2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela. Electronic Journal of Differential Equations, Conference 13, 2005, pp. 29-34. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

CRITICAL POINTS OF THE STEADY STATE OF A FOKKER-PLANCK EQUATION

JORGE GUÍÑEZ, ROBERT QUINTERO, ANGEL D. RUEDA

Abstract. In this paper we consider a set of vector fields over the torus for which we can associate a positive function v_{ϵ} which define for some of them in a solution of the Fokker-Planck equation with ϵ diffusion:

ϵΔv_{ϵ} − div(v_{ϵ}X) = 0 .

Within this class of vector fields we prove that X is a gradient vector field if and only if at least one of the critical points of v_{ϵ} is a stationary point of X, for an ϵ > 0. In particular we show a vector field which is stable in the sense of Zeeman but structurally unstable in the Andronov-Pontriaguin sense. A generalization of some results to other kind of compact manifolds is made.

1. Vector fields in covering spaces

Let π : M̃ → M be a covering space of a Riemannian and oriented manifold M. In M̃ there exists one and only one Riemannian structure such that

dπ_{y} : T_{y}(M̃) → T_{π(y)}(M)

is an isometry for all y ∈ M̃. Then it is able to associate to every C^{r} vector field X in M, another C^{r} vector field X̃ in M̃ in the following way:

X̃(y) = (d(π)(y))^{−1}(X(π(y))).

It is easy to verify the following theorem:

Theorem 1.1.

({ \atop X + Y}) = X̃ + Ỹm  (1.1)
˜∇f = ∇(π ∘ f)  (1.2)
div(X̃(y) = div(X)(π(y))  (1.3)

Definition A vector field X is called almost gradient respect to the projection π, if and only if X̃ is a gradient in M. This set will be denoted by V_{ag}(π). Particular we can write

grad(M) = V_{ag}(1_{M})

2000 Mathematics Subject Classification. 58J60, 37C20. Key words and phrases. Almost gradient vector fields. \begin{matrix}c \\ ◯ \end{matrix}2005 Texas State University - San Marcos. Supported by CONDES University of Zulia.

There are non trivial projections for which is true the preceding statement, so we have the following theorem.

Theorem 1.2. If π : M̃ → M is a finite covering and M is compact, then V_{ag} is the set of gradient vector fields in M.

Proof. Let X be a vector field in M and let

X = ∇f + W

be its Hodge’s decomposition. If X̃ is gradient, then there exists a C^{∞} function g such that:

∇g = X̃ = ∇(f ∘ π) + W̃.

Then by Theorem 1.1 it follows that W̃ is a gradient vector field in M̃ and we can write W̃ = ∇h. Finally from Theorem 1.1 we get, div(W̃) = div(W) ∘ π = 0. Thus ∇h = 0 and by compactness of M̃ it follows h to be a constant. So W = 0 and W = 0. □

2. Vector Fields in T_{n}

Let π : ℝ^{n} → T_{n} = ℝ^{n}/ℤ^{n} be the universal covering space of the torus T_{n}. So there exists a Riemannian structure in T_{n} such that T̃_{n} = ℝ^{n}, where ℝ^{n} is considered with the usual Riemannian structure. It is easy to realize that V_{ag}(π) is different from grad(M). More precisely we have the following statement.

Theorem 2.1. X ∈ V_{ag}(π) if and only if X is in the form X = ∇f + λ, where λ ∈ ℝ^{n}.

Proof. Let X = ∇f + W be the Hodge’s decomposition of X. Then X̃ ∈ V_{ag}(π) implies

X̃ = ∇g,

but

X̃ = ˜∇f + W̃ = ∇(f ∘ π) + W̃

and W̃ = ∇h with h = g − f ∘ π. By the periodicity of W̃ it follows that ∥W̃∥ ≤ K. Then

|h(x)| ≤ K∥x∥ ∀x ∈ ℝ^{n}.  (2.1)

Because W has divergence zero so it does W̃, and h is an harmonic function in whole ℝ^{n}. From the estimate (4) h is a linear function; i.e.,

h(x) = a + λ · x  (2.2)

with λ ∈ ℝ^{n} so ∇h = W̃ = λ, consequently W = λ. □

3. The function v_{ϵ}

For the rest of this article, we consider X in T_{n} to be of the form X = ∇f + λ. Let us consider

v_{ϵ}(x) = ∫_{T_{n}} exp (\frac{h(x, z)}{ϵ})dz  (3.1)

where

g(x) = f(x) + λ · x ,  (3.2)
h(x, z) = g(x) − g(x + z)  (3.3)

Lemma 3.1. X is a gradient if λ = 0 and we have

v_{ϵ} = L(ϵ) exp(f/ϵ)

Proof. If λ = 0, we have

v_{ϵ} = exp(\frac{f}{ϵ}) ∫_{T_{n}} exp(\frac{−f(x + z)}{ϵ})dz = exp(\frac{f}{ϵ}) ∫_{T_{n}} exp(\frac{−f(z)}{ϵ})dz .

