A Stochastic Pitchfork Bifurcation in a Reaction-Diffusion Equation

A stochastic pitchfork bifurcation in a reaction-diffusion equation By Tom´ as Caraballo, Jos´ e A. Langa, & James C. Robinson† Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla. Spain We study in some detail the structure of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise, du= (−Au +βu −u3) dt+σu ◦dWtx∈D⊂Rm. First we prove, for m≤5, a lower bound on the dimension of the random attractor which is of the same order in βas the upper bound we derived in an earlier paper, and is the same as that obtained in the deterministic case. Then we show, for m= 1, that as βpasses through λ1(the first eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the first example of such a stochastic bifurcation in an infinitedimensional setting. Central to our approach is the existence of a random unstable manifold. Keywords: Random attractors, stochastic bifurcation, Hausdorff dimension 1. Introduction Attractors for infinite-dimensional dynamical systems have proven very useful tool in the study of the asymptotic behaviour of many partial differential equations (see for example Hale (1988), Ladyzhenskaya (1991), Temam (1988, 2nd ed. 1997)). Much attention has been focused on the dimension of these attractors, since in many cases one can prove that this is finite, and deduce that the long-time behaviour of the system depends on only a finite number of degrees of freedom. The definition of an attractor has been generalized to the stochastic case by Crauel & Flandoli (1994) and Schmalfuß (1992), and has once again been fruitful in furthering our understanding of the associated random and stochastic differential equations, in particular in finite dimensions (Arnold 1998). Roughly speaking, a random attractor is a family of compact random sets, which are invariant for the stochastic flow and attract all solutions ‘from t=−∞’. Once again, it is possible to show that certain of these attractors are finite-dimensional, although the number of examples is much more limited than in the deterministic case. Debussche (1997) has adapted the most powerful deterministic technique to †Permanent address: Mathematics Institute, University of Warwick, Coventry CV4 7AL. UK. Article submitted to Royal Society T EX Paper 2T. Caraballo, J.A. Langa, & J.C. Robinson treat the stochastic case, and in a previous paper (Caraballo et al. 2000) we used his method to obtain an upper bound on the Hausdorff dimension of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative noise in the sense of Stratonovich, du= (−Au +βu −u3) dt+σu ◦dWtx∈D⊂Rm.(1.1) Different techniques are required, for both stochastic and deterministic systems, to obtain lower bounds on the dimension of these attractors. Such bounds are generally based on finding some invariant manifolds which must be subsets of the attractor (Babin & Vishik 1983). Here we adopt this approach, using some ideas due to DaPrato & Debussche (1996) to show (in §4) that there is an n-dimensional unstable manifold near the zero solution when λn< β < λn+1 (the λjare the eigenvalues of the negative Laplacian, see §3). This gives a lower bound on the attractor dimension which is of the same order as our previous upper bound. Remarkably these bounds do not depend on the level of the noise (σin (1.1)), and are of the same order as the bounds in the deterministic case. In the last decade there has been some research in stochastic bifurcation theory, although it still seems to be unclear how this theory can be set up in general (Arnold 1998, 