Traveling waves dans le modèle de Kuramoto quenched

Transcription

1 Traveling waves dans le modèle de Kuramoto quenched Eric Luçon MAP5 - Université Paris Descartes Grenoble - Journées MAS août 2016 Travail en commun avec Christophe Poquet (Lyon 1) [Giacomin, L., Poquet, 2014] [L., Poquet, 2015, arxiv : ] Eric Luçon Modèle de Kuramoto 31 août / 23

2 Outline 1 The model 2 Stable invariant manifold for the McKean-Vlasov limit 3 Long-time dynamics of the empirical measure Eric Luçon Modèle de Kuramoto 31 août / 23

3 The stochastic Kuramoto model Consider the system of N mean-field interacting diffusions dθ i,t = ω idt+ K N N sin(θ j,t θ i,t)dt+db i,t, i = 1,,N, j=1 θ i S := R/2π (each particle is an angle), K > 0 : interaction strength, {B i} i : i.i.d. Brownian motions, {ω i} i : i.i.d. λ (local frequency for each particle, random environment). Introduced in 70 to describe synchronisation phenomena. Eric Luçon Modèle de Kuramoto 31 août / 23

4 dθ i,t = ω K N idt+ sin(θ j,t θ i,t)dt+db i,t, i = 1,,N, N j=1 If for example, the frequencies(ω i) are chosen along λ = 1 (δ 1 +δ+1), The Kuramoto model is invariant by rotation : if {θ j(t)} j=1...n is a solution, so is {θ j(t)+ψ} j=1...n, for all ψ S. Eric Luçon Modèle de Kuramoto 31 août / 23

5 A particular case of mean-field interacting diffusions The interacting kernel is a function of the (disordered) empirical measure µ N,t = 1 N δ (θi,t,ω N i ), N 1, t > 0, i=1 so that the dynamics can be rewritten as dθ i,t = ω idt K(sin µ N,t)(θ i,t,ω i)dt+db i,t, i = 1,,N, π Eric Luçon Modèle de Kuramoto 31 août / 23

6 The nonlinear process A good candidate for the limiting process as N for each particle is d θ t = ωdt+k sin( θ t θ)l( θ t)(d θ,dω)dt+db t, E S R Using only the Lipchitz continuity of the coefficients, one obtains [Sznitman] [ ] θi,t θ t 2 sup t [0,T] ect N. As it is, this result is only useful up to time of order clog(n)! Rq : propagation of chaos uniform in time possible under supplementary convexity assumptions on the kernels (e.g. [Bolley, Guillin, Malrieu, Gentil]). Eric Luçon Modèle de Kuramoto 31 août / 23

7 General framework for the asymptotics of mean-field systems Aim : understand the behavior of the empirical measure µ N,t, in large population (N ) and in large time (t ). N with t fixed( propagation of chaos) : the limiting object solves a nonlinear Fokker-Planck PDE, (quite general) [Sznitman, McKean, Gartner, Méléard, Jourdain, Malrieu, Bolley, Guillin, etc.] then t : stationary solutions of this PDE? (system dependent), [Giacomin, Poquet, Bertini, Pellegrin, Pakdaman, L.] The purpose is to look at the asymptotics of µ N for (t,n) (τ N,N), for N. Eric Luçon Modèle de Kuramoto 31 août / 23

8 General framework for the asymptotics of mean-field systems Aim : understand the behavior of the empirical measure µ N,t, in large population (N ) and in large time (t ). µ N,t N µ,t N with t fixed( propagation of chaos) : the limiting object solves a nonlinear Fokker-Planck PDE, (quite general) [Sznitman, McKean, Gartner, Méléard, Jourdain, Malrieu, Bolley, Guillin, etc.] then t : stationary solutions of this PDE? (system dependent), [Giacomin, Poquet, Bertini, Pellegrin, Pakdaman, L.] The purpose is to look at the asymptotics of µ N for (t,n) (τ N,N), for N. Eric Luçon Modèle de Kuramoto 31 août / 23

9 General framework for the asymptotics of mean-field systems Aim : understand the behavior of the empirical measure µ N,t, in large population (N ) and in large time (t ). µ N,t N µ,t t µ, N with t fixed( propagation of chaos) : the limiting object solves a nonlinear Fokker-Planck PDE, (quite general) [Sznitman, McKean, Gartner, Méléard, Jourdain, Malrieu, Bolley, Guillin, etc.] then t : stationary solutions of this PDE? (system dependent), [Giacomin, Poquet, Bertini, Pellegrin, Pakdaman, L.] The purpose is to look at the asymptotics of µ N for (t,n) (τ N,N), for N. Eric Luçon Modèle de Kuramoto 31 août / 23