□

Definition.X will be called without coupling, if

((i ≠ j) and (\frac{∂X_{i}}{∂x_{j}} ≠ 0)) ⇒ λ_{i} = 0.

Theorem 3.2. Let X be a vector field without coupling. Then v_{ϵ} is a solution of the Fokker-Planck equation

ϵΔv − div(vX) = 0.  (3.4)

Proof. Let I be the set of indices for which λ_{i} ≠ 0. Then the i-component of the vector field X is

X_{i} = f′_{i}(x_{i}) + λ_{i}

with f_{i} a function in the variable x_{i}. Therefore, X = ∇f + λ with

f = ∑(f_{i}(x_{i})) + p(x)

where p(x) is a periodic function and

\frac{∂}{∂x_{i}} (p(x)) = 0, i ∈ I.

Then

h(x, z) = ∑_{iϵI} h_{i}(x_{i}, z_{i}) + p(x) − p(x + z).

Because X is without coupling, applying Lemma 3.1 to ∇p,

v_{ϵ} = ∫_{T_{n}} exp (\frac{h(x, z)}{ϵ})dz = K( ∏_{iϵI} v_{ϵ}^{i}(x_{i})) exp (\frac{p(x)}{ϵ}) ,  (3.5)

where

v_{ϵ}^{i}(x_{i}) = ∫_{0}^{1} exp (\frac{h_{i}(x_{i}, z_{i})}{ϵ})dz_{i}

is associated with the vector field ∇f_{i} + λ_{i}. If i ∈ I it follows that

ϵ(∇v_{ϵ})_{i} = (X_{i}(x_{i})v_{ϵ}^{i} − ϵR_{i}) ∏_{kϵ(I−\{i\})} v_{ϵ}^{k} exp (\frac{p(x)}{ϵ})

where

R_{i} = ∫_{0}^{1} \frac{X_{i}(x_{i} + z_{i})}{ϵ} exp (\frac{f_{i}(x_{i}) − f(x_{i} + z_{i}) − λ_{i}z_{i}}{ϵ} )dz_{i} − exp ( − \frac{−λ_{i}}{ϵ} ) + 1.

For i ∉ I,

(ϵ∇v_{ϵ})_{i} = (∇p(x))_{i}v_{ϵ} = X_{i}v_{ϵ}

thus v_{ϵ} is solution of (3.4). □

4. Dynamics and Steady State

We begin this section with some definitions: Definition Let X be a vector field in T_{n} and let u_{ϵ} = ∑_{0}^{∞} \frac{F_{i}}{ϵ^{i}} be a series with a positive ratio of convergence. Suppose that u, is a solution of (3.4). We will denote:

C_{ϵ} = {x ∈ M : ∇u_{ϵ} = 0}  (4.1)
E(X) = {x ∈ M : X(x) = 0}  (4.2)
D(X) = \{x ∈ M : det (\frac{∂X_{i}}{∂x_{j}} ) = 0\}  (4.3)

In [1] and [2], we have such series on T_{n} and S_{n}.

Theorem 4.1. Consider X ∈ V_{ag}(T_{n}) such that

(i) There exists a convergent series u_{ϵ} = ∑_{i=0}^{∞} \frac{F_{i}}{ϵ^{i}} solving (3.4) for \frac{1}{ϵ} ≤ r, r > 0 (ii) There exists an infinity set S ⊂ [r_{1}, r], r1 > 0 and a point x in T_{n} such

that

x ∈ C_{ϵ} ∩ E(X) ∀\frac{1}{ϵ} ∈ S  (4.4)

Then X is a gradient vector field.

Proof. Because X ∈ V_{ag}(T_{n}), by Theorem 2.1 we can write

X = ∇f + λ ,  (4.5)
∇u_{ϵ} = 0 = ∑_{i=0}^{∞} ∇F_{i}(x)(\frac{1}{ϵ})^{i} ∀ϵ ∈ S.  (4.6)

Then ∇F_{i}(x) = 0, for every i. In particular ∇F_{1}(x) = ∇f(x) = 0 and by (15) λ = 0. □

Theorem 4.2. Let X be a vector field without coupling. Then the following statements are equivalent.

