1. Synchronization of large number of nonidentical electrochemical oscillators of S-NDR type
Adrian Birzu
University of A. I. Cuza, Iaşi, Romania
Vilmos Gáspár
University of Debrecen, Debrecen, Hungary
1st Workshop, Haslev, Denmark, May 2-5, 2007
Hungarian Research Found 60417, Romanian-Hungarian S&T Programme

2. Motivation
During the last few decades, the motivation for studying nonlinear chemical dynamics has originated – partially – from our hope to model similar behavior of living systems (rhythm of heart, neural activity in the brain, etc.).
However, it is characteristics of biological tissues that
they are built of large number of cells
global and/or local coupling of the units must play an essential role in generating the collective dynamics
With the present project we plan to
investigate nonlinear dynamics of coupled chemical systems,
learn about the general laws governing the emergence of coherent dynamics
develop algorithms for achieving synchronized (controlled) behavior.
To reach these goals coupled electrochemical systems are studied experimentally and numerically.

3. Previous results with an HN-NDR type electrochemical oscillator
Potenciostat
Rcoll
Rext 1 2 3 4 5 6 7 8
Rind C R
Pt electrode
Counter electrode
Ni wires
Working electrodes
Hg/Hg2SO4 Reference electrode
Synchronization and Control of Chaos on Coupled Electrochemical Oscillators
I. Z. Kiss, V. Gáspár, J. L. Hudson: J. Phys. Chem. B, 2000, 104, 7554.

4. Polarization curve of one Ni electrode in H2SO4 electrolyte (284 K, Rcoll = 0 W )
HN-NDR: HN-type of Negative Differential Resistance N
I (mA)
V (V)

5. Polarization curve of one Ni electrode in H2SO4 electrolyte (284 K, Rcoll = 200 W )SL
H - Hopf
C – Chaos
SL – Saddle-Loop
I (mA)
V (V)

6. Chaotic current oscillations of 8 Ni electrodes (weak global coupling)
t /s
i (mA)

7. Chaotic current oscillations of 8 Ni electrodes (weak global coupling + local feedback)
i (mA)
electrode
t /s
Individual resistors are varied as:
Synchronized chaos

8. An S-NDR type electrochemical system
Anodic deposition of Zn from ZnCl2 solution S
M. G. Lee, J. Jorné: J. Electrochem. Soc., 1992, 139, 2843.

9. An S-NDR type electrochemical oscillator
S-NDR type systems may oscillate only at large Cd’ values
Recursive derivative control:
I. Z. Kiss, Z. Kazsu, V. Gáspár: J. Phys. Chem. A, 2005, 109, 9521.

10. Synchronization
First observed and described by Christiaan Huygens in1665
“I finally found that this happened due to a sort of sympathy”

11. Coupled pendulum clocks

12. Simple modes of synchronization in-phase anti-phase

13. Synchronization cronoz - chronos (time)  - syn (same, common)
“synchronous” - „sharing the common time”, „occurring in the same time”
Universal behavior occurring in physical, chemical, biological, economical etc. systems.
SYNC: adjustment of rhythms of oscillating objects due to their weak interactions.

14. Anodic deposition of Zn
Zn2+ (aq) + 2e- ⇌ Zn (s)
Mechanism: Zn2+ + e- ⇌
Zn+ad
(1)
Zn+ad + e- ⇌ Zn
(2)
Kinetic study proved that the first step is autocatalytic
Zn2+ + Zn+ads + e- ⇌
2Zn+ads
M. G. Lee, J. Jorné: J. Electrochem. Soc., 1992, 139, 2843.

15. Detailed mechanism of Zn electrodeposition
M. G. Lee, J. Jorné: J. Electrochem. Soc., 1992, 139, 2843. K1
H+ + e-
Had K2
H+ + Had + e- H2 K3
Zn2+ + Zn+ad + e-
2Zn+ad
K’3 K4
Zn+ad + Had
H+ + Zn K5
Zn+ad + e- Zn K6
Zn2+ + Had
Zn+ad + H+

16. M. G. Lee, J. Jorné: J. Electrochem. Soc., 1992, 139, 2843.
Had
Zn+ad
1 and 2: fractional surface coverage
1 and 2: surface capacities (mol cm-2)
A1 … A6: complex functions of the potential through K1 … K6

17. Circuit of an array of Zn electrodes (n)
The electrolyte (through which the global coupling occurs) is not shown

18. Model equations: n electrodes + global coupling
local charge balance
Had
Zn+ad
Faradaic current density
current of the i-th electrode

19. Strength of global coupling (κ)
The strength of global coupling κ is varied by changing the individual (Rind) and/or collective resistances (Rcoll)
For simplicity, we consider unit surface area (A = 1,0 cm2) for each electrode.