2000). However, to our knowledge there are as yet no studies of bifurcations in infinite-dimensional stochastic differential equations. We adopt here the dynamical concept of a stochastic bifurcation, which is understood as a qualitative change in the invariant sets (or invariant measures) of the system (see §5). For the one-dimensional case m= 1 we use the manifold structure, which gave us the lower bound on the dimension, to investigate in more detail what happens to the random attractor as βpasses through λ1from below. For β < λ1we showed previously that the attractor is just the point {0}, and our lower bound show that for β > λ1the Hausdorff dimension of the attractor is at least 1. We show that for λ1< β < λ2the unstable manifold is in fact tangent to the space spanned by the first eigenfunction of the Laplacian, and thus actually has one branch in the cone of positive solutions and the other one in the cone of negative solutions (§5). It is only a short step from here to the required pitchfork bifurcation, using the theory of monotone random dynamical systems developed by Arnold & Chueshov (1998a & b). We end with some conclusions and open problems. 2. Formulation of the problem and deterministic results Let D⊂Rm,m≤5, be an open bounded set with regular boundary. We consider the following Chafee-Infante reaction-diffusion equation in Dperturbed by a linear multiplicative white noise du= (∆u+βu −u3)dt+σu ◦dWt,(2.1) with u(x, t) = 0 for x∈∂D, and where W·(ω) : Ω →C0(R,R) is a one-dimensional Wiener process on the probability space (Ω,F,P). We rewrite (2.1) as the following differential equation on L2(D), du= (−Au +βu −u3) dt+σu ◦dWt(2.2) Article submitted to Royal Society A stochastic pitchfork bifurcation 3 where A=−∆ on Dwith the appropriate (Dirichlet) boundary conditions. The operator Ais positive, linear, self-adjoint and with compact inverse. Thus there exists a sequence {λj}of positive eigenvalues, whose associated eigenfunctions wj (with Awj=λjwj) form an orthonormal basis for H(e.g. Renardy & Rogers 1992). We order these so that λn+1 ≥λn. (a)The deterministic case (σ= 0) When σ= 0, existence and uniqueness results are proved for the equation in Marion (1987), Temam (1988), and Robinson (2001). There exists a unique weak solution u(t;u0)∈L2(0, T;H1 0(D)) ∩L4((0, T)×D)∩C([0, T]; L2(D)), so that in particular we can use the solutions to define a semigroup S(t) on L2(D), via S(t)u0=u(t;u0). S(t) satisfies the usual semigroup properties, S(0) = id, S(t)S(s) = S(t+s),and S(t)u0continuous in tand u0.(2.3) It is shown in all three of the above references that the equation also enjoys the existence of a global attractor A, that is, a compact invariant set which attracts the orbits of all bounded sets, i.e. S(t)A=Afor all t∈R(2.4) and dist(S(t)B, A)→0 as t→ ∞,(2.5) where Bis any bounded subset of L2(D) and dist(A, B) is the Hausdorff semidistance between Aand B, dist(A, B) = sup a∈A inf b∈B|a−b|.(2.6) Estimates of the Hausdorff dimension of this attractor in terms of βcan also be obtained, namely dH(A)≤Cβm/2. (Recall that D⊂Rm.) By showing the existence of an unstable manifold near the origin, Babin & Vishik (1983) show that a similar lower bound holds, so that in fact there exists a constant csuch that cβm/2≤dH(A)≤Cβm/2 (see also Temam 1988). We write this more compactly as dH(A) = O(βm/2).