Pattern formation in 4:1 resonance of the periodically forced CO oxidation on Pt(110)

Periodically forced oscillatory reaction-diffusion systems may show complex spatiotemporal patterns. At high-frequency resonant forcing, multiple-phase patterns can be found. In the present work, the dynamics of turbulent CO oxidation on Pt(110), forced with the fourth harmonic of the system’s natural frequency, is investigated. Experiments result in subharmonic entrainment, where the system locks to a quarter of the forcing frequency. Cluster patterns are observed, where different parts of the pattern show a defined phase difference. The experimental results are compared with numerical simulations using the realistic Krischer-Eiswirth-Ertl model for catalytic CO oxidation. Using the fourth harmonic of an uncoupled surface element’s natural frequency, we find 3:1 entrainment with three-phase cluster patterns in a wide parameter range of forcing amplitudes and frequency detuning. Numerical analysis of the spatially extended, turbulent system reveals a remarkable upshift of the mean oscillation frequency compared to homogeneous oscillations. Using the fourth harmonic of the most prominent frequency found in the turbulent system results in four-phase patterns with partial or full 4:1 entrainment, depending on the forcing parameters chosen.

Figure 1

Four-phase cluster formation and entrainment at 4:1 resonant forcing. (a) Snapshots of PEEM images (size $300\times 300\mu {\text{m}}^2$). (b) Space-time plots showing the pattern evolution along the $AB$ line indicated in (a). (c) Averaged intensity of a local area of the surface (solid line), indicated by the square in (a), and the forcing signal (dotted line). Scaling is chosen to show both signals overlaid. The reaction parameters are $T=543\text{K}$, $p_{{\text{O}}_2}=1.7\times {10}^{-4}\text{mbar}$, $p_0=5.88\times {10}^{-5}\text{mbar}$, ${\nu }_0=0.72\text{Hz}$, ${\nu }_f=2.88\text{Hz}$, and $a=0.068$.

Figure 2

Phase and amplitude representations of the cluster patterns shown in Fig. 1. (a) Phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase histogram.

Figure 3

Phase and amplitude representations of the cluster pattern, using the KEE model with forcing parameters ${\nu }_f=1.567\text{Hz}$ and $a=0.065$. (a) Phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase histogram.

Figure 4

Spatial distribution (a) and histogram (b) of the most prominent frequency ${\nu }_{\text{max}}$ of the unforced KEE model in the turbulent regime. ${\nu }_{0,single}$ denotes the natural frequency of a single oscillator using the same reaction parameters.

Figure 5

Oscillation frequency of a single oscillator (circles) and mean oscillation frequency of the extended system (squares) using the KEE model with different $p_0$ within the turbulent regime. The end points, where both frequencies coincide, indicate the onset of homogeneous oscillations.

Figure 6

Four-phase cluster and entrainment at 4:1 resonant forcing, KEE model. (a) Snapshots of CO coverage (size $400\times 400\mu {\text{m}}^2$). (b) Space-time plot along the $AB$ line (see top panel). (c) Local CO coverage at point $A$ and forcing signal (dotted line). Model parameters: ${\nu }_f=2.04\text{Hz}$ and $a=0.086$.

Figure 7

Phase and amplitude representations of the cluster patterns shown in Fig. 6. (a) Phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase histogram.

Figure 8

Stroboscopic space-time plot showing the pattern evolution along a vertical line on the surface every fourth forcing cycle. For model parameters, see Fig. 6.

Figure 9

Four-phase cluster and entrainment at 4:1 resonant forcing, KEE model. (a) Snapshots of CO coverage (size $400\times 400\mu {\text{m}}^2$). (b) Space-time plot along the $AB$ line (see top panel). (c) Local CO coverage at point $A$ and forcing signal (dotted line). Model parameters: ${\nu }_f=2\text{Hz}$ and $a=0.086$.

Figure 10

Phase and amplitude representations of the cluster patterns given in Fig. 9. (a) Phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase histogram.