Obtaining certificates for complete synchronisation of coupled oscillators

In this paper, we provide a novel reformulation of sufficient conditions that guarantee global complete synchronisation of coupled identical oscillators to make them computationally implementable. To this end, we use semidefinite programming techniques. For the first time, we can efficiently search for and obtain certificates for synchronisability and, additionally, also optimise associated cost functions. In this paper, a Lyapunov-like function (certificate) is used to certify that all trajectories of a networked system consisting of coupled dynamical systems will eventually converge towards a common one, which implies synchronisation. Moreover, we establish new conditions for complete synchronisation, which are based on the so called Bendixson’s Criterion for higher dimensional systems. This leads to major improvements on the lower bound of the coupling constant that guarantees global complete synchronisation. Importantly, the certificates are obtained by analysing the connection network and the model representing an individual system only. In order to illustrate the strength of our method we apply it to a system of coupled identical Lorenz oscillators and to coupled van der Pol oscillators.     

Regular and Chaotic Vibrations of a Parametrically and Self-Excited System Under Internal Resonance Condition

Vibrations of a parametrically and self-excited system with two degrees of freedom have been analysed in this paper. The system is constituted by two parametrically coupled oscillators characterised by self-excitation and nonlinear Duffing’s type nonlinearities. Synchronisation phenomenon has been determined near the principal resonances in the neighbourhood of the first p₁ and the second p₂ natural frequencies, and near the combination resonance (p₁+p₂)/2. Vibrations have been investigated for parameters which satisfy the internal resonance condition p₂/p₁=3. The existence and break down of the synchronisation phenomenon have been revealed analytically by the multiple time scale method, whilst transition of the system to chaotic motion has been carried out numerically.     

Symbolic dynamics of two coupled Lorenz maps: From uncoupled regime to synchronisation

The bounded dynamics of a system of two coupled piecewise affine and chaotic Lorenz maps is studied over the coupling range, from the uncoupled regime where the entropy is maximal, to the synchronized regime where the entropy is minimal. By formulating the problem in terms of symbolic dynamics, bounds on the set of orbit codes (or the set itself, depending on parameters) are determined which describe the way the dynamics is gradually affected as the coupling increases. Proofs rely on monotonicity properties of bounded orbit coordinates with respect to some partial ordering on the corresponding codes. The estimates are translated in terms of (bounds on the) entropy, which are monotonically decreasing with coupling and which are compared to the numerically computed entropy. A good agreement is found which indicates that these bounds capture the essential features of the transition from the uncoupled regime to synchronisation.     

Synchronisation of chaos and its applications

Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical networks can exhibit behaviour in which deterministic chaos, exhibiting unpredictability and disorder, coexists with synchronisation, a classical paradigm of order. We survey the main theory behind complete, generalised and phase synchronisation phenomena in simple as well as complex networks and discuss applications to secure communications, parameter estimation and the anticipation of chaos.     

Chaos synchronization on the N-torus and cryptography

A class of chaotic dynamical systems on the N-dimensional torus is proposed for masking some information in secure communications. The information is then recovered thanks to a chaos synchronization process. To cite this article: L. Rosier et al., C. R. Mecanique 332 (2004).

Résumé

Nous proposons une classe de systèmes chaotiques sur le tore N-dimensionnel pour masquer une information à transmettre dans une communication sécurisée. Cette information est ensuite reconstruite à l'aide d'un mécanisme de synchronisation du chaos. Pour citer cet article : L. Rosier et al., C. R. Mecanique 332 (2004).     

Driven dissipative dynamics and topology of quantum impurity systems

In this review, we provide an introduction to and an overview of some more recent advances in real-time dynamics of quantum impurity models and their realizations in quantum devices. We focus on the Ohmic spin–boson and related models, which describe a single spin-1/2 coupled with an infinite collection of harmonic oscillators. The topics are largely drawn from our efforts over the past years, but we also present a few novel results. In the first part of this review, we begin with a pedagogical introduction to the real-time dynamics of a dissipative spin at both high and low temperatures. We then focus on the driven dynamics in the quantum regime beyond the limit of weak spin–bath coupling. In these situations, the non-perturbative stochastic Schrödinger equation method is ideally suited to numerically obtain the spin dynamics as it can incorporate bias fields $h_z(t)$ of arbitrary time-dependence in the Hamiltonian. We present different recent applications of this method: (i) how topological properties of the spin such as the Berry curvature and the Chern number can be measured dynamically, and how dissipation affects the topology and the measurement protocol, (ii) how quantum spin chains can experience synchronization dynamics via coupling with a common bath. In the second part of this review, we discuss quantum engineering of spin–boson and related models in circuit quantum electrodynamics (cQED), quantum electrical circuits, and cold-atoms architectures. In different realizations, the Ohmic environment can be represented by a long (microwave) transmission line, a Luttinger liquid, a one-dimensional Bose–Einstein condensate or a chain of superconducting Josephson junctions. We show that the quantum impurity can be used as a quantum sensor to detect properties of a bath at minimal coupling, and how dissipative spin dynamics can lead to new insight in the Mott–superfluid transition.     

Computing Arnol′d tongue scenarios

A famous phenomenon in circle-maps and synchronisation problems leads to a two-parameter bifurcation diagram commonly referred to as the Arnol′d tongue scenario. One considers a perturbation of a rigid rotation of a circle, or a system of coupled oscillators. In both cases we have two natural parameters, the coupling strength and a detuning parameter that controls the rotation number/frequency ratio. The typical parameter plane of such systems has Arnol′d tongues with their tips on the decoupling line, opening up into the region where coupling is enabled, and in between these Arnol′d tongues, quasi-periodic arcs. In this paper, we present unified algorithms for computing both Arnol′d tongues and quasi-periodic arcs for both maps and ODEs. The algorithms generalise and improve on the standard methods for computing these objects. We illustrate our methods by numerically investigating the Arnol′d tongue scenario for representative examples, including the well-known Arnol′d circle map family, a periodically forced oscillator caricature, and a system of coupled Van der Pol oscillators.     

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

Spontaneous synchronisation is a remarkable collective effect observed in nature, whereby a population of oscillating units, which have diverse natural frequencies and are in weak interaction with one another, evolves to spontaneously exhibit collective oscillations at a common frequency. The Kuramoto model provides the basic analytical framework to study spontaneous synchronisation. The model comprises limit-cycle oscillators with distributed natural frequencies interacting through a mean-field coupling. Although more than forty years have passed since its introduction, the model continues to occupy the centre stage of research in the field of non-linear dynamics and is also widely applied to model diverse physical situations. In this brief review, starting with a derivation of the Kuramoto model and the synchronisation phenomenon it exhibits, we summarise recent results on the study of a generalised Kuramoto model that includes inertial effects and stochastic noise. We describe the dynamics of the generalised model from a different yet a rather useful perspective, namely, that of long-range interacting systems driven out of equilibrium by quenched disordered external torques. A system is said to be long-range interacting if the inter-particle potential decays slowly as a function of distance. Using tools of statistical physics, we highlight the equilibrium and nonequilibrium aspects of the dynamics of the generalised Kuramoto model, and uncover a rather rich and complex phase diagram that it exhibits, which underlines the basic theme of intriguing emergent phenomena that are exhibited by many-body complex systems.