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# Complexity and transition to chaos in coupled Adler-type oscillators.

D. Estevez-Moya
Facultad de Física, Universidad de La Habana, San Lazaro y L. CP 10400. La Habana. Cuba.
E. Estevez-Rams
[estevez@fisica.uh.cu](mailto:estevez@fisica.uh.cu)Facultad de Física-Instituto de Ciencias y Tecnología de Materiales(IMRE), Universidad de La Habana, San Lazaro y L. CP 10400. La Habana. Cuba.
H. Kantz
MPI for the Physics of Complex Systems. Nöthnitzer Strasse
38\. D-01187 Dresden.

###### Abstract

Coupled non-linear oscillators are ubiquitous in dynamical studies. A wealth of behaviors have been found mostly for globally coupled systems. From a complexity perspective, less studied have been systems with local coupling, which is the subject of this contribution. The phase approximation is used, as weak coupling is assumed. In particular, the so called needle region, in parameter space, for Adler-type oscillators with nearest neighbors coupling is carefully characterized. The reason for this emphasis is that in the border of this region to the surrounding chaotic one, computation enhancement at the edge of chaos has been reported. The present study shows that different behaviors within the needle region can be found, and a smooth change of dynamics could be identified. Entropic measures further emphasize the region’s heterogeneous nature with interesting features, as seen in the spatiotemporal diagrams. The occurrence of wave-like patterns in the spatiotemporal diagrams points to non-trivial correlations in both dimensions. The wave patterns change as the control parameters change without exiting the needle region. Spatial correlation is only achieved locally at the onset of chaos, with different clusters of oscillators behaving coherently while disordered boundaries appear between them.

## I Introduction

Coupled non-linear oscillators under weak coupling conditions have been studied, at least since the pioneering works of Winfree [winfre67](https://ar5iv.labs.arxiv.org/html/2310.12166#bib.bib1 ""). In his approach, the oscillator phase is the relevant parameter in the dynamics instead of the oscillator amplitude. Earlier studies by Adler [adler46](https://ar5iv.labs.arxiv.org/html/2310.12166#bib.bib2 "") of locking occurrence in feedback circuits lead to an equation for a type of non-linear oscillators (since then known as Adler-type oscillators) with phase equation

|     |     |     |     |
| --- | --- | --- | --- |
|  | d​θd​t=−γ​sin⁡θ+Δ​ω,𝑑𝜃𝑑𝑡𝛾𝜃Δ𝜔\\frac{d\\theta}{dt}=-\\gamma\\sin\\theta+\\Delta\\omega, |  | (1) |

where θ𝜃\\theta is the oscillator phase, γ𝛾\\gamma a control parameter, and Δ​ω=ω0−ω1Δ𝜔subscript𝜔0subscript𝜔1\\Delta\\omega=\\omega\_{0}-\\omega\_{1} is the difference between the natural frequency of the oscillator and the imposed frequency on the circuit. Adler’s work has been generalized to systems of coupled non-linear oscillators that describe a wide range of phenomena in diverse fields ranging from physics to chemistry

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