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Dynamics on Mδ: analysis of the active rotators case
Let us use the results of the previous section to tackle the questions we have raised in the introduction for the active rotators case and that, ultimately, boil down to: what is the relation between the Isolated Deterministic one dimensional System ˙ψ = −V ′(ψ) (IDS) and the behavior of the associated N dimensional diffusion, for N large? So we focus on (2.2.20) with G[p] = (pV ′)′ and regularity assumptions on V ′ are going to appear along the way. Theorem 2.2.1 tells us that if kV ′k∞ < ∞, at least when δ is small enough, the N → ∞ limit system – ruled by (2.2.20) – is described by a dynamics on a one dimensional smooth and compact manifold Mδ equivalent to a circle and, via Theorem 2.2.2 and Theorem 2.2.3, we have a sharp control on this dynamics. In order to be precise on this issue let us speed up time by 1/δ in (2.2.24). If we keep just the leading terms we are dealing with the dynamics ˙ψ = −f(ψ) .
Noise and interaction induce arbitrary generic dynamics
It is practical and sufficient to work with V ′(·) that is a trigonometric polynomial, that is V ′(θ) = a0 + Xn j=1 (aj cos(jθ) + bj sin(jθ)) .
Theorem 2.3.1. For any generic dynamics on the circle ˙ψt = −f(ψt) with f ∈ C1(S;R) and for any value of K > 1 there exists a trigonometric polynomial V ′(·) (see Remark 2.3.2 for an explicit expression) such that for δ small enough, the phase dynamics on Mδ (2.3.3) is δ-close to ˙ψ = −f′(ψ). Proof. Let f be a generic function in C1. By the Stone-Weierstrass Theorem, for every ε > 0 there exists a trigonometric polynomial P(·) such that kf′ −Pk∞ ≤ ε. If c0 is such that R 2π 0 (P − c0) = 0 then, since R 2π 0 f′ = 0, |c0| ≤ ε. Thus if we define the trigonometric polynomial Q(ψ) := f(0) + R ψ 0 (P(θ) − c0) dθ we have kQ−fkC1 = kQ−fk∞+kP −c0−f′k∞ ≤ (2π+1)kP −c0−f′k∞ ≤ (4π+2)ε , (2.3.5) so it suffices to consider functions f which are trigonometric polynomials: f(θ) = A0 + Xn k=1 (Ak cos(kθ) + Bk sin(kθ)) .
On the persistence of normally hyperbolic manifolds
In this section we prove theorem 2.2.1. The proof in a more general case can be found in [99] but we pay more attention on the relation between the various small parameters that enter the proof. We first give a lemma which defines a parametrisation in a neighbourhood of M using the scalar structure given by the operators Lq. The proof of this lemma is in [99, p. 501].
Collective phenomena in noisy coupled oscillators
Coupled oscillator models are omnipresent in the scientific literature because the emergence of coherent behavior in large families of interacting units that have a periodic behavior, that we generically call oscillators, is an extremely common phenomenon (crickets chirping, fireflies flashing, planets orbiting, neurons firing,…). It is impossible to properly account for the literature and the various models proposed for this kind of phenomena, but while a precise description of each of the different instances in which synchronization emerges demands specific, possibly very complex, models, the Kuramoto model has emerged as capturing some of the fundamental aspects of synchronization [1]. It can be introduced via the system of N stochastic differential equations dϕωj (t) = ωj dt − K N XN i=1 sin(ϕωj (t) − ϕωi (t)) dt + σ dBj(t) ,
Table of contents :
1 Introduction
1.1 Mod`ele de Kuramoto
1.1.1 Mod`ele de Kuramoto et r´eduction `a un syst`eme de phases
1.1.2 Pr´esentation du mod`ele de Kuramoto
1.2 Mod`ele de Kuramoto sans d´esordre
1.2.1 R´eversibilit´e
1.2.2 Limite du nombre infini de particules en temps fini
1.2.3 Solutions stationnaires et synchronisation
1.2.4 Comportement en temps long