(2.7) We will obtain, below, the same behaviour in the stochastic case. Article submitted to Royal Society 4T. Caraballo, J.A. Langa, & J.C. Robinson In the particular case m= 1 the structure of this attractor is extremely well understood: the attractor consists of a collection of stationary points, which bifurcate from the origin as βpasses through each eigenvalue λj, and the unstable manifolds joining them (Hale 1988; Henry 1984). In particular, as βpasses through λ1the attractor changes from a single stable fixed point at u≡0 to a set, homeomorphic to an interval, which consists of the one-dimensional unstable manifold of the origin. This has two distinct components, one of which lies in the cone of positive solutions and on which all solutions approach a new positive fixed point, and one which lies in the cone of negative solutions and on which all solutions approach a new negative fixed point. It is this pitchfork bifurcation that we will seek to mirror in the stochastic case. 3. Random attractors We now discuss briefly the definition of a random dynamical system and a random attractor, unsurprisingly using our equation (2.2) as an illustrative example. (a)Random dynamical systems Let (Ω,F,P) be a probability space and {θt: Ω →Ω, t ∈R}a family of measure preserving transformations such that (t, ω)7→ θtωis measurable, θ0= id, and θt+s=θtθsfor all s, t ∈R. The flow θttogether with the corresponding probability space, (Ω,F,P,(θt)t∈R) is called a (measurable) dynamical system. In our case we take (Ω,F,P) to be the probability space which generates the one-dimensional Wiener process dWt. The shift θtacts on Ω so that Wt(θsω) = Wt+s(ω)−Ws(ω),(3.1) the additional subtracted term ensuring that W·(θsω) is still a Brownian motion. For this example it also follows that the shift θtis ergodic (Arnold 1998). A continuous random dynamical system (RDS) on a Polish space (X, d) with Borel σ-algebra Bover θon (Ω,F,P) is a measurable map ϕ:R+×Ω×X→X (t, ω, x)7→ ϕ(t, ω)x such that P−a.s. i) ϕ(0, ω) = id on X ii) ϕ(t+s, ω) = ϕ(t, θsω)◦ϕ(s, ω) for all t, s ∈R+(cocycle property) iii) ϕ(t, ω) : X→Xis continuous. When σ6= 0 it is known (Pardoux 1975) that for each u0∈L2(D) and T > 0, there exists a unique solution u(t;x0) of (2.2), with u(t;x0)∈L2(Ω ×(0, T); H1 0(D)) ∩L4(Ω ×(0, T)×D)∩L2(Ω; C(0, T;L2(D)). Article submitted to Royal Society A stochastic pitchfork bifurcation 5 In particular, it follows that the solutions of (2.2) can be used to generate a random dynamical system if we define ϕ(t, ω)u0=u(t;ω, u0), where u(t;ω, u0) is the solution of (2.2) with noise ωand initial condition u(0) = u0. (b)Random attractors A random set A(ω) is said to be a random attractor for the RDS ϕif i) A(ω) is a random compact set, that is, P−a.s.,A(ω) is compact and for all x∈X, and the map ω7→ dist(x, A(ω)) is measurable with respect to F. ii) P−a.s. ϕ(t, ω)A(ω) = A(θtω) for all t≥0 (invariance) and iii) for all B⊂Xbounded (and non-random), P−a.s., lim t→∞ dist(ϕ(t, θ−tω)B, A(ω)) = 0, where dist( . , . ) denotes the Hausdorff semidistance in X(cf. (2.6)). Since ϕ(t, θ−tω)u0can be interpreted as the position at t= 0 of the trajectory which was at u0at time −t, this pullback convergence property is essentially attraction ‘from t=−∞’. In Caraballo et al. (2000) we proved the existence of a random attractor for our equation, using a theorem due to Crauel & Flandoli (1994). We also showed that if β < λ1then A(ω) = {0}, and more generally (using a result of Debussche 1998) that if β < 1 d d X j=1 λj then dH(A(ω)) < d. Since one can bound Pd j=1 λj≤Cd(m+2)/m, this implies the upper bound dH(A(ω)) ≤c1βm/2.(3.2) Note that this is of the same order as the deterministic bound in (2.7), and of course is extremely suggestive of the fact that the attractor becomes more complicated as βincreases through λ1. 