2026

I. Leyva, I. Sendiña-Nadal, R. Sevilla-Escoboza, V. P. Vera-Ávila & C. Letellier
From chaotic itinerancy to intermittent synchronization in complex networks
Physical Review Research, 8, 013328, 2026. Online

• Abstract
  Although synchronization has been extensively studied, key processes underlying its emergence in complex networks have remained hidden by the use of global order parameters. Here, we uncover how the route to synchronization unfolds through a sequential transition between two well-known but previously unconnected phenomena : chaotic itinerancy CI and intermittent synchronization IS. Using a symbolic dynamics approach, we show that CI develops as a collective yet unsynchronized exploration of different domains of the high-dimensional attractor of the network, whose dimension is reduced as the coupling increases, ultimately collapsing into the reference chaotic attractor of an individual unit. At this stage, the IS emerges as irregular alternations between synchronous and asynchronous phases. These phenomena are mutually exclusive, each dominating a distinct coupling interval and governed by different mechanisms. Furthermore, network structural heterogeneity enhances itinerant behavior since access to different domains of the attractor depends on the nodes’ topological roles. The CI-IS crossover occurs within a consistent coupling interval across models and topologies. Experiments on electronic oscillator networks confirm this two-step process, establishing a unified framework for understanding the route to synchronization in complex systems.

Eduardo Mendes, Claudia Lainscsek & Christophe Letellier
Classification of chaotic systems by using canonical (jerk) forms : the case of Lorenz-like systems
Chaos, 36 (5), in print.

• Abstract
  When a \(d\)-dimensional system is investigated through one of its variables \(v\), a natural space for characterizing its dynamics is provided by the v-induced differential embedding spanned by that variable and its \((d − 1)\) successive derivatives. The corresponding governing equations are called the canonical form of that system. Canonical forms — often called jerk equations when, in addition, the system is three-dimensional — can be used for classifying chaotic systems by comparing, not only their algebraic structures, as done in previous works, but also by considering their parameter values, helping in identifying system’s parameter values for producing equivalent dynamics, that is, for producing a given type of attractor. Two types of equivalence can therefore be distinguished : (i) structural, when only the algebraic structures of canonical forms match, and (ii) dynamical, when the canonical parameter values coincide. Despite their potential, canonical forms have been only partially exploited for systematically relating and identifying chaotic systems. In this work, we address this gap by using the \(x\)-induced canonical form of the Lorenz system to investigate its relationships with other Lorenz-like systems. In particular, we show how canonical forms can be used to determine parameter values that reproduce a Lorenz attractor, even when the underlying algebraic structures differ. This provides a novel perspective in which the algebraic structure of the governing equations is explicitly leveraged to generate prescribed chaotic attractors, opening new ways for system identification and design.

Christophe Letellier
A Lorenz-like system with an inversion symmetry:topology of some of its attractors
Chaos, 36 (5), 053140, 2026. Online

• Abdstract
  In a study devoted to discrete Lorenz systems, Gonchenko and coworkers Chaos, 32, 121107, 2022 briefly mentioned a three-dimensional system that would be a Lorenz-like system if a quadratic monomial was not replaced with a cubic one. Such a modification introduces an in- version symmetry, rending this system atypical and whose dynamics deserves to be investigated. This is here performed along a line of the parameter space in terms of template for some of its attractors. There are similarities with the classifical bifurcation diagram produced by the Lorenz system with \(R\) as the bifurcation parameter, and some differences that are exhibited. In particular, it is shown that an attractor merging crisis with the saddle point located at the origin of the state space allows to produce an attractor that is similar to the one observed in the Moore-Spiegel system long ago Letellier, PhD, Paris vii University, 1994 and that is rarely observed in Lorenz-like system. We also took the opportunity to show how it may be useful to investigate the structure of the dynamics that can be safely characterized with a first-return map built on \(\Delta \tau_n = \tau_n - \tau_{n−1}\)where τn is the duration of the return to a surface of section. This paper was written with a warm memory of the time spent with Robert Gilmore in discovering the effect of symmetries in chaotic systems.

This paper was written with a warm memory of the time spent with Robert Gilmore in discovering the effect of symmetries in chaotic systems.