# Time Correlations in Mode Hopping of Coupled Oscillators

## Abstract

We study the dynamics in a system of coupled oscillators when Arnold Tongues overlap. By varying the initial conditions, the deterministic system can be attracted to different limit cycles. Adding noise, the mode hopping between different states become a dominating part of the dynamics. We simplify the system through a Poincare section, and derive a 1D model to describe the dynamics. We explain that for some parameter values of the external oscillator, the time distribution of occupancy in a state is exponential and thus memoryless. In the general case, on the other hand, it is a sum of exponential distributions characteristic of a system with time correlations.

## Keywords

Coupled oscillators Mode hopping Arnold tongues Poincare sections Time correlations## Notes

### Acknowledgements

SK thanks the Simons Foundation for funding. MLH and MHJ acknowledge support from the Danish Council for Independent Research and Danish National Research Foundation through StemPhys Center of Excellence, grant number DNRF116.

## References

- 1.Jensen, M.H., Kadanoff, L.P., Krishna, S.: Universal behaviour of coupled oscillators: lessons for biology preprint (2016)Google Scholar
- 2.Stavans, J., Heslot, F., Libchaber, A.: Fixed winding number and the quasiperiodic route to chaos in a convective fluid. Phys. Rev. Lett.
**55**, 596–599 (1985)ADSCrossRefGoogle Scholar - 3.Jensen, M.H., Bak, P., Bohr, T.: Complete devil’s staircase, fractal dimension and universality of mode-locking structure in the circle map. Phys. Rev. Lett.
**50**, 1637–1639 (1983)ADSCrossRefGoogle Scholar - 4.Jensen, M.H., Kadanoff, L.P., Libchaber, A., Procaccia, I., Stavans, J.: Global universality at the onset of Chaos: results of a forced Rayleigh Benard experiment. Phys. Rev. Lett.
**55**, 2798–2801 (1985)ADSCrossRefGoogle Scholar - 5.Jensen, M.H., Bak, P., Bohr, T.: Transition to chaos by interaction of resonances in dissipative systems. I. Circle maps. Phys. Rev. A
**30**, 1960–1969 (1984)ADSMathSciNetCrossRefGoogle Scholar - 6.Huygens, C.: A copy of the letter on this topic to the Royal Society of London appears in 1893. In: Nijhoff, M. (ed.) Ouevres Completes de Christian Huygens, vol. 5, p. 246. Societe Hollandaise des Sciences, The Hague (1989)Google Scholar
- 7.Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
- 8.Ramirez, J.P., Olvera, L.A., Nijmeijer, H., Alvarez, J.: The sympathy of two pendulum clocks: beyond Huygens’ observations. Sci. Rep.
**6**, 23580 (2016)ADSCrossRefGoogle Scholar - 9.Arnold, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. Addison-Wesley, Boston (1989)MATHGoogle Scholar
- 10.Brown, S.E., Mozurkewich, G., Gruner, G.: Subharmonic shapiro steps and devil’s-staircase behavior in driven charge-density-wave systems. Phys. Rev. Lett.
**52**, 2277–2380 (1984)ADSCrossRefGoogle Scholar - 11.Gwinn, E.G., Westervelt, R.M.: Frequency Locking, quasiperiodicity, and Chaos in Extrinsic Ge. Phys. Rev. Lett.
**57**, 1060–1063 (1986)ADSCrossRefGoogle Scholar - 12.Tsai, T.Y., Choi, Y.S., Ma, W., Pomerening, J.R., Tang, C., Ferrell Jr., J.E.: Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science
**321**, 126–129 (2008)ADSCrossRefGoogle Scholar - 13.Goldbeter, A.: Computational approaches to cellular rhythms”. Nature
**420**, 238–245 (2002)ADSCrossRefGoogle Scholar - 14.Woller, A., Duez, H., Staels, B., Lefranc, M.: A mathematical model of the liver circadian clock linking feeding and fasting cycles to clock function. Cell Rep.
**17**, 1087–1097 (2016)CrossRefGoogle Scholar - 15.Mondragon-Palomino, O., Danino, T., Selimkhanov, J., Tsimring, L., Hasty, J.: Entrainment of a population of synthetic genetic oscillators. Science
**333**, 1315–1319 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar - 16.Mengel, B., Hunziker, A., Pedersen, L., Trusina, A., Jensen, M.H., Krishna, S.: Modeling oscillatory control in NF-kB, p53 and Wnt signaling. Curr. Opin. Genet. Dev.
**20**, 656–664 (2010)CrossRefGoogle Scholar - 17.Hoffmann, A., Levchenko, A., Scott, M.L., Baltimore, D.: The I\(\kappa \)B-NF-\(\kappa \)B signaling module: temporal control and selective gene activation. Science
**298**, 1241–1245 (2002)ADSCrossRefGoogle Scholar - 18.Nelson, D.E., et al.: Oscillations in nf-\(\kappa \)b signaling control the dynamics of gene expression. Science
**306**, 704–708 (2004)ADSCrossRefGoogle Scholar - 19.Krishna, S., Jensen, M.H., Sneppen, K.: Spiky oscillations in NF-kappaB signalling. Proc. Natl. Acad. Sci.
**103**, 10840–10845 (2006)ADSCrossRefGoogle Scholar - 20.Tay, S., Kellogg, R.: Noise facilitates transcriptional control under dynamic inputs. Cell
**160**, 381–392 (2015)CrossRefGoogle Scholar - 21.Heltberg, M.L., Kellogg, R.A., Krishna, S., Tay, S., Jensen, M.H.: Noise induces hopping between NF-\(\kappa \)B entrainment modes, Cell Syst. (2016)Google Scholar
- 22.Jensen, M.H., Krishna, S.: Inducing phase-locking and chaos in cellular oscillators by modulating the driving stimuli. FEBS Lett.
**586**, 1664–1668 (2012)CrossRefGoogle Scholar - 23.Strogatz, S.H.: Dynamical Systems and Chaos, pp. 278–279. Westview Press, Cambridge (2000)Google Scholar
- 24.Sun, H., Scott, S.K., Showalter, K.: Uncertain destination dynamics. Phys. Rev. E
**60**(4), 3876 (1999)ADSCrossRefGoogle Scholar - 25.Hens, C., Dana, S.K., Feudel, U.: Extreme multistability: Attractor manipulation and robustness. Chaos
**25**(5) (2015)Google Scholar - 26.Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.
**81**(25), 2340–2361 (1977). doi: 10.1021/j100540a008 CrossRefGoogle Scholar - 27.Fisher, R.A.: The use of multiple measurements in taxonomic problems. Ann. Eugen.
**7**, 179–188 (1936)CrossRefGoogle Scholar