4. A lower bound on the attractor dimension We now turn to proving the promised lower bound on the dimension of the random attractor A(ω). We make use of an idea from the deterministic theory, which is to show that the attractor must contain an unstable manifold of a certain dimension. However, this approach usually requires various differentiability properties of the flow, and such technicalities are by no means straightforward in the stochastic case. Furthermore, there is currently no well-developed theory of unstable manifolds for general stochastic PDEs. Article submitted to Royal Society 6T. Caraballo, J.A. Langa, & J.C. Robinson So we proceed in a manner which we believe is slightly unusual, and could prove useful not only for other stochastic examples but also in the deterministic case, by proving not the existence of a C1unstable manifold, but of a Lipschitz manifold in a neighbourhood of the origin. The theory of inertial manifolds, introduced by Foias et al. (1988) in the deterministic case, and developed by various authors for stochastic equations (Bensoussan & Flandoli 1994; Chueshov & Girya 1994), is well suited to this, and we follow some ideas from a paper of DaPrato & Debussche (1996) in our proof. In this way, we prove the following theorem. Theorem 4.1. If λn< β < λn+1 then there exists a set Mδ(ω), a (locally) invariant n-dimensional Lipschitz manifold, which is part of the unstable set of the origin, i.e. lim t→∞ dist(ϕ(−t, ω)u, 0) = 0 (4.1) for all u∈ M(ω). In (4.1), ϕ(−t, ω)uis the point v∈ M(θ−tω) such that ϕ(t, θ−tω)v=u(the fact that M(ω) is a finite-dimensional invariant manifold ensures that such a point exists – see proposition 4.6 for more details). We now appeal to a result of Crauel (1999), which guarantees that the unstable set of the origin (defined precisely as in (4.1)) must be a subset of the random attractor, and so obtain the lower bound on the attractor dimension given in the following theorem. Theorem 4.2. Provided that m≤5, if λn< β < λn+1 then dH(A(ω)) ≥n. In particular, the dimension of the attractor satisfies dH(A(ω)) = O(βm/2). (a)Truncating the equation We prove the existence of Mδ(ω) in a series of propositions. We apply a version of the theory in DaPrato & Debussche (1996) developed to prove the existence of inertial manifolds for stochastic PDEs with a general multiplicative noise term. However, here we do not look for an inertial manifold, but in fact an unstable manifold in a neighbourhood of the origin. The first task is to truncate the equation in a neighbourhood of the origin, to ensure that the nonlinear term is globally Lipschitz, with a small Lipschitz constant. [Our approach is, in fact, similar to that of Boxler (1991) for centre manifolds, although here we can truncate our equation in a manner which is independent of ω.] Although this is a standard approach in the deterministic theory, in which one has a compact absorbing ball, it is perhaps the main weakness of the theory of inertial manifolds for stochastic equations, where in general the radius of the absorbing ball will depend on ω, and as yet there is no proof which allows the Lipschitz constant of the nonlinearity to vary in this way. However, since we require only a local unstable manifold, we will be able to truncate the equation in a consistent fashion. We restrict to the case m≤5, since then we can use the embedding H2α⊂L6 with 3/4< α < 1 to show that F(u) = u3is Lipschitz from H2αinto L2(we suppress the Din all our function spaces from now on), |u3−v3|L2≤C(kuk2 H2α+kvk2 H2α)ku−vkH2α.