1. EXPERIMENTAL PHASE SYNCHRONIZATION OF CHAOS IN A PLASMA DISCHARGE By Catalin Mihai Ticos A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Coral Gables, Florida May 2002 UNIVERSITY OF MIAMI

2. 1. Introduction 2. Dynamics of Continuous Chaotic Systems The Lorenz Attractor The Rössler Attractor The Chua Attractor Chaotic Semiconductor Lasers Chaotic Plasma Discharges 3. Control and Synchronization of Chaotic Systems Methods of Controlling Chaos The Ott-Grebogi-Yorke Method Pyraga’s Method Pecora and Carrol Method Synchronization of Identical Systems 4. Phase Synchronization of Chaos Phase of a Periodic Oscillator Phase of Noisy Oscillators Phase of Chaotic Oscillators Dynamics of the Phase of Chaotic Oscillators Phase Synchronization with an External Periodic Force 5. Phase Synchronization of Chaos in a Plasma Discharge Phase Synchronization of the Discharge Light Intensity Real-Time Phase Synchronization of the Current Discharge Driving a Chaotic Plasma with an Information Signal 6. Phase Synchronization of Chaos Between a Plasma Discharge and the Chua Circuit

3. Chaos Theory James A. Yorke: “In this paper we analyze a situation in which the sequence…is non-periodic and might be called chaotic” “Period Three Implies Chaos”, Amer. Math. Soc., 82 (1975) 985-992 Chaos theory is the study of apparently random behavior of deterministic systems. Where do we find Chaos? In Physics, Engineering, and Biology : Plasma discharges, Electronic circuits, Lasers, Fluid flows, Granular media, Mechanical tools, Neurons, Heart Muscle, Oscillations of Lakes, Weather models

4. Ed Lorenz found : The solution is called a strange attractor. -It oscillates irregularly, never exactly repeating. -It is bounded.

5. O.E. Rössler (1976) simplified the Lorenz system and obtained an attractor with a single spiral:

6. Aperiodic: A chaotic signal looks like a burst of irregular pulses

7. Sensitive dependence on initial conditions: Two initially nearby trajectories (green and red) separate exponentially fast in time.

8. Lyapunov Exponents Largest Lyapunov Exponent: For a 3-dim system we have 3 exponents: Chaos: λ1 >0 λ2 =0, where | λ3| > | λ1| => λ1 + λ2 + λ3 <0 λ3 <0 Periodic Torus : λ1 = 0 λ2 = 0 λ3 < 0 Periodic Cycle : λ1 = 0 λ2 < 0 λ3 < 0 Chaos Characterization

9. Lyapunov Dimension If k is the largest integer for which then we define Fractal Dimension For regular geometrical shapes D 0 is an integer. For 3-dim Chaotic Attractors: 2 < D 0 < 3

10. Chua Circuit

11. Chua System Power Spectrum Attractor Change of Variables: x=V C1, y=V C2, z=Ri L, α=C 2 /C1, β=R 2 (C 2 /L)

12. Nonlinear and Chaotic Plasma Discharges  Nonlinear spatial structures (striations) and stability analysis in RF Ar plasmas: R. A. Goldstein et al, Phys. Fluids 22 (1979), 231 M. A. Huerta et al, Phys. Rev. A 26 (1982), 539 J. A. Walkenstein et al., Phys. Lett. A261 (1999), 183  Chaotic oscillations of the current, light flux, and bifurcation route to chaos in glow discharges in Ar, He and Ne, and thermoionic discharges in Ar: P.Y. Cheung et al., Phys. Rev. Lett.59 (1987), 551 T. Braun et al., Phys. Rev. Lett. 59 (1987), 613 T. Mausbach et al., Phys. Plasmas 6 (1999), 3817

13. Models for a chaotic plasma: Metastables are important in obtaining self sustained oscillations in the positive column of a dc discharge: V. O. Papanyan et al., Int. J. Bifurc. Chaos 4 (1994), 1495 Where n, m are the electron and metastable concentrations,  n and  m are the lifetimes; P is the rate of metastable production Z is the effective ionization rate where m 0, n 0 are the equilibrium values N i+1 =  N i (1-N i ) (bifurcation similar to the logistic map)

14. Control of Chaos  OGY method (Ott-Grebogi-Yorke ‘89) Stabilization on a periodic orbit by a small time-dependent perturbation of one systems’ parameter  Pyragas’ method Stabilization on a periodic orbit by delayed feedback  Complete Synchronization of Chaos (Pecora and Carroll ‘90) Synchronization trough coupling between identical systems

15.  Complete Synchronization of Chaos (Pecora and Carroll ‘90)

16. V (2) C2 (t) and V (1) C2 (t) for R=878  V (2) C2 vs. V (1) C2 :

17. V (2) C2 (t) and V (1) C1 (t) for R=1116 

18. For the Rössler attractor an angle coordinate can be introduced as the oscillator phase Φ. A sharp peak in the power spectrum indicates the presence of a dominant frequency of oscillation f=0.164.

