Multifaceted Dynamics of Janus Oscillator Networks

“power Janus oscillators are "phase-phase oscillators" introduced in our recent PRX paper, which, when coupled through a network, can exhibit myriad synchronization patterns. Their name is by analogy with the homonymous two-faced particles and ancient Roman deity, since the components of a Janus oscillator have different natural frequencies. Janus oscillator networks can exhibit explosive synchronization, extreme multi-stability, chimera states, asymmetry-induced synchronization, and inverted synchronization transitions. The source code and an animated summary of the paper are both available. See also “Janus Bunch” in Complexity Explorables.

Predicting Growth Rate from Gene Expression

network analog butterfly The quantitative relationship between functional traits like growth rate and their genetic underpinnings is an outstanding problem in molecular biophysics. In our recent PNAS article, we introduce a method, dubbed MI-POGUE, to directly map transcriptional profiles to growth rates. The method uses a machine-learning technique called k-nearest-neighbors regression and is demonstrated for both bacteria and yeast. A ready to use implementation of our method, which includes the relevant metadata and source code, is available here. The method can be easily adapted to different organisms and different traits.


Cascade vulnerability

Cascade Vulnerability of the North American Power Grid

Janus oscillators video

Janus Oscillator Networks


Levitation of Heavy Particles in Downward Flows

Network Analog Butterfly Effect

Network Analog of the Butterfly Effect








Transient chaos

Fractal Geometry of Doubly Transient Chaos

Adilson Motter video

Outreach: What is Entropy?

Dance Outreach video

Outreach: Syncing Up Without Sameness

Self-siphoning beads

Outreach: Self-Siphoning Beads








Recent Publications

J.D. Hart, Y. Zhang, R. Roy, and A.E. Motter,
Topological control of synchronization patterns: Trading symmetry for stability,
Phys. Rev. Lett. 122, 058301 (2019).
arXiv:1902.03255 - doi:10.1103/PhysRevLett.122.058301

Z.G. Nicolaou, D. Eroglu, and A.E. Motter,
Multifaceted dynamics of Janus oscillator networks,
Phys. Rev. X 9, 011017 (2019).
arXiv:1810.06576 - doi:10.1103/PhysRevX.9.011017

T.P. Wytock and A.E. Motter,
Predicting growth rate from gene expression,
Proc. Natl. Acad. Sci. USA 116(2), 367-372 (2019).
arXiv:1901.05010 - doi:10.1073/pnas.1808080116

Z. G. Nicolaou, B. Xu, and A.E. Motter,
Minimal scattering entanglement in one-dimensional trapped gases,
Phys. Rev. A 99, 012316 (2019).
arXiv:1901.04513 - doi:10.1103/PhysRevA.99.012316

A. Haber, F. Molnar, and A.E. Motter,
State observation and sensor selection for nonlinear networks,
IEEE Trans. Control Netw. Syst. 5(2), 694 (2018).
arXiv:1706.05462 - doi:10.1109/TCNS.2017.2728201

A.E. Motter and M. Timme,
Antagonistic phenomena in network dynamics,
Annu. Rev. Condens. Matter Phys. 9, 463 (2018).
arXiv:1808.00165 - doi:10.1146/annurev-conmatphys-033117-054054 (Access without subscription)

Y. Zhang and A.E. Motter,
Identical synchronization of nonidentical oscillators: When only birds of different feathers flock together,
Nonlinearity 31, R1 (2018).
arXiv:1712.03245 - doi:10.1088/1361-6544/aa8fe7

T.P. Wytock, A. Fiebig, J.W. Willett, J. Herrou, A. Fergin, A.E. Motter, and S. Crosson,
Experimental evolution of diverse Escherichia coli metabolic mutants identifies genetic loci for convergent adaptation of growth rate,
PLoS Genetics 14(3), e1007284 (2018).
arXiv:1812.05623 - doi:10.1371/journal.pgen.1007284

Z.G. Nicolaou, H. Riecke, and A.E. Motter,
Chimera states in continuous media: Existence and distinctness,
Phys. Rev. Lett. 119, 244101 (2017).
arXiv:1712.00458 - doi:10.1103/PhysRevLett.119.244101

Y. Yang and A.E. Motter,
Cascading failures as continuous phase-space transitions,
Phys. Rev. Lett. 119, 248302 (2017).
arXiv:1712.04053 - doi:10.1103/PhysRevLett.119.248302

T. Nishikawa, J. Sun, and A.E. Motter,
Sensitive dependence of optimal network dynamics on network structure,
Phys. Rev. X 7, 041044 (2017).
arXiv:1611.01164 - doi:10.1103/PhysRevX.7.041044