(4.2) Article submitted to Royal Society A stochastic pitchfork bifurcation 7 Indeed, we have |u3−v3|2 L2=ZD (u3(x)−v3(x))2dx =ZDµZu(x) v(x) 3s2ds¶2 dx ≤9ZD |u(x)−v(x)|2(u(x)2+v(x)2)2dx ≤9µZD |u(x)−v(x)|2pdx¶1/pµZD |u(x)2+v(x)2|2qdx¶1/q ≤9|u−v|2 L2p(|u|L4q+|v|L4q)4 ≤9|u−v|2 L6(|u|L6+|v|L6)4, taking (p, q) = (3,3/2). For a C1cut-off function θ:R+→[0,1], such that θ(r) = ½1r≤1 0r≥2 with |θ0(r)| ≤ 2, it is straightforward to verify (cf. Temam, 1988) that F(u) = −θµkukH2α R¶u3(4.3) is globally bounded, |F(u)| ≤ M0for all u∈H2α, and globally Lipschitz, |F(u)−F(v)| ≤ ˜ Lfku−vkH2αfor all u, v ∈H2α.(4.4) with ˜ Lf≤c0R2for some constant c0. Note in particular that F(u) = −u3in BH2α(0, R), and so the two equations agree in a small H2α-neighbourhood of the origin. We will find it more convenient to use the norm equivalence between the H2α norm and the norm in D(Aα). Writing |Aαu|=|u|α, where Ais the negative Laplacian on Dwith Dirichlet boundary conditions, we have kukH2α≤c|u|α.(4.5) We can re-write (4.4) as |F(u)−F(v)| ≤ Lf|u−v|αfor all u, v ∈D(Aα),(4.6) where Lf=c˜ Lf(with cas in (4.5)). Note that we can make Lfas small as we wish provided that Rin (4.3) is chosen small enough. Article submitted to Royal Society 8T. Caraballo, J.A. Langa, & J.C. Robinson (b)An invariant manifold for the truncated equation We now work with the truncated equation du= (−Au +βu +F(u)) dt+σu ◦dWt,(4.7) and prove the existence of an invariant manifold of dimension nwhen λn< β < λn+1. In the statement of the theorem we write Pnfor the orthogonal projection onto the first neigenfunctions of A, ordered so that the corresponding eigenvalues are non-decreasing in n, and Qnfor its orthogonal complement, Qn=I−Pn. Theorem 4.3. If λn< β < λn+1, and Rin (4.3) is chosen small enough that Lf satisfies β−λn>4Lfλα n(4.8) and λn+1 −β > (2KLf)1/(1−α),(4.9) then (4.7) has an n-dimensional invariant manifold M(ω), ϕ(λ, ω)M(ω) = M(θλω), λ ≥0. Furthermore, M(ω)is given as the graph of a Lipschitz function Φω:PnH→ QnH∩D(Aα), |Φω(p1)−Φω(p2)|α≤2|p1−p2|α.(4.10) We analyse the equation ω-by-ω. To this end we consider not equation (2.2) for u, but, setting v= e−σWtuand observing that dv= e−σWtdu−e−σWtσu ◦dWt, we can treat the non-autonomous equation for v, dv dt=−Av +βv + e−σWtF(eσWtv).(4.11) We write this as dv dt=−Av +βv +Fσ(v), where Fσ(v) = e−σWtF(eσWtv). Note that |Fσ(u)−Fσ(v)| ≤ e−σWt|F(eσWtu)−F(eσWtv| ≤e−σWtLf|eσWtu−eσWtv|α =Lf|u−v|α,(4.12) Article submitted to Royal Society A stochastic pitchfork bifurcation 9 so that the Lipschitz property of Fin (4.6) is transferred to Fσ. Furthermore it follows, since the support of Fis contained in a bounded set in D(Aα), and F(0) = 0, that |Fσ(u)| ≤ M1 for some constant M1. The solution operator corresponding to the transformed equation we write as ψ(t, ω). Clearly, ψ(t, ω) = e−σWt(ω)ϕ(t, ω).(4.13) For a function y: (−∞,0] →PnH(4.14) we define a norm kykE= sup t∈(−∞,0] |y(t)|α,(4.15) and similarly for z: (−∞,0] →QnD(Aα).(4.16) For a pair (y, z), we define k(y, z)kE= max(kykE,kzkE).(4.17) The space of all functions (y, z) as in (4.14) and (4.16) with finite Enorm (the norm in (4.17)) we denote by E. We will need the following standard bound on the operator norm of Aαe−AtQn from L2into L2(see Temam 1988, for example), kAαe−AtQnkop ≤˜ K(t−α+λα n+1)e−λn+1tfor all t≥0.