While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyze large, hybrid systems of interconnected Kuramoto oscillators and Hopfield switches with positive feedback. In this system, oscillator synchronization promotes switches to turn on. In turn, when switches turn on, they enhance the synchrony of the oscillators to which they are coupled. Depending on the choice of parameters, we find theoretically coexisting stable solutions with either (i) incoherent oscillators and all switches permanently off, (ii) synchronized oscillators and all switches permanently on, or (iii) synchronized oscillators and switches that periodically alternate between the on and off states. Numerical experiments confirm these predictions. We discuss how transitions between these steady state solutions can be onset deterministically through dynamic bifurcations or spontaneously due to finite-size fluctuations.
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September 2013
Research Article|
September 25 2013
Dynamics in hybrid complex systems of switches and oscillators
Dane Taylor;
Dane Taylor
a)
1
Department of Applied Mathematics, University of Colorado
, Boulder, Colorado 80309, USA
2
Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA
3
Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina 27709, USA
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Elana J. Fertig;
Elana J. Fertig
b)
4
Oncology Biostatistics, John Hopkins University
, Baltimore, Maryland 21205, USA
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Juan G. Restrepo
Juan G. Restrepo
c)
1
Department of Applied Mathematics, University of Colorado
, Boulder, Colorado 80309, USA
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a)
Electronic mail: dane.taylor@colorado.edu
b)
Electronic mail: ejfertig@jhmi.edu
c)
Electronic mail: juanga@colorado.edu
Chaos 23, 033142 (2013)
Article history
Received:
June 08 2013
Accepted:
September 10 2013
Citation
Dane Taylor, Elana J. Fertig, Juan G. Restrepo; Dynamics in hybrid complex systems of switches and oscillators. Chaos 1 September 2013; 23 (3): 033142. https://doi.org/10.1063/1.4822017
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