Figure 1.
Experimental setup for nickel electrodissolution.
a: Schematic diagram of electrochemical cell for a three locally coupled elements. Ref: Hg/HgSO4/Sat.K2SO4 reference electrode, CE: Pt electrode, Rind: Individual resistors, R: Coupling resistors. b: Coupling topology induced by the cross resistors.
Figure 2.
Spatially organized partial synchronization of six smooth oscillators in extended triangle network.
Left panel: Current time series of the electrodes: the core oscillators (2–4) are phase locked while the peripheral oscillators (1,5,6) are not synchronized. Right panel: coupling topology and schematic of synchronization pattern. V=1095 mV, Rind= 1200 Ω, R=40 kΩ.
Figure 3.
Spatially organized partial synchronization induced by ‘chimera’ mechanism of in nonlocally coupled regular (NLR) network.
a: Coupling topology of the NLR network. (Each oscillator is coupled to 14 nearest neighbors.) b: Snapshot of currents of the electrodes: the core oscillators (1–4,17–20) have similar currents. c: Frequencies of the oscillations of the elements. Open circles: Natural frequencies measured without coupling. Black circles: frequencies of the coupled oscillators. d. Global mean field order parameter vs time for the chimera dynamics. e: Grayscale plot of phase of oscillators relative to mean field phase. V=1094 mV, Rind= 1000 Ω, R= 499 kΩ, C=4.7 μΦ.
Figure 4.
Cluster dynamics of relaxation oscillators on extended triangle and NLR networks.
Left panels: current vs time of the electrodes. Right panels: coupling topology and schematic of synchronization patterns. a: (3–3) two-cluster state in which the synchronized center (2–4) oscillators (black) are in anti-phase configuration to the periphery (1,5,6) oscillators (red). V=1290 mV, Rind= 1200 Ω, R= 1500 Ω. b: (2,4) two-cluster states in anti-phase synchronization. V=1300 mV, Rind=1200 Ω, R= 1000 Ω. c: (10–10) condensed two-cluster cluster state with the NLR network. Oscillators in one half of the circle (red) are anti-phase synchronized with oscillators in the other half (black). V=1335 mV, Rind=1200 Ω, R=5000 Ω. d: (10–10) two-cluster condensed state with combined resistive-capacitive coupling. V=1190 mV, Rind= 1000 Ω, C=33 μΦ. e: (10–10) flip-flop two-cluster synchronized state in which the red elements are anti-phase synchronized with black elements. V=1190 mV, Rind= 1000 Ω, C=33 μΦ.
Figure 5.
Identical synchronization of chaotic oscillators in small networks.
a: Current time series of two chaotic oscillators without any coupling. V=1295 mV, Rind= 1395 Ω. b: Current time series of two identically synchronized oscillators with strong coupling. V=1295 mV, Rind=1395 Ω, R=900 Ω. c: Order parameter (r) vs coupling strength (K) for various networks. d: Order parameter vs rescaled coupling strength (κ). e. Schematics of network topologies. For each network, the experimental conditions were set to exhibit chaotic behavior: V=1295-1345 mV, Rind= 1350-1520 Ω.
Figure 6.
Cluster dynamics of chaotic oscillators.
Left panels (1–1,2): three cluster dynamics in an extended triangle configuration with four oscillators. V=1335 mV, Rind= 1390 Ω, R=800 Ω. Right panels (1–1,2): three cluster dynamics in a star configuration with four oscillators. V=1335 mV, Rind=1380-1460 Ω, R=750 Ω. Top row: schematic of coupling topology and synchronization pattern along with reconstructed phase space showing the chaotic attractor of one oscillator and a snapshot of the elements in the state space. b-e and g-j: distances of phase points as a function of time for the corresponding cluster states. In panels a and f identical symbols (circle, square, triangle) indicate clustered elements.