(4.18) We also set K=˜ KZ∞ 0 (t−α+ 1)e−tdt. (4.19) ¿From now on we write P=Pnand Q=Qn. Roughly speaking, we follow the proof in DaPrato & Debussche (1996). However, here the argument is somewhat different, since our particular choice of multiplicative noise allows us to consider the transformed equation (4.11) ω-by-ω, and we do not have to deal with the problems which arise when trying to solve a more general stochastic equation backwards in time. We also need to tailor the analysis to make sure that the dependence on βremains explicit. Proposition 4.4. For each ω∈Ω, there exists a unique solution (y, z)of the coupled equations dy dt=−Ay +βy +PFσ(y+z)y(0) = y0 dz dt=−Az +βz +QFσ(y+z)z(t)→0 as t→ −∞,(4.20) on the interval (−∞,0]. Article submitted to Royal Society 16 T. Caraballo, J.A. Langa, & J.C. Robinson (b)Differentiability of M(ω)at 0 First we prove the differentiability property of the manifold. In particular, the cubic bound (5.1) will be used to show that a portion of M(ω) lies in K+. Proposition 5.2. The manifold M(ω)is tangent to PnHat the origin, and moreover |Φω(p)|α≤Cω|p|3 α.(5.1) Proof. First, observe that Fσ(u) = −e−σWtθµ|ueσWt|α R¶(ueσWt)3 is essentially cubic, so that |Fσ(u)| ≤ e2σWtµZu6dx¶1/2 ≤e2σWtkuk3 L6 ≤Ce2σWt|u|3 α. Now, Φω(y0) is z(0) from the coupled equations in proposition 4.4, so we know (cf. (4.22)) that z(0) = Z0 −∞ e(A−βI)sQFσ(y(s) + z(s)) ds, and hence |z(0)|α≤˜ KZt −∞ ((−s)−α+ (λn+1 −β)α)e(λn+1−β)sCe2σWs|u|3 αds. Since |u|3 α≤(|y|α+|z|α)3≤8(|y|3 α+|z|3 α), we have, using (4.29), |z(0)|α≤Kω[kyk3 E+kzk3 E].(5.2) Now, we know that z(t) = Zt −∞ e−(A−βI)(t−s)QFσ(y+z) ds, and we can estimate |z(t)|αas in the proof of proposition 4.4, using the fact that Fσ(0) = 0, |z(t)|α≤˜ KLfZt −∞ ((t−s)α+ (λn+1 −β)α)e−(λn+1−β)(t−s)|y+z|αds ≤µ˜ KLfZt −∞ ((t−s)α+ (λn+1 −β)α))e−(λn+1−β)(t−s)ds¶ Article submitted to Royal Society A stochastic pitchfork bifurcation 17 ×(kykE+kzkE). Now, we saw in proposition 4.4 that the integral expression here is bounded by 1 2, and so we have kzkE≤1 2(kykE+kzkE), i.e. kzkE≤ kykE. Equation (5.2) now becomes |z(0)|α≤2Kωkyk3 E,(5.3) so it only remains to estimate kykE. However, this is the solution of the governing equation which has y(0) = y0and z(0) = Φω(y0), and so which lies on M(ω). The inequality (4.28) from proposition 4.6 shows that in fact |y(t)|α≤ |y0|αfor all t≤0, and so kykE≤ |y0|α. Combining this with (5.3) gives the required bound. (c)The bifurcation theorem We now prove our second main result. Theorem 5.3. Let m= 1. Then (2.2) undergoes a stochastic pitchfork bifurcation at β=λ1. In particular, for β > λ1there exist positive and negative random fixed points ±a(ω), and a(ω)→0as β↓λ1. Note that one could easily recast this result in terms of the appearance of two new invariant measures, namely the random Dirac measures concentrated at ±a(ω). Proof. Choose and fix β > λ1. Since the first eigenfunction of the Laplacian is positive, we use the cubic estimate to show that Mδ(ω) must contain a positive function. More precisely, we let K+={u∈L2(D) : u(x)≥0 almost everywhere}, and K−={u∈L2(D) : u(x)≤0 almost everywhere}. The main idea is to show that one portion of the unstable manifold must be a subset of K+, and another a subset of K−. On a one-dimensional domain [0, L] we know that the eigenfunctions of the Laplacian with Dirichlet boundary conditions are proportional to wn(x) = sin(nπx/L). Since the manifold Mδ(ω) is given as a graph over the space spanned by the first eigenfunction, for each fixed ωthe manifold consists of a family of functions (here parametrised by ²) given in the form u(x, ²) = ²sin(πx/L) + ∞ X j=2 cjsin(jπx/L), where cj=cj(², ω). Article submitted to Royal Society 18 T. Caraballo, J.A. Langa, & J.C. Robinson Since ¯¯¯¯ sin(jπx/L) sin(πx/L)¯¯¯¯ ≤j (simply rewrite the sine terms