Y. Yang, T. Nishikawa, and A.E. Motter,
Small vulnerable sets determine large network cascades in power grids,
Science 358 (6365), eaan3184 (2017).
arXiv:1804.06432 - doi:10.1126/science.aan3184 - PDF

Y.S. Cho, T. Nishikawa, and A.E. Motter,
Stable chimeras and independently synchronizable clusters,
Phys. Rev. Lett. 119, 084101 (2017).
arXiv:1707.06657 - doi:10.1103/PhysRevLett.119.084101

Y. Zhang, T. Nishikawa, and A.E. Motter,
Asymmetry-induced synchronization in oscillator networks,
Phys. Rev. E 95, 062215 (2017).
arXiv:1705.07907 - doi:10.1103/PhysRevE.95.062215

X. Chen, T. Nishikawa, and A.E. Motter,
Slim fractals: The geometry of doubly transient chaos,
Phys. Rev. X 7, 021040 (2017).
arXiv:1705.02349 - doi:10.1103/PhysRevX.7.021040

L Zhang, A.E. Motter, and T. Nishikawa,
Incoherence-mediated remote synchronization,
Phys. Rev. Lett. 118, 174102 (2017).
arXiv:1703.10621 - doi:10.1103/PhysRevLett.118.174102

J.-R. Angilella, D.J. Case, and A.E. Motter,
Levitation of heavy particles against gravity in asymptotically downward flows,
Chaos 27, 031103 (2017).
arXiv:1703.03296 - doi:10.1063/1.4978386

Y. Yang, T. Nishikawa, and A.E. Motter,
Vulnerability and cosusceptibility determine the size of network cascades,
Phys. Rev. Lett. 118, 048301 (2017).
arXiv:1701.08790 - doi:10.1103/PhysRevLett.118.048301

A.E. Motter and Y. Yang,
The unfolding and control of network cascades,
Physics Today 70(1), 32 (2017).
arXiv:1701.00578 - doi:10.1063/PT.3.3426

Adilson E. Motter

Motter photo

Photo by Eileen Molony

Professor Motter's research is focused on the dynamical behavior and control of complex systems and networks. More...

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YouTube Channel

“Janus Bunch” Complexity Explorable

Explore the wealth of dynamical states exhibited in a network of Janus oscillators. “Janus




Multidisciplinary Complex Systems Research

Report from an NSF Workshop in May 2017. Released May 2018. K.A. Gray and A.E. Motter (co-chairs). PDF for download.

Network Control

See the talk Advances on the Control of Nonlinear Network Dynamics by Adilson E. Motter at the 2015 SIAM Conference on Applications of Dynamical Systems and check out our featured control projects page to see a summary of our recent work in this area.

Group News

June 2019: Fiona Brady receives the Outstanding Undergraduate in Graduate Coursework Award from the Northwestern University Department of Physics and Astronomy.

December 2018: Yuanzhao Zhang is first place in the 2018 Northwestern Scientific Images Contest.

September 2018: Takashi Nishikawa is elected APS Fellow.

December 2017: Group organizes the 4th edition of the Network Frontier Workshop.

May 2017: Vicky Yang receives Red Sock Award (SIAG/Dynamical Systems) for joint work at DS17.

Selected Press

‘The Big Bang Theory’ finale: Sheldon and Amy’s fictional physics parallels real science
The Conversation (May 16, 2019)

Grid unlock: simple fixes, better modelling are keys to reliable energy supply
Cosmos Magazine (April 17, 2019)

Scientists Discover Exotic New Patterns of Synchronization
Quanta Magazine (April 4, 2019)


All Systems Go in Shanghai for Connecting Innovators
Nature (December 20, 2018)

Two Phases, Two Faces
Physics Buzz (February 7, 2019)

The Complex Dance of Two-Faced Oscillators
APS Physics Synopsis (January 30, 2019)

The subtle success of a complex mindset
Nature Physics (December 4, 2018)

Sensitive Dependence on Network Structure: Analog of Chaos and Opportunity for Control
SIAM News (April 2, 2018)

Student Videos

Watch members of the Motter group explain complex systems research to general audiences.

Rosangela Follmann video

Rosangela Follmann
Unveiling Mysteries of Synchronization

Joo Sang Lee video

Joo Sang Lee
Sunja's Two Dreams

Yang Yang video

Yang Yang
Say No to Blackouts

Phalguni Shah video

Phalguni Shah
Untangling the nets: The paradox of network complexity and stability

Yuanzhao Zhang video

Yuanzhao Zhang
Network, Synchronization, and the Paradox of Heterogeneity

Thomas Wytock video

Thomas Wytock
Anarchists or Authoritarians? Probing the Internal Politics of E. coli

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