### Abstract

We discuss the behavior of the largest Lyapunov exponent $\lambda$ in the incoherent phase of large ensembles of heterogeneous, globally-coupled, phase oscillators. We show that the scaling with the system size $N$ depends on the details of the spacing distribution of the oscillator frequencies. For sufficiently regular distributions $\lambda \sim 1/N$, while for strong fluctuations of the frequency spacing, $\lambda \sim \ln N/N$ (the standard setup of independent identically distributed variables belongs to the latter class). In spite of the coupling being small for large $N$, the development of a rigorous perturbative theory is not obvious. In fact, our analysis relies on a combination of various types of numerical simulations together with approximate analytical arguments, based on a suitable stochastic approximation for the tangent space evolution. In fact, the very reason for $\lambda$ being strictly larger than zero is the presence of finite size fluctuations. We trace back the origin of the logarithmic correction to a weak synchronization between tangent and phase space dynamics.

Original language | English |
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Article number | 012203 |

Journal | Physical Review. E, Statistical, Nonlinear and Soft Matter Physics |

Volume | 97 |

Issue number | 1 |

DOIs | |

Publication status | Published - 10 Jan 2018 |

### Keywords

- nlin.CD
- cond-mat.stat-mech

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## Cite this

Carlu, M., Ginelli, F., & Politi, A. (2018). Origin and scaling of chaos in weakly coupled phase oscillators.

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*,*97*(1), [012203]. https://doi.org/10.1103/PhysRevE.97.012203