using complex exponentials) we have u(x, ²)≥sin(πx/L)·²− ∞ X j=2 |cj|j¸. Using the cubic bound on Φω, we know that ¯¯¯¯ ∞ X j=2 cjsin(jπx/L)¯¯¯¯α =Cµ∞ X j=2 |cj|2j4α¶1/2 ≤Cω¯¯²sin(πx/L)|3 α=Kω²3. Since α > 3/4 we can now write ∞ X j=2 |cj|j≤µ∞ X j=2 j4α|cj|2¶1/2µ∞ X j=2 j2(1−2α)¶1/2 ≤C²3, and so u(x, ²)≥sin(πx/L)[²−C²3]. Thus for ² > 0 small enough, u(x, ²)≥0 on [0, L]; similarly for ² < 0 and small enough, u(x, ²)≤0 on [0, L]. In other words, there are portions of Mδ(ω) near the origin which intersect non-trivially with K+and K−. Now, Kotelenez (1992) shows that the cocycle generated by the equation is order-preserving on L2(D), so that if u0≥v0almost everywhere then ϕ(t, ω)u0≥ϕ(t, ω)v0. In particular, since zero is a fixed point of the equation, K±are invariant subspaces for ϕ. (These two facts are easy to check if u0and v0are continuous functions by applying the standard deterministic theory (essentially the maximum principle, see Smith 1995) to the equation for v(t) = e−σWtu(t).) By following the analysis in Caraballo et al. (2000) one can show that the flow in each of these subspaces has a positively invariant compact absorbing set, namely a bounded set in H1(D) with radius r(ω), where r(ω) is a tempered random variable. In particular therefore, using the theory in Flandoli & Schmalfuss (1996) the flow in each of these subspaces has its own random attractor, A+(ω) and A−(ω), and these attract all tempered random sets. Since Mδ(ω)∩K±forms part of the unstable set of the origin, it must be a subset of A±(ω), and so in particular A±(ω) is non-trivial. In their 1998 paper Arnold & Chueshov develop an extensive theory for orderpreserving random dynamical systems, and in particular one of their results (theorem 2) provides the existence of new stochastic fixed points in this example (essentially we follow Chueshov, 2000): if ϕis an order-preserving random dynamical system for which there exists a random interval J(ω) = {u:b(ω)≤u≤c(ω)} which is attracted to and contains A(ω), then there exist random fixed points u−(ω)≤u+(ω) such that u−(ω)≤u≤u+(ω) for all u∈ A(ω). Article submitted to Royal Society A stochastic pitchfork bifurcation 19 Since A(ω) lies within a random ball in H1(D) of radius r(ω), and since H1(D)⊂ C0(D) with kuk∞≤C(D)kuk1 H, it follows in particular that 0≤u≤c(D)r(ω) for all u∈ A+(ω). Using the above result we can deduce that there exist two random fixed points u−(ω) and u+(ω) such that u−(ω)≤u≤u+(ω) for all u∈ A+(ω). Clearly u−(ω) = 0, but since A+(ω) is non-trivial this proves the existence of a new random (positive) fixed point a(ω) = u+(ω). By symmetry there is also a new random (negative) fixed point −a(ω). (For a similar argument see Chueshov (2000).) That a(ω)→0 as β↓λ1follows from a slight variant of the general result on the upper-semicontinuity of random attractors to be found in Caraballo & Langa (2001, theorem 3), which guarantees under conditions which are easily verified in our case (convergence of the corresponding cocyles; convergence of the compact absorbing sets; and positive invariance of the compact absorbing set for the limit equation (β=λ1)) that P-a.s. lim β↓λ1 dist(A+ β(ω), Aλ1(ω)) = 0. Since a(ω)⊂ A+(ω) and Aλ1(ω) = {0}we have a(ω)→0. In fact, we know a little more than is stated in the theorem. For β > λ1the attractor contains the two distinct, non-trivial, invariant subsets A+(ω) and A−(ω), which attract all initial conditions in K+and K−respectively. One can think of this, loosely, as a transfer of stability of the origin to A+(ω) and A−(ω). For a more concrete stability