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1. Strong coupling yields abrupt synchronization transitions in coupled oscillators

   https://doi.org/10.1103/PhysRevResearch.6.033328  

   Ocampo-Espindola, Jorge L; Kiss, István Z; Bick, Christian; Wedgwood, Kyle_C A (September 2024, Physical Review Research)

   Coupled oscillator networks often display transitions between qualitatively different phase-locked solutions—such as synchrony and rotating wave solutions—following perturbation or parameter variation. In the limit of weak coupling, these transitions can be understood in terms of commonly studied phase approximations. As the coupling strength increases, however, predicting the location and criticality of transition, whether continuous or discontinuous, from the phase dynamics may depend on the order of the phase approximation—or a phase description of the network dynamics that neglects amplitudes may become impossible altogether. Here we analyze synchronization transitions and their criticality systematically for varying coupling strength in theory and experiments with coupled electrochemical oscillators. First, we analyze bifurcations analysis of synchrony and splay states in an abstract phase model and discuss conditions under which synchronization transitions with different criticalities are possible. In particular, we show that such conditions can be understood by considering the relative contributions of higher harmonics to the phase dynamics. Second, we illustrate that transitions with different criticality indeed occur in experimental systems. Third, we highlight that the amplitude dynamics observed in the experiments can be captured in a numerical bifurcation analysis of delay-coupled oscillators. Our results showcase that reduced order phase models may miss important features that one would expect in the dynamics of the full system. Published by the American Physical Society2024 

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2. Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin-1/2 chain

   https://doi.org/10.1016/j.nuclphysb.2024.116624  

   Gehrmann, Sascha; Kotousov, Gleb A; Lukyanov, Sergei L (July 2024, Nuclear physics B)

   It is known that for the Heisenberg XXZ spin - chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schrödinger equation for the anharmonic oscillator. 

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3. A global bifurcation organizing rhythmic activity in a coupled network

   https://doi.org/10.1063/5.0089946  

   Medvedev, Georgi S.; Mizuhara, Matthew S.; Phillips, Andrew (August 2022, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   We study a system of coupled phase oscillators near a saddle-node on invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using Ott–Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation from noncontractibile ones after bifurcation. Both families are stable for the model at hand. 

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4. Localized states in passive and active phase-field-crystal models

   https://doi.org/10.1093/imamat/hxab025  

   Holl, Max Philipp; Archer, Andrew J; Gurevich, Svetlana V; Knobloch, Edgar; Ophaus, Lukas; Thiele, Uwe (July 2021, IMA Journal of Applied Mathematics)

   null (Ed.)

   Abstract The passive conserved Swift–Hohenberg equation (or phase-field-crystal [PFC] model) describes gradient dynamics of a single-order parameter field related to density. It provides a simple microscopic description of the thermodynamic transition between liquid and crystalline states. In addition to spatially extended periodic structures, the model describes a large variety of steady spatially localized structures. In appropriate bifurcation diagrams the corresponding solution branches exhibit characteristic slanted homoclinic snaking. In an active PFC model, encoding for instance the active motion of self-propelled colloidal particles, the gradient dynamics structure is broken by a coupling between density and an additional polarization field. Then, resting and traveling localized states are found with transitions characterized by parity-breaking drift bifurcations. Here, we briefly review the snaking behavior of localized states in passive and active PFC models before discussing the bifurcation behavior of localized states in systems of (i) two coupled passive PFC models with common gradient dynamics, (ii) two coupled passive PFC models where the coupling breaks the gradient dynamics structure and (iii) a passive PFC model coupled to an active PFC model. 

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5. Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability

   https://doi.org/10.1016/j.physd.2024.134058  

   Chekroun, Mickaël D; Liu, Honghu (April 2024, Physica D: Nonlinear Phenomena)

   Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics. To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems. These dynamical insights enable us in turn to design a stochastic model whose solutions---as the delay parameter drifts slowly through its critical values---produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping ‘‘points’’ beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena. 

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