(i) There exists ϵ such that C_{ϵ} ∩ E(X) ≠ ∅

(ii) For all ϵ, C_{ϵ} ∩ E(X) ≠ ∅

(iii) X is gradient.

Proof. For x ∈ C_{ϵ} ∩ E(X) and i ∈ I we have a contradiction:

0 = (v_{ϵ}^{i})′(x) = e^{\frac{λ_{i}}{ϵ}} − 1 + \frac{λ_{i}}{ϵ} v_{ϵ}(x) .

wich completes the proof. □

Remark. The main idea here is that for non-gradient cases critical points of a steady state are different from stationary points of the vector field. This fact enable us to find a vector field X with an associated u_{ϵ} which has not generated critical points, even when X has degenerated stationary points.

Lemma 4.3. Let’s suppose that X = ∇f + λ is without coupling and let I_{+} be the set of index such that λ_{i} > 0 and let I_{−} be the set of index such that λ_{i} < 0. Then

C_{ϵ} ⊂ ∩_{iϵI_{+}}X_{i}^{−1}((0, +∞)) ∩_{iϵI_{−}} X_{i}^{−1}((−∞, 0))

EJDE/CONF/13  CRITICAL POINTS OF THE STEADY STATE  33

Proof. For a such f we can write

f = ∑(f_{i}(x_{i})) + p(x)

where p(x) is not depending of x_{i} for i ∈ I = I_{−} ∪ I_{+} and

v_{ϵ} = K( ∏_{iϵI} v_{ϵ}^{i}(x_{i})) exp (\frac{p(x)}{ϵ})  (4.7)

where

v_{ϵ}^{i} = ∫_{0}^{1} exp (\frac{f_{i}(x_{i}) − f(x_{i} + z_{i}) − λ_{i}z_{i}}{ϵ} )dz_{i}  (4.8)

So for every i ∈ I, we have

\frac{∂v_{ϵ}}{∂x_{i}} = (\frac{X_{i}(x_{i})v_{ϵ}^{i}}{ϵ} − R_{i}) ∏_{kϵ(I−\{i\})} v_{ϵ}^{k} exp (\frac{p(x)}{ϵ}) ,  (4.9)

where R_{i} = − exp(−λ_{i}/ϵ) + 1. So if x ∈ C_{ϵ},

\frac{X_{i}(x_{i})}{ϵ}v_{ϵ}^{i} = − exp ( − \frac{λ_{i}}{ϵ} ) + 1, i ∈ I  (4.10)

Then for i ∈ I_{+} we have X_{i}(x_{i}) > 0 and for i ∈ I_{−} we have X_{i}(x_{i}) < 0. □ Lemma 4.4. Under the hypothesis of Lemma 4.3, the set of degenerated critical points of u_{ϵ} is a subset of D_{1}(X) = ∪_{iϵI_{+}}[D(X_{i}) ∩ (X_{i}^{−1}(0, +∞))] ∪_{iϵI_{−}} [D(X_{i}) ∩ (X_{i}^{−1}(−∞, 0))] ∪ D(∇_{p}) Here x ∈ D(X_{i}), means X′(x_{i}) = 0 and x ∈ D(∇_{p}) means det(\frac{∂^{2}p}{∂x_{i}x_{j}}) = 0. Proof. With the notation of Lemma 4.3 and by (4.9) and (4.10), for every x ∈ C_{ϵ},

\frac{∂^{2}u_{ϵ}}{∂x_{i}^{2}} (x) = X′(x_{i})u_{ϵ}, i ∈ I  (4.11)
\frac{∂^{2}u_{ϵ}}{∂x_{i}∂x_{j}} (x) = 0, i, j ∈ I, i ≠ j  (4.12)
\frac{∂^{2}u_{ϵ}}{∂x_{i}∂x_{j}} (x) = \frac{∂^{2}p}{∂x_{i}∂x_{j}} (x)u_{ϵ}, i, j ∈ I′  (4.13)

where I′ = {1, 2, …, n} − I. So

det (\frac{∂^{2}u_{ϵ}}{∂x_{i}∂x_{j}} (x)) = ( ∏_{iϵI} X′_{i}(x_{i}))( det (\frac{∂^{2}p}{∂x_{i}∂x_{j}} (x))_{j,iϵI′})(u_{ϵ}(x))^{n}  (4.14)