1.3 Active Rotators et excitabilit´e
1.3.1 Syst`emes excitables bruit´es en interaction
1.3.2 Le mod`ele des Active Rotators
1.3.3 Sous-vari´et´es normalement hyperboliques stables
1.3.4 Probl`eme g´en´eral et persistance des sous-vari´et´es normalement hyperboliques stables
1.3.5 Dynamique de phases sur Mδ
1.3.6 P´eriodicit´e induite par le bruit
1.4 Mod`ele de Kuramoto d´esordonn´e
1.4.1 Limite du nombre infini de particules
1.4.2 D´esordre sym´etrique et solutions stationnaires
1.4.3 Asymptotique de faible d´esordre
1.4.4 Hyperbolicit´e normale
1.4.5 Solutions p´eriodiques
1.4.6 Retour sur les Actives Rotators
1.4.7 Stabilit´e lin´eaire dans le cas du d´esordre sym´etrique
1.5 Probl`eme de sortie de domaine et r´eduction
1.5.1 Probl`eme de sortie de domaine
1.5.2 Pr´esentation du mod`ele
1.5.3 Hyperbolicit´e normale
1.5.4 R´eduction `a un syst`eme de phases
2 Synchronization and excitable systems
2.1 Introduction
2.1.1 Coupled excitable systems
2.1.2 Active rotator models
2.1.3 Informal presentation of approach and results
2.2 Mathematical set-up and main results
2.2.1 On the reversible Kuramoto PDE
2.2.2 The full evolution equation
2.3 Dynamics on Mδ: analysis of the active rotators case
2.3.1 Noise and interaction induce arbitrary generic dynamics
2.3.2 Active rotators with V (θ) = θ − a cos(θ)
2.3.3 Active rotators with V (θ) = θ − a cos(jθ)/j, j = 2, 3,
2.4 Perturbation arguments
2.5 On the persistence of normally hyperbolic manifolds
2.A On a norm equivalence
2.B Erratum
3 Kuramoto model : the effect of disorder
3.1 Introduction
3.1.1 Collective phenomena in noisy coupled oscillators
3.1.2 The Fokker-Planck or McKean-Vlasov limit
3.1.3 About stationary solutions to (3.1.4)
3.1.4 An overview of the results we present
3.2 Mathematical set-up and main results
3.2.1 The reversible and the non-disordered PDE
3.2.2 Synchronization: the main result without symmetry assumption
3.2.3 Symmetric disorder case
3.2.4 Organization of remainder of the paper
3.3 Hyperbolic structures and periodic solutions
3.3.1 Stable normally hyperbolic manifolds
3.3.2 M0 is a SNHM
3.3.3 The spectral gap estimate (proof of Proposition 3.2.1)
3.4 Perturbation arguments
3.5 Active rotators
3.6 Symmetric case: stability of the stationary solutions
3.6.1 On the non-trivial stationary solutions (proof of Lemma 3.2.3)
3.6.2 On the linear stability of non-trivial stationary solutions
3.A Regularity in the non-linear Fokker-Planck equation
4 Random long time behavior
4.1 Introduction
4.1.1 Overview
4.1.2 The model
4.1.3 The N → ∞ dynamics and the stationary states
4.1.4 Random dynamics on M: the main result
4.1.5 The synchronization phenomena viewpoint
4.1.6 A look at the literature and perspectives
4.2 More on the mathematical set-up and sketch of proofs
4.2.1 On the linearized evolution
4.2.2 About the manifold M
4.2.3 A quantitative heuristic analysis: the diffusion coefficient
4.2.4 The iterative scheme
4.3 A priori estimates: persistence of proximity to M
4.3.1 Noise estimates
4.4 The effective dynamics on the tangent space
4.5 Approach to M
4.6 Proof of Theorem 4.1.1
4.A The evolution in H−1
4.A.1 Second order estimates of the projection
4.B Spectral estimates
5 Escape problem and phase reduction
5.1 Introduction
5.1.1 Phase reduction and escape problem
5.1.2 Mathematical set-up and main result
5.2 Preliminary results of geometrical nature
5.2.1 Projection and local coordinates
5.2.2 Stable Normally Hyperbolic Manifolds
5.2.3 Persistence of hyperbolic manifolds
5.2.4 Choice of projection
5.3 Quasipotential and optimal path
5.4 Proof of Theorem 5.1.1 and Corollary 5.1.2
5.4.1 Sketch of the proof
5.4.2 Preliminary results
5.4.3 Proof of Theorem 5.1.1
5.4.4 Proof of Corollary 5.1.2
