Supplementary Informations for Papers

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Transitions from Trees to Cycles in Adaptive Flow Networks

Basins of Attraction for Chimera States

Chimera states in networks with mechanical dynamics

Controlling Chimeras / Traveling Chimeras

A Model for Polygonal Hydraulic Jumps

Transitions from Trees to Cycles in Adaptive Flow Networks, with Konstantin Klemm

Reference: Martens E. A., Klemm, K., Transitions from Trees to Cycles in Adaptive Flow Networks (under review), or arXiv:1711.00401 for a pre-print.
Simulations of the adaptive dynamics for networks with height H = 4. The flow at the top of the tree-like structure (inlet) is constant, while the outlets in the leaves experience load fluctuations of strength a. During the simulation, the parameter is quasi-adiabatiacally decreasing from 1.0 in steps of size 0.01.
Legend for edge thickness and conductances C_kl:

Supplementary Video 1: Augmented tree with cross-edges in minimal triangular subgraphs

Dynamics for a tree augmented with cross-edges allowed only within triangular minimal subgraphs (Fig. 1 c)).

Supplementary Video 2: Augmented tree with all cross-edges.

Dynamics for a tree augmented with all possible cross-edges allowed (Fig. 1 d))

Supplementary Video 3: Damaged network, augmented tree with all cross-edges, realization #1.

Damaged versions of the network in Video 2, but where roughly half of all cross edges have been randomly selected and removed. These simulations confirm the observation that the cross edges begin to conduct in order of the tree hierarchy, starting from sinks and moving towards the source.

Supplementary Video 4: Damaged network, augmented tree with all cross-edges, realization #2.

Another instance of a damaged network in Video 2, but where roughly half of all cross edges have been randomly selected and removed. These simulations confirm the observation that the cross edges begin to conduct in order of the tree hierarchy, starting from sinks and moving towards the source.

Basins of Attraction for Chimera States, with Mark Panaggio and Danny M. Abrams

Reference: Martens E. A., Panaggio, M., Abrams, D.M. Basins of Attraction for Chimera States, New Journal of Physics, 2016 arXiv:1507.01457 for a pre-print.
Description of Supplementary Videos 1,2,3 and 4.

Supplementary Video 1: Destination maps at s=s_c.

Destination maps as a function of 0.1≤s≤1 (A=0.2,β=0.025). Even though A=0.2 is fairly large with regards to our perturbative calculus, numerical results match the predicted motion qualitatively well. As s increases from zero, basins merge and pinch-off in an alternating fashion, so that the basin boundaries rotate counter-clockwise about R₀ (d,ψ)=(0,0). Once s reaches s_c≈√(1-A), this rotation stops, demonstrating that knowledge of the trajectory position in the s=s_c plane is sufficient for determining the final fate of the trajectory.

Basins are color-coded, where blue/red/yellow denote the DS/SD/SS₀ basins, respectively, where S and D denote which one of the populations is synchronized.

Supplementary Video 2: Double helical structure of trajectories near R₀-manifold.

Twisting motion of trajectories in a double helical structure following the R₀-manifold (A=0.1,β=0.025). Initial conditions of 31 trajectories are equally spaced with s=0.1045; -0.0345≤d≤0.0345; ψ=0. Trajectories are color-coded, depending on which state they converge to (blue/red/yellow denote the DS/SD/SS₀ basins, respectively, where S and D denote which one of the populations is synchronized).

Supplementary Video 3: Double helical structure of trajectories near R_π-manifold.

Twisting motion of trajectories in a double helical structure following the R_π manifold (A=0.1,β=0.025). Initial conditions of the 3 trajectories are s=0.4487,d ∈ {-0.6,-0.2,0.6}*10^-3,ψ=π.

Supplementary Video 4: 3D-visualization of separatrices.

Three dimensional visualization of the separatrices emanating from the chimera saddle points near the R₀-manifold (A=0.1, β=0.025).

Controlling Chimeras

Reference: New Journal of Physics 17:033030 (2015), doi:10.1088/1367-2630/17/3/033030; arXiv:1402.6363

Supplementary Movie: Traveling Chimera with Control.

Asymmetry in the coupling kernel results in a traveling chimera. Control on the asymmetry parameter allows the chimera to move to a target location.

Chimera states in networks with mechanical dynamics, with S. Thutupalli, A. Fourrière and O. Hallatschek

Supplementary Movie 1: In-phase synchronization on two swings.

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Supplementary Movie 2: Chimera state on two swings.

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Supplementary Movie 3: Anti-phase synchronization on two swings.