If x is a degenerated critical point of u_{ϵ}, by Lemma 4.3, we get x ∈ D_{1}(X). □ Theorem 4.5. Let X = ∇f + λ be a vector field without coupling and suppose

I_{+} = {i : λ_{i} > 0}, I_{−} = {i : λ_{i} < 0}, I = I_{+} ∪ I_{−}, k = card(I) Let also suppose:

(i) For every i ∈ I_{+} the set D_{1}(X_{i}) = D(X_{i}) ∩ X_{i}^{−1}(0, +∞) is finite and for x_{i} ∈ D_{1}(X_{i}) there exits z_{i} ∈ (0, 1) such that f(x_{i}) − f(x_{i} + z_{i}) − λ_{i}z_{i} > 0. (ii) For i ∈ I_{−} the set D_{1}(X_{i}) = D(X_{i}) ∩ X_{i}^{−1}(−∞, 0) is finite and for every

z_{i} ∈ (0, 1) we have f(x_{i}) − f(x_{i} + z_{i}) − λ_{i}z_{i} ≤ 0. (iii) Considering p as a function in T_{n−k} do not has critical points which are

degenerated.

Then there exists ϵ_{0} > 0 such that u_{ϵ} does not have degenerated critical points for 0 < ϵ < ϵ_{0}.

Proof. Suppose there exists a sequence of values ϵ_{n} with ϵ_{n} > 0 and lim_{n→∞} ϵ_{n} = 0 and a sequence of point x_{n} in such way that x_{n} is a critical degenerated point of u_{ϵ_{n}}. Then by the proceeding Lemma and under conditions (i), (ii) and (iii) we can find a sequence of (ϵ_{n_{k}}) such that lim_{k→∞} x_{n_{k}} = x with x_{i} ∈ D_{1}(X_{i)} for some index i ∈ I. Clearly (x_{n_{k}})_{i} = x_{i} for k > k_{0} because D_{1}(X_{i}) is finite set. Then for that index i, it follows:

X_{i}(x_{i})u_{ϵ_{n_{k}}}^{i} (x_{i}) = ϵ_{n_{k}}( − exp (\frac{λ_{i}}{ϵ_{n_{k}}} ) + 1)  (4.15)

then for (i) or (ii) we have a contradiction when k → ∞. □

Example. Consider the vector field

X(x) = \begin{cases}α exp ( − \frac{1}{sin(2πx)}) & 0 ≤ x ≤ 1/2, \\ −β exp ( − \frac{1}{sin(2πx)}) & 1/2 ≤ x ≤ 1 \end{cases}

It is a C^{∞} vector field on T_{1}. We put

H = ∫_{0}^{1/2} exp ( − \frac{1}{sen2πx}),

X = ∇f + λ,

\begin{aligned}h(x, z) & = f(x_{i}) − f(x + z) − λz \\ & = ∫_{0}^{x} X(t)dt − ∫_{0}^{x+z} X(t)dt \end{aligned}

Then we have

h(1/4, 3/4) = (β − \frac{α}{2})H, λ = (α − β)H

Then if α > β > \frac{α}{2}, λ = (α − β) > 0, D_{1}(X) = \{1/4\} and by theorem 4.5, we have ϵ_{0} > 0 such that u_{ϵ} does not have degenerated critical points. In this case, v_{ϵ} is a Morse function for ϵ < ϵ_{0} and X is Zeeman Stable vector field [3].

References

[1] J. Guíñez, R. Quintero, A. D. Rueda; Calculating steady state for a Fokker-Planck equation. Act. Math. Hungar. 91 (2001), No. 4, 311-323.

[2] J. Guíñez and A. D. Rueda; Steady state for a Fokker-Planck equation on S_{n}. Act. Math. Hungar. Vol. 94 (2002), No. 3, 211-221.

[3] E. C. Zeeman; Stability of dinamical systems, Nonlinearity, Vol. 1 (1988), No. 1, 115–155.

Centro de Investigación de Matemática Aplicada (C.I.M.A.), Facultad de Ingeniería, Universidad del Zulia, Apartado 10482, Maracaibo, Venezuela

E-mail address, J. Guíñez: jguinez@luz.edu.ve

E-mail address, R. Quintero: rquintero@luz.edu.ve

E-mail address, A. D. Rueda: ad-rueda@cantv.net