Natural dynamical system: Chaos; order and randomness

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

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 http://2.bp.blogspot.com/-73jjFxeZhjc/T-IRe264OII/AAAAAAAADXU/2PBDgyH2Ufk/s400/Brain_Highways.png Kitzbichler (2009) PLoS Comput Biol 5(3)

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 http://www.efoodsdirect.com/blog/wp-content/uploads/2013/10/power-grid-drill.jpg

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

Experimental Four-Node Network Topologies Summer 2014 Previously studied CRS Williams et al. CHAOS 23, 043117 (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

The Experiment CRS Williams et al. CHAOS 23, 043117 (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...

Dynamics of a Node Changing the feedback strength β of a node varies the dynamics of the node. CRS Williams et al. CHAOS 23, 043117 (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)

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.

Bidirectional Ring

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

Bidirectional Chain with Unidirectional Links offset the two pairs

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

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

For More Information... Synchronization states and multistability in a ring of periodic oscillators: Experimentally variable coupling delays CRS Williams et al. CHAOS 23, 043117 (2013) Experimental Observations of Group Synchrony in a System of Chaotic Optoelectronic Oscillators CRS Williams et al. PRL 110, 064104 (2013) Cluster Synchronization and Isolated Desynchronization in Complex Networks with Symmetries L Pecora et al. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5079 (2014)