) (upper frame), y = sin(x) + sin(πx)(lower frame). e Cross recurrence plot between signals from two logistic equations, xn + 1 = αxn(1 − xn), with α = 3.7 (upper frame) and α = 3.8 (lower frame). The cross recurrence plot is dominated by chaotic transitions (vertical and horizontal lines) but regions of periodic behaviour occur characterised by equally spaced diagonal lines. f Cross recurrence plot for y = sin(x) (upper frame) and y = sin(20x) (lower frame). The large frequency difference between the two signals means that the LOSs are almost vertical. g Joint recurrence plot between sin(x) (upper trace) and cos(x) (lower trace). High synchronisation of recurrences is represented by the dark blue regions of synchronisation. h Joint recurrence plot between sin(x) (upper trace) and sin(2x) (lower trace). Lines of high synchronisation between recurrences are represented by the dark blue lines. i Joint recurrence plot between sin(x) (upper trace) and sin(x) + sin(5x2) (lower trace). Again high synchronisation of recurrences is represented by the dark blue lines

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