Using wavelet analysis to investigate synchronization

Wavelet analysis is shown to be a more robust technique than previously used methods in the investigation of synchronization. The highlight of the technique is that it encompasses most of the information obtained by conventional methods into a single picture, while giving a deeper insight into the dynamics of the system. Order parameters derived from continuous wavelet transform coefficients are proposed, which can be used in the quantification of measure synchronization in Hamiltonian systems and identical synchronization in dissipative systems, irrespective of the nature of coupling, the nature of synchronization (complete or partial, quasiperiodic or chaotic), and the number of coupled subsystems.

Figure 1

Time series of the $y$ and $z$ components of the drive-response coupled Lorenz systems (indexed as 1 and 2, respectively). $y_1\left(t\right)$ and $z_1\left(t\right)$ are represented by solid lines, and $y_2\left(t\right)$ and $z_2\left(t\right)$ by dot-dashed lines.

Figure 2

CWT spectra of the $y$ and $z$ components of the drive-response coupled Lorenz systems (indexed as 1 and 2, respectively).

Figure 3

Color plot showing the variation of ${\mathcal{M}}_D$ (represented by colors) together with that of $e\left(t\right)$ for the drive-response coupled Lorenz systems. Values of ${\mathcal{M}}_D\ge 50$ are assigned the highest color index.

Figure 4

CWT spectra of the $x$ and $z$ components of the drive-response coupled Rössler systems (indexed as 1 and 2, respectively).

Figure 5

Color plot showing the variation of ${\mathcal{M}}_D$ (represented by colors) together with that of $e\left(t\right)$ for the drive-response coupled Rössler systems. Values of ${\mathcal{M}}_D\ge 4$ are assigned the highest color index.

Figure 6

Color plot showing the variation of ${\mathcal{M}}_D$ (represented by colors) together with that of $e\left(t\right)$ for the Lorenz systems coupled in the APD scheme. Values of ${\mathcal{M}}_D\ge 50$ are assigned the highest color index.

Figure 7

Variation of ${\mathcal{M}}_{\mathrm{UPO}}$ (represented by colors) at different scales and time, for the coupled $x$-driven Lorenz systems.

Figure 8

Time series of the oscillators $X$ and $Y$ of the nonlinearly coupled two-oscillator HS. (a) Left: $\gamma =0.76(<{\gamma }_{cq})$, right: $\gamma =0.78(>{\gamma }_c)$; (b) left: $\gamma =18.2$ (chaotic unsynchronized), right: $\gamma =30$ (chaotic MS).

Figure 9

CWT spectra of the nonlinearly coupled two-oscillator HS. (a) Left: $\gamma =0.76(<{\gamma }_{cq})$, right: $\gamma =0.78(>{\gamma }_{cq})$; (b) left: $\gamma =18.2$ (chaotic unsynchronized), right: $\gamma =30$ (chaotic MS).

Figure 10

Variation of ${\mathcal{M}}_H(a,\gamma )$, represented by colors, for the nonlinearly coupled two-oscillator HS: (a) QPMS to QP unsynchronized transition; (b) QP and chaotic unsynchronized, and CMS states. Values of ${\mathcal{M}}_H\ge 1.5$ are assigned the highest color index.

Figure 11

Variation of ${\mathcal{M}}_H(a,\gamma )$, represented by colors, for the HS of three nonlinearly coupled oscillators, showing the presence of: (a) $X-Z$ PMS, (b) $X-Y$ PMS for $p_x\left(0\right)=0.3,p_y\left(0\right)=0.35,p_z\left(0\right)=0.4,$ and (c) complete MS for $p_x\left(0\right)=0.3,p_y\left(0\right)=0.5,p_z\left(0\right)=0.7$. Values of ${\mathcal{M}}_H>0.15,0.08$, and 0.4 have been assigned the highest color index in panels (a), (b), and (c), respectively.