0. Synchronized Chaos in Coupled Optical Feedback Networks
Briana E. Mork, Gustavus Adolphus College
Katherine R. Coppess, University of Michigan

1. Natural dynamical system: Chaos; order and randomness
Oscillators (Nodes)
Synchrony
Synchrony in chaos and why it’s important
Dynamical networks
Man made: power grids
Brain
Can’t experiment with those, can experiment with oscillators in the lab
natural and artificial-- chaotic and periodic
Connect to our experiment:
Topology is important
Kitzbichler (2009) PLoS Comput Biol 5(3)

2. Man-made dynamical system:
Periodic (mostly)
Oscillators (Nodes)
Synchrony
Synchrony in chaos and why it’s important
Dynamical networks
Man made: power grids
Brain
Can’t experiment with those, can experiment with oscillators in the lab
natural and artificial-- chaotic and periodic
Connect to our experiment:
Topology is important

3. Examples of Four-Node Network Topologies
REMOVE feedback loop in topology
We pick two to study… because they show different behaviors
Variations of symmetries… AENNC has what # of symmetries,
Bidir has what #
Expectations of clustering and global

4. Experimental Four-Node Network Topologies
Summer 2014
Previously studied
CRS Williams et al. CHAOS 23, (2013)
REMOVE feedback loop in topology
We pick two to study… because they show different behaviors
Variations of symmetries… AENNC has what # of symmetries,
Bidir has what #
Expectations of clustering and global

5. The Experiment
CRS Williams et al. CHAOS 23,
(2013)
Four nodes form a delay-coupled system with weighted and directed links.
Weight is determined by the coupling strength ε as implemented by the DSP board.
To experimentally investigate synchrony in small networks, four identical optoelectronic oscillators are coupled together to form a configurable network with several variable parameters, allowing for the exploration of different dynamics and synchronization states. Keeping other parameters constant, feedback and coupling strengths are varied for multiple topologies to investigate the synchrony of the network for different node dynamics (e.g. periodic and chaotic).
Feedback- which is why we see different behaviors… electrical signal modulates...

6. Dynamics of a Node
Changing the feedback strength β of a node varies the dynamics of the node.
CRS Williams et al. CHAOS 23, (2013)
x(t) (A.U.)
REMOVE +
Experimental Setup for a single node: an optoelectronic, nonlinear oscillator with time delayed feedback.
Make everything bigger ~ words
time (ms)

7. Experimental Network Topologies
Bidirectional ring
Bidirectional chain with unidirectional links
Talk about why we chose these two; what’s interesting about them
laplacian coupling
as coupling increases, feedback decreases
allows global synchronization, while some other methods do not
Bottom: Laplacian coupling matrices for the two networks, respectively.

8. Bidirectional Ring

9. Bidirectional Ring
Left: Simulated (left) and experimental (right) results for the bidirectional ring. Lower epsilon values yield non-synchronous states. As epsilon is increased, the network transitions to global synchrony and then, to clustering of opposite nodes (1&3 and 2&4).
We investigate with simulation and experimentally demonstrate stable states of global and cluster synchrony, where synchrony is defined as identical waveforms observed in two or three nodes (cluster) or all nodes (global). We compare experimental and simulated results with symmetry analysis predictions of topology-dependent states of global and cluster synchrony.
show time traces with epsilon, then error graphs

10. Bidirectional Chain with Unidirectional Links
offset the two pairs

11. Conclusions Future work
- The synchronous states that arise depend on topology of the network.
- Transitions between synchronous states depend on coupling strength.
Future work
- Stability analysis for synchronous states
- Many-node networks
- Comparison of convergence rates between global and cluster synchrony

12. Caitlin R. S. Williams, Washington and Lee University
Acknowledgments
Caitlin R. S. Williams, Washington and Lee University
Aaron M. Hagerstrom, University of Maryland
Louis Pecora, Naval Research Laboratory
Francesco Sorrentino, University of New Mexico
Thomas E. Murphy, University of Maryland
Rajarshi Roy, University of Maryland

13. For More Information...
Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays CRS Williams et al. CHAOS 23, (2013)
Experimental Observations of Group Synchrony in a System of Chaotic Optoelectronic Oscillators CRS Williams et al. PRL 110, (2013)
Cluster Synchronization and Isolated Desynchronization in Complex Networks with Symmetries L Pecora et al. NATURE COMMUNICATIONS | DOI: /ncomms5079 (2014)