20. Characterizing synchronization
phase diagram
(Hilbert transform)
sH(t)
Pk(t)
s(t)
order parameter: r(t) = |Z(t)|

21. Order parameter
From the book “SYNC” with the permission of the author S. Strogatz

22. Order parameter vs. coupling strength
From the book “SYNC” with the permission of the author S. Strogatz

23. Two nonidentical Zn electrodes κ = 0 Ω-1 cm-2
Independent oscillations (V = V and Cd = 10 F cm-2, A  B)

24. Two nonidentical Zn electrodes κ = 0.8 Ω-1 cm-2
In-phase oscillations (V = V and Cd = 10 F cm-2, A  B)

25. Two nonidentical Zn electrodes κ = 1.1 Ω-1 cm-2
Anti-phase oscillations (V = V and Cd = 10 F cm-2, A  B)

26. Order parameter vs. coupling strength

27. Two nonidentical Zn electrodes κ = 0.2 Ω-1 cm-2
Partial synchronization (V = V and Cd = 15 F cm-2, A  B)

28. Two nonidentical Zn electrodes κ = 0.2 Ω-1 cm-2
Partial synchronization (V = V and Cd = 15 F cm-2, A  B)

29. Two nonidentical Zn electrodes κ = 0.5 Ω-1 cm-2
Period-2 synchronization (V = V and Cd = 10 F cm-2, A  B)

30. Two nonidentical Zn electrodes κ = 0.5 Ω-1 cm-2
Period-2 synchronization (V = V and Cd = 10 F cm-2, A  B)

31. 128 nonidentical oscillators time evolution vs. κ (Ω-1 cm-2)
V = V
Cd = 10 F cm-2

32. 128 nonidentical oscillators < r > vs. κ (Ω-1 cm-2)
V = V
Cd = 10 F cm-2

33. 128 nonidentical oscillators κ = 0 Ω-1 cm-2
V = V
Cd = 10 F cm-2
Amplitude
Phase

34. 128 nonidentical oscillators κ = 1.0 Ω-1 cm-2
V = V
Cd = 10 F cm-2
Amplitude
Phase
Partial synchronization

35. 128 nonidentical oscillators time evolution vs. κ (Ω-1 cm-2)
V = V
Cd = 10 F cm-2

36. 128 nonidentical oscillators < r > vs. κ (Ω-1 cm-2)
V = V
Cd = 10 F cm-2

37. 128 nonidentical oscillators κ = 0.3 Ω-1 cm-2
V = V
Cd = 10 F cm-2
Amplitude
Phase
clusters

38. 128 nonidentical oscillators κ = 4.2 Ω-1 cm-2
V = V
Cd = 10 F cm-2
Amplitude
Phase
clusters

39. 128 nonidentical oscillators time evolution vs. κ (Ω-1 cm-2)
V = V
Cd = 15 F cm-2

40. 128 nonidentical oscillators order parameter vs. time
V = V
Cd = 15 F cm-2

41. 128 nonidentical oscillators < r > vs. κ (Ω-1 cm-2)
V = V
Cd = 15 F cm-2

42. 128 nonidentical oscillators κ = 0.15 Ω-1 cm-2
V = V
Cd = 15 F cm-2
Amplitude
Phase
“travelling waves”
“1D spiral”

43. 128 nonidentical oscillators κ = 0.5 Ω-1 cm-2
V = V
Cd = 15 F cm-2
Amplitude
Phase
“travelling waves”
partial SYNC

44. 128 nonidentical oscillators κ = 3.0 Ω-1 cm-2
V = V
Cd = 15 F cm-2
Amplitude
Phase
SYNC + swinging

45. No summary but …
SYNC + SWINGING = SYMPATHY

46. Extras X

47. To be continued ...

48. A letter of Huygens to his father

49. Exploring the phase space
Two parameter bifurcation diagram of the Lee-Jorné model for Zn electrodeposition showing the locus of Hopf-bifurcation.

50. The electrodes were made nonidentical
by decreasing and increasing the
individual surface capacity (mol/cm2)
by 25 % as follows:

51. Two nonidentical Zn electrodes κ = 0 Ω-1 cm-2
Independent oscillations (V= V and Cd = 10 F cm-2, A  B)

52. Two nonidentical Zn electrodes κ = 0.8 Ω-1 cm-2
In-phase oscillations (V= V and Cd = 10 F cm-2, A  B)

53. Two nonidentical Zn electrodes κ = 1.1 Ω-1 cm-2
Anti-phase oscillations (V= V and Cd = 10 F cm-2, A  B)

54. 128 nonidentical oscillators κ = 0 Ω-1 cm-2
V = V
Cd = 15 F cm-2
Amplitude
Phase