1. Dynamics of Coupled Oscillators-Theory and Applications
Alexandra Landsman, Naval Research Laboratory
Ira B. Schwartz, Naval Research Laboratory
Research supported by the Office of Naval Research

2. Outline Brief intro Types of synchronization
Phase synchronization (frequency locking)
Complete
Generalized
Synchronization of coupled lasers
Phase synchronization of limit cycle oscillators
Summary and conclusions

3. Synchronization – What is it?
Many things in nature oscillate
Many things in nature are connected
Definition: Synchronization is the adjustment of rhythms of oscillating objects due to their weak interactions*
*A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization, Cambridge Univ. Press 2001

4. Why Synchronization is Interesting
Physical systems: Clocks; Pattern formation;
Dynamics of coherent structures
in spatially extended systems (Epidemics, Neurons,
Lasers, continuum mechanics,…)
Engineering: Communication systems;
Manufacturing processes
Coupled fiber lasers for welding
Coupled chemical reactors for etching
Biological systems: Healthy dynamical rhythms;
Dynamical diseases;
Population dynamics
Defense Applications: New tunable radiation sources
THz sources for IED detection
Secure communications
Communicating autonomous vehicles

5. Complete Synchronization
Complete or identical synchronization (easiest to understand)
The difference between states of systems goes asymptotically to zero as time goes to infinity.
Amplitudes and phases are identical
X(t)
Y(t)
x-y t x t y

6. Phase synchronization
Unidirectional Coupling in a Laser (Meucci)
Synchronization phases of one oscillator to an external oscillator
Phases have a functional relationship
If phases are locked, or entrained,
Then dynamics is in phase synchrony
Frequency locking
Georgiou and Schwartz, SIAM J. Appli Math, 1999
Schwartz et al, CHAOS, 2005

7. Generalized synchronization
Systems exhibit quite different temporal evolutions,
There exists a functional relation between them.
N. F. Rulkov et al. Phys. Rev. E51 980, (1995)
Detecting generalized synchronization is difficult to implement in experiments
Good for large changes in time scales

8. Generalized synchronization
The auxiliary system method:
Two or more replicas of the response system are available ( i.e. obtained starting from different initial conditions)
Complete synchronization between response systems implies generalized synchronization between response and drive systems.
Experimental evidence of NIS
CO2 laser (Meucci)
Start of the common noise signal

9. Application 1: coupled arrays of limit cycle oscillators
How diffusive coupling leads to different types of phase-locked synchronization
The effect of global coupling and generalized synchronization via bifurcation analysis X1 Y1 x1 X2 Y2 x2 X5 Y5 X3 Y3 x3 X4 Y4 x4
Landsman and Schwartz, PRE 74, (2006)

10. Application 2: coupled lasers
Mutually coupled, time-delayed semi-conductor lasers
Generalized synchronization can be used to understand complete synchronization of a group of lasers
Laser1
Laser2
Laser3
A.S. Landsman and I.B. Schwartz, PRE 75, (2007),

11. Coherent power through delayed coupling architecture
Experiments with Delayed Coupling – N=2
Coupled lasers do not have a stable coherent in-phase state
Two delay coupled semiconductor lasers: experiment showing stable out-of-phase state
Time series synchronized after being shifted by coupling delay
Each bar is a snap shot of a prediction of the flow.
Time is flowing downward.
The y-axis is distance to the wall from the center of the cylinder. The x-axis is the length to the wall.
Blue is the measured signal, which is increased when the pump is perturbed stochastically around a given frequency.
Heil et al PRL, (2001)
Leaders and followers switch over time

12. 1 2 3 Chaotic Synchronization of 3 semiconductor
lasers with mutual, time delayed coupling 1 2 3
Scaled equations of a single,
uncoupled laser: y x
y - intensity
Weak dissipation:
x - inversion

13. Problem: Explain synchronization of outside lasers in
a diffusively coupled, time-delayed, 3-laser system,
with no direct communication between outside lasers
Log(I1)
Log(I2)
time
Log(I2)
Log(I1)
Log(I3)
time
Log(I3)
time
Log(I1)

14. 3 mutually coupled lasers, with delays2
Laser 3
coupling strengths:
delay:
dissipation:
detuning:
y - intensity
x - inversion

15. Synchronized state Dynamics can be reduced to two coupled lasers 1 2 3
1,3
Above dynamics
equivalent to
Detuning:
Laser 2 leads
Laser 2 lags

16. Synchronization over the delay time is similar to generalized synchronization
Outside lasers can be viewed as identical, dissipative
driven system during the time interval
Laser 1
Laser 2
Laser 3 1 2 3
Stable synchronous
state:

17. Analysis of dynamics close to the synchronization manifold
Symmetry:
Outer lasers identical
Synchronized solution:
The outer lasers synchronize
if the Lyapunov exponents
transverse to the synchronization
manifold are negative

18. Linearized dynamics transverse to the synchronization manifold
The synchronous state,
is not affected by
over the time interval of
acts like a
driving signal for
Phase-space volume
Abel’s
Formula

19. Transverse Lyapunov exponents
for sufficiently long delays:
Contracting
phase-space
volume
Lyapunov exponents
linear dependence of Lyapunov exponents on
Synchronization due to dissipation in the outer lasers!

20. Effect of dissipation on synchronization: Numerical results
Sum of Lyapunov exponents
Correlations
0.05
1.2
1.0
-0.05
-0.1
0.8
-0.15
0.6
-0.2
0.02
0.03
0.04
0.05
0.06
0.07
0.02
0.03
0.04
0.05
0.06
0.07
dissipation
dissipation

21. Dependence of synchronization on parameters:
Condition for negative
Lyapunov exponents
Maximum fluctuations in depend on
Less synchronization for increased coupling strengths,
Better synchronization for longer delays,
Better synchronization with increased dissipation,

22. Numerical results for synchronization as a function of delay
Correlations
between the outer
and the middle laser
Correlations between
outer lasers
Sum of transverse
Lyapunov exponents
delay

23. Synchronization as a function of coupling strength
Correlations
Sum of Lyapunov exponents
Coupling strength
Coupling strength

24. Laser Results
Synchronization on the time scale of the delay, similar to generalized synchronization of driven dissipative systems
Outer lasers become a function of the middle one
Improved synchronization with increased dissipation
“washes out” the difference in initial conditions
Improved synchronization for longer delays
Need sufficiently long times to average out fluctuations
Less synchronization with increase in coupling strength
Greater amplitude fluctuations, requiring longer delays for the outer lasers to synchronize

25. Discussion Conclusion
Synchronization phenomena observed in many systems (chaotic and regular)
Chaotic Lasers
Limit-cycle oscillators
Phase-locking
Complete synchronization
Generalized synchronization
Conclusion
basic ideas from synchronization
useful in studying a wide variety of nonlinear
coupled oscillator systems

26. References
A.S. Landsman and I.B. Schwartz, "Complete Chaotic Synchronization in mutually coupled time-delay systems", PRE 75, (2007),
“A.S. Landsman and I.B. Schwartz, "Predictions of ultra-harmonic oscillations in coupled arrays of limit cycle oscillators”, PRE 74, (2006),
A.S. Landsman, I.B. Schwartz and L. Shaw, “Zero Lag Synchronization of Mutually Coupled Lasers in the Presence of Long Delays”, to appear in a special review book on “Recent Advances in Nonlinear Laser Dynamics: Control and Synchronization”, Research Signpost, Volume editor: Alexander N. Pisarchik