# Origin and scaling of chaos in weakly coupled phase oscillators

Mallory Carlu, Francesco Ginelli, Antonio Politi

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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 012203 Physical Review. E, Statistical, Nonlinear and Soft Matter Physics 97 1 https://doi.org/10.1103/PhysRevE.97.012203 Published - 10 Jan 2018

### Fingerprint

Tangent Space
tangents
Spacing
chaos
Chaos
oscillators
spacing
Scaling
Fluctuations
scaling
Largest Lyapunov Exponent
Stochastic Approximation
Identically distributed
synchronism
Phase Space
Logarithmic
Ensemble
Synchronization
Strictly
Trace

### Keywords

• nlin.CD
• cond-mat.stat-mech

### Cite this

Origin and scaling of chaos in weakly coupled phase oscillators. / Carlu, Mallory; Ginelli, Francesco; Politi, Antonio.

In: Physical Review. E, Statistical, Nonlinear and Soft Matter Physics, Vol. 97, No. 1, 012203, 10.01.2018.

Research output: Contribution to journalArticle

title = "Origin and scaling of chaos in weakly coupled phase oscillators",
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.",
keywords = "nlin.CD, cond-mat.stat-mech",
author = "Mallory Carlu and Francesco Ginelli and Antonio Politi",
note = "Acknowledgments AP wishes to acknowledge D. Paz ́o for a preliminary analysis. All the authors thank H. Chat ́e for useful discussions. We acknowledge support from EU Marie Curie ITN grant n. 64256 (COSMOS).",
year = "2018",
month = "1",
day = "10",
doi = "10.1103/PhysRevE.97.012203",
language = "English",
volume = "97",
journal = "Physical Review. E, Statistical, Nonlinear and Soft Matter Physics",
issn = "1539-3755",
publisher = "AMER PHYSICAL SOC",
number = "1",

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TY - JOUR

T1 - Origin and scaling of chaos in weakly coupled phase oscillators

AU - Carlu, Mallory

AU - Ginelli, Francesco

AU - Politi, Antonio

N1 - Acknowledgments AP wishes to acknowledge D. Paz ́o for a preliminary analysis. All the authors thank H. Chat ́e for useful discussions. We acknowledge support from EU Marie Curie ITN grant n. 64256 (COSMOS).

PY - 2018/1/10

Y1 - 2018/1/10

N2 - 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.

AB - 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.

KW - nlin.CD

KW - cond-mat.stat-mech

U2 - 10.1103/PhysRevE.97.012203

DO - 10.1103/PhysRevE.97.012203

M3 - Article

VL - 97

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 1

M1 - 012203

ER -