10 General framework for the asymptotics of mean-field systems Aim : understand the behavior of the empirical measure µ N,t, in large population (N ) and in large time (t ). µ N,t N ց µ,t t?? (t,n) (, ) N with t fixed( propagation of chaos) : the limiting object solves a nonlinear Fokker-Planck PDE, (quite general) [Sznitman, McKean, Gartner, Méléard, Jourdain, Malrieu, Bolley, Guillin, etc.] then t : stationary solutions of this PDE? (system dependent), [Giacomin, Poquet, Bertini, Pellegrin, Pakdaman, L.] The purpose is to look at the asymptotics of µ N for (t,n) (τ N,N), for N. Eric Luçon Modèle de Kuramoto 31 août / 23

11 Large population limit for bounded times : µ N,t µ,t, for t [0,T]. Proposition (McKean-Vlasov equation (L. 11)) When E( ω ) < +, for a.e. (ω i) i 1, for any T > 0, (µ N,t) t [0,T] converges, as N to µ,t(dθ,dω) = p t(θ,ω)dθλ(dω), where p t solves tp t(θ,ω) = θp t(θ,ω) θ (p t(θ,ω) ( ω K )) sin( ) p t(, ω)λ(d ω). p t(θ,ω) : density of particles with θ and frequencyω, when N =. For now, suppose that the random environment is chosen in {±1}. In this case, write p + t (θ) = p t(θ,+1), p t (θ) = p t(θ, 1). Eric Luçon Modèle de Kuramoto 31 août / 23

12 Outline 1 The model 2 Stable invariant manifold for the McKean-Vlasov limit 3 Long-time dynamics of the empirical measure Eric Luçon Modèle de Kuramoto 31 août / 23

13 The large population limit Recall that the random environment is chosen in {±1}. The stationary solutions of the McKean-Vlasov PDE writes 0 = θp ± (θ) θ (p ± (θ) ( Ksin ( p + +p 2 ) )) (θ). (q +,q ) = ( 1, 1 ) is always a stationary solution (incoherence), 2π 2π there exists K c > 0, if K > K c, there exists a whole circle of synchronized stationary solutions M = {q ψ ( ) = q 0( ψ),ψ S} q 1 (θ) q 1 (θ) θ Eric Luçon Modèle de Kuramoto 31 août / 23

14 The large population limit Recall that the random environment is chosen in {±1}. The stationary solutions of the McKean-Vlasov PDE writes 0 = θp ± (θ) θ (p ± (θ) ( Ksin ( p + +p 2 ) )) (θ). (q +,q ) = ( 1, 1 ) is always a stationary solution (incoherence), 2π 2π there exists K c > 0, if K > K c, there exists a whole circle of synchronized stationary solutions M = {q ψ ( ) = q 0( ψ),ψ S}. M ψ q ψ ( ) := q 0 ( ψ) ψ q 0 ( ) 1 2π Eric Luçon Modèle de Kuramoto 31 août / 23

15 Local stability of the circle M Let q = (q +,q ) M. Add a small perturbationu t = (u + t,u t ), with S u± t (θ)dθ = 0. The evolution of u is given by tu t = L qu t +R q(u t), where L qu ± t = 1 ( ) ( 2 2 θu ± t θ (±ω 0u ± t u ± q + +q t Ksin q ± u + t +u t Ksin 2 2 is the linearization of q and R q is quadratic. )), Eric Luçon Modèle de Kuramoto 31 août / 23

16 Definition For q M, let H 1 q be the dual of H 1 q, weighted Sobolev space endowed with u 1,q := ( 1 ( θ u σ (θ)) 2 q (θ)dθ)12 σ. 2 σ=± S Theorem [Bertini, Giacomin, Pakdaman, 2010], [Giacomin, L., Poquet, 2014] : For q M, L q is self-adjoint in Hq 1 and the following decomposition holds : = T q M q where H 1 q T q : L qq = 0 N q : e tlq u 1,q Ce λt u 1,q q M The neutral directiont q reflects the invariance by rotation of the system. Eric Luçon Modèle de Kuramoto 31 août / 23

17 Idea of proof : perturbing the case without disorder. 1 Without disorder (ω i 0), dθ i,t = K N N sin(θ j,t θ i,t)dt+db i,t, i = 1,,N, j=1 the system is reversible with invariant measure π N,K (XY model) ( ) K N π N,K(dθ) exp cos(θ i θ j) dθ N i,j=1 The McKean-Vlasov equation can be written in a gradient form : [ ( )] δf(qt) tq t(θ) = q t(θ), δq t(θ) for F(q) = 1 2 S q(θ)logq(θ)dθ K 2 [Bertini, Giacomin, Pakdaman, 2010] S 2 cos(θ θ )q(θ)q(θ )dθdθ. 2 This invariant manifold is stable by small perturbation. Eric Luçon Modèle de Kuramoto 31 août / 23

18 Outline 1 The model 2 Stable invariant manifold for the McKean-Vlasov limit 3 Long-time dynamics of the empirical measure Eric Luçon Modèle de Kuramoto 31 août / 23