19. We sample the trajectory at f=0.164. The points are scattered all over the attractor due to chaotic phase diffusion.

20. We introduce in the system a small perturbation P = A sin (2πft), called pacer : where f=0.164 is the dominant frequency and A=0.007 We sample the trajectory of the paced System at the pacer frequency and we get points that are located nearby each other, within a limited range. Phase Synchronization of Chaos: -E. Rosa, Jr., et al (1998) Phys. Rev. Lett. 80, 1642-1645; -M.G. Rosenblum et al (1996) Phys. Rev. Lett. 76, 1804-1807 ;

22. C. M. Ticos et al (2000) Phys. Rev. Lett. 85, 2929-2932; Phase synchronization of the plasma discharge current

23. Attractor of the chaotic plasma Lyapunov exponents and dimension: The plasma power spectrum shows a peak at f=6960 Hz

24. Stroboscopic sections in the plasma attractor at 6960 Hz Sampled points in (r,  ) plane

25. Phase synchronization of plasma Stroboscopic sections in the paced plasma attractor; Pacer amplitude and frequency A=0.4 V, f=6960 Hz Sampled points in (r,  ) plane of the paced plasma

26. Rosa, Jr., E. et al (2000) Int. J. Bifurc. Chaos 10, 2551-2563 ; Phase synchronization of the plasma light flux

27. Attractor of the local light intensity reconstructed by time-delay embedding Fractal Dimension D 0 = 2.18 Power spectrum of the light intensity f peak = 3850 Hz

28. Stroboscopic sections in the plasma attractor at 3850 Hz Stroboscopic sections in the paced plasma attractor; Pacer amplitude and frequency A~1V, f=3850 Hz

29. We explore the region of phase synchronization (green circles) in the pacer parameter space (frequency - amplitude), the Arnold Tongue.

30. Real-time power spectrum of the plasma oscillation in LabVIEW (resolution of 1 Hz)

31. Real-time phase synchronization in LabVIEW Sampling at the frequency of the Pacer Real-time unsynchronized plasma in LabVIEW

32. Real-time phase synchronization in LabVIEW Sampling at twice the frequency of the Pacer

33. We apply Kirchoff’s law on the two loops: Plasma-Resistor-Source, and Resistor-Inductor-Capacitor -L p is the parasitic inductance of the discharge; -V p (I 1 ) is the nonlinear voltage-current characteristic of the plasma; -E is the high voltage. Numerical Model for the Experimental Set-up

35. and the system becomes : where The values corresponding to our experiment are: R=30KΩ, C=3.5 pF, L=30mH, L p =4mH, E=850V, m 0 =-136.53 V/mA, m 1 = -7.46 V/mA, Change of variables:

36. Numerical attractor of the plasma in the our specific set-up

37. W.B. Pardo et al. (2001) Phys. Lett. A 284, 259-265 ; A harmonic signal, from a CD player, drives the plasma - We retrieve the driving signal in the negative light of the discharge when the plasma oscillates with period 2 -In the chaotic regime the chaos covers the driving signal

38. At a discharge voltage of U=802.1V we measure steady oscillations with a period 2 Attractor of the local light intensity emitted in the cathode region reconstructed by time-delay embedding

39. The driving signal modulates the local light intensity After subtraction of the driving signal we recover the original period-2 plasma signal

40. Power spectrum of the periodic plasma oscillation Power spectrum of the driven plasma (plasma is periodic)

41. Chaotic plasma local light intensity (grey) and driving signal (black) Attractor of the plasma at U=830.1 V in chaotic regime

42. Power spectrum of the chaotic plasma oscillation Power spectrum of the chaotic driven plasma

43. The Chua circuit is driving the chaotic plasma -Variable coupling through R c -One way coupling using OP-AMP LM741 -Measured signals: Plasma Light Flux Chua Voltage V C2 E. Rosa, Jr., et al, To be published

44. We tune the two uncoupled systems until they show nearly the same dominant frequency in their power spectrum. The frequency mismatch of the peaks is about Δf ≈ 50 Hz.

45. Time delay embedding is used to reconstruct the attractors of the two uncoupled systems. The delay is 2T S, where T S is the acquisition rate T S = 50µs.

46. We then couple the systems at Rc=0.7 KΩ. The peaks in the power spectrum are at exactly the same frequency.

47. We compute the phase of the acquired signals by using the Hilbert Transform (H): u (t)= arbitrary signal H (u(t)) = phase shift with π/2 of each component in the power spectrum of u (t), at any moment t. Instantaneous phase: Condition for Phase Synchronization:

48. We compute the phase difference ΔΦ = Φ Chua - Φ plasma of the time series acquired for different coupling strengths Rc Average time  of chaotic transients :

49. During phase synchronization, the phase difference between the two signals is constant while their amplitudes remain chaotic Plasma Light Flux vs. Chua Voltage VC2 During a phase jump with 2  we observe on the scope a burst perpendicular on the 45º line

50. CONCLUSIONS  Showed phase synchronization between the plasma discharge current and a periodic low-voltage sine wave  Showed phase synchronization between the plasma discharge light flux and a periodic low-voltage sine wave  Found the whole region of phase synchronization, in the space of amplitude and frequency (A, f) of the pacing voltage  Drove the plasma discharge with a harmonic signal  Showed phase synchronization between a plasma discharge and the Chua electronic circuit. 3 published papers, 1 submitted

51. Special Thanks to : Prof. Rosa, Prof. Pardo Prof. Alexandrakis Prof. Huerta Prof. Voss Prof. Wang Jonathan Walkenstein Marco Monti Robert Heyman