result, theorem 2 in Arnold & Chueshov (1998) guarantees a limited type of stability for the new fixed points. In particular, a(ω) is stable from above, in that if u(ω)≥a(ω) (note that this “initial condition” cannot in general be taken to be deterministic) then lim t→∞ ϕ(t, θ−tω)u(θ−tω) = a(ω) (stability “from below” holds for −a(ω)). Since trajectories near zero in K+move away from the origin, it seems reasonable to expect that a(ω) is attracting in K+. If one could show that solutions eventually enter K+or K−this would give the stability of the fixed points. (A general result guaranteeing the existence of stable fixed points is given by Schmalfuß (1996), but we have not been able to apply this in our case.) 6. Conclusion We have shown that the structure of the random attractor for our example, the Chafee-Infante equation with a multiplicative noise term, is surprisingly close to that of the deterministic equation. In particular, we have shown that the dimension of the attractor, asymptotically in β, exhibits the same behaviour as in the deterministic case, and is independent of the level of noise (σ). Article submitted to Royal Society 20 T. Caraballo, J.A. Langa, & J.C. Robinson Analysing further the structure of this attractor we have shown that as βpasses through λ1the attractor grows and two new random fixed points appear. Noting that a±(ω)∈ A±(ω), what we expect is that in fact, for λ1< β < λ2, A(ω) = A+(ω)∪A−(ω), with A±(ω) consisting of a one-dimensional manifold (the image of Mδ(ω)∩ K± under the flow) joining the origin to ±a(ω). This would ensure that the attractor’s structure is exactly that in the deterministic case. Some support for this picture, at least for small σ, is given by the upper semicontinuity result in Caraballo et al. (1998). We proved there that as σ→0, for each ω, dist(Aσ(ω),A)→0. Thus the random attractor, a set which is at least one-dimensional, must lie within a small neighbourhood of A, which is itself homeomorphic to a line when λ1< β < λ2. Finally, we comment that we would not expect similar ideas to work in the case of an additive noise term (e.g. +²φ dWt, for some φ∈H). Without the fixed point {0}there is no clear way to show that an invariant manifold for a truncated version of the equation is a subset of the random attractor. Indeed, the one-dimensional example in Crauel & Flandoli (1998) suggests that with an additive noise the attractor will be very much simpler than in the deterministic case. We hope to investigate this further in a future paper. Many thanks to Igor Chueshov, Hans Crauel, Franco Flandoli, and Antonio Su´arez for some interesting and helpful conversations. James Robinson is currently a Royal Society University Research Fellow, and would like to thank the Society for their support. He would also like to thank EDAN for their hospitality, and Iberdrola for their financial assistance during his visit. Tom´as Caraballo and Jos´e Langa have been partially supported by DGICYT Project PB98-1134. References Arnold, L. 1998 Random dynamical systems. New York: Springer. Arnold, L. 1998 Recent progress in stochastic bifurcation theory. Report no. 439, University of Bremen. Arnold, L. & Boxler, P. 1992 Stochastic bifurcation: instructive examples in dimension one. In Diffusion processes and related problems in analysis, Vol II (eds. Pinsky & Wihstutz), pp. 241–256. Stuttgart: Birkh¨auser. Arnold, L. & Chueshov, I. 1998 Order-preserving random dynamical systems: Equilibria, attractors, applications. Dyn. Stab. 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