19 µ = 1 2 (δ 1 +δ 1 ). Eric Luçon Modèle de Kuramoto 31 août / 23

20 Symmetric disorder Suppose that (ω i) i 1 is chosen alongλ = 1 2 δ δ Eric Luçon Modèle de Kuramoto 31 août / 23

21 Symmetric disorder Suppose that (ω i) i 1 is chosen alongλ = 1 2 δ δ Here, as N =, there is no asymmetry red/blues, so no reasons to rotate in any direction whatsoever : the limit µ,t gives no information. But for N large but finite, fluctuations of the sample (ω 1,...,ω N) induce a microscopic asymmetry of order N. One expects random traveling waves on a time scale N. Eric Luçon Modèle de Kuramoto 31 août / 23

22 Reformulation of the system Let N 1 and N + (resp. N ) the random size of the population with disorder +1 (resp. 1). Hence, one can rewrite dθ + j,t = +1+ K N + N sin(θ + l,t θ + N j,t)+ sin(θ l,t θ j,t) + dt+db j,t, l=1 l=1 dθ j,t = 1+ K N + N sin(θ + l,t θ N j,t)+ sin(θ l,t θj,t) dt+db j,t l=1 l=1 The empirical measure µ N is identified with (µ + N,µ N ) µ ± N,t = 1 N ± δ N ± θ ±. j,t Within this reformulation, the random environnement consists in the random sizes (N +,N ). j=1 Eric Luçon Modèle de Kuramoto 31 août / 23

23 Quenched traveling waves in long time Definition A sequence(ω i) i 1 is admissible if for all ζ > 0, there exists N 0 such that for N N 0, max( ξ + N, ξ N ) N ζ, with ( N ξ ± N := N 1/2 ± N 1 ). 2 Theorem [L., Poquet (2015)] : Let τ f > 0, ψ 0 S and an admissible sequence (ω i) i 1. If for all ε > 0 lim P( µ N,0 q ψ0 N 1 ε ) = 1, then where b is linear. ( ) lim P sup µ N,N N 1/2 τ q ψ 0 +b(ξ N )τ 1 ε τ [0,τ f ] = 1, Eric Luçon Modèle de Kuramoto 31 août / 23

24 We want to quantify the proximity of µ N w.r.t. the circle M. Define ν N,t := µ N,t q ψ, where q ψ M. Introducing the semigroup e tlq ψ, there exists a mild equation for ν N in H 1 : ν N,t = e tlq ψν N,0 + t 0 e (t s)lq ψ (D q,n +R q,n(ν N,s))ds+Z N,t, where ( {( ( }) N D q,n = θ q + ψ )(Ksin q 1 +ψ )+ N )(Ksin q 1 ψ ) is N 2 N 2 the drift induced by the disorder. R q,n(ν N) is a quadratic term, Z N,t is the noise part. Eric Luçon Modèle de Kuramoto 31 août / 23

25 ν N,t = e tlq ψν N,0 + t There is a competition between 0 e (t s)lq ψ (D q,n +R q,n(ν N,s))ds+Z N,t. the drift D N and the noise Z N that move µ N away from the circle M the semigroup e tlq that keeps µ N close to M along the normal direction N q, Hence : the dynamics of µ N is essentially along the neutral direction T q : one obtains a traveling wave whose speed depends only on the drift D N. ( ) N Since D N is a linear functional of + 1, this term is of order 1 N 2 N. We need to wait a time of order N to see these traveling waves. Eric Luçon Modèle de Kuramoto 31 août / 23

26 The procedure is to consider the dynamics of ν N on [0, NT], by discretization on subintervals[nt,(n+1)t], n = 0,..., N. Using the semimartingale decomposition, one proves recursively that if d(µ N,nT,M) = O(N 1/2+2ζ ) for a certain n, then with high probability as N, on the interval [nt,(n+1)t], The drift D N,t and the noise Z N,t are of order O(N 1/2+ζ ) the quadratic term R N is of order O(N 1+4ζ ), ν N,t is of order O(N 1/2+2ζ ) One can recursively define the projectionq ψn of µ N,nT on M q ψn = P ψn (µ N,nT). The process ν n,t := µ N,nt q ψn, t [0,T] is well defined and one has Proposition (a priori bound on ν N ) With a probability close to 1 as N, sup sup 1 n N t [0,T] ν n,t 1 CN 1/2+2ζ. Eric Luçon Modèle de Kuramoto 31 août / 23

27 Mean-field model invariant by rotation, Stable invariant manifold in the N = -limit, When N and t fixed, the population is balanced : no traveling wave at the level of µ,t. Quenched disorder induces traveling waves at the scale of fluctuations for a finite population on a time scale of order N. Questions : more complicated dynamics (FitzHugh-Nagumo)? more general graphs of interactions (Erdös-Renyi, locally tree-like graphs)? [Delattre, Giacomin, L., 2016] Eric Luçon Modèle de Kuramoto 31 août / 23

28 Merci de votre attention! E. Luçon and C. Poquet Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model, arxiv : Eric Luçon Modèle de Kuramoto 31 août / 23