GLOBAL DYNAMICS OF STATIONARY SOLUTIONS OF THE EXTENDED FISHER-KOLMOGOROV EQUATION

. In this paper we study the fourth order diﬀerential equation d 4 u dt 4 + q d 2 u dt 2 + u 3 − u = 0 , which arises from the study of stationary solutions of the Extended Fisher–Kolmogorov equation. Denoting x = u, y = dudt ,z = d 2 u dt 2 ,v = d 3 u dt 3 this equation becomes equivalent to the polynomial system ˙ x = y, ˙ y = z, ˙ z = v, ˙ v = x − qz − x 3 with ( x,y,z,v ) ∈ R 4 and q ∈ R . As usual, the dot denotes derivative with respect to the time t. Since the system has a ﬁrst integral we can reduce our analysis to a family of systems on R 3 . We provide the global phase portrait of these systems in the Poincar´e ball (i.e. in the compactiﬁcation of R 3 with the sphere S 2 of the inﬁnity).


Introduction and statement of main results
The classical equations of mathematical physics are typically linear second order differential equations. However, many problems in the sciences and in engineering are intrinsically nonlinear. The Fisher-Kolmogorov equation we brought into the form of the canonical equation 3 we get the polynomial differential system (2) with (x, y, z, v) ∈ R 4 and q ∈ R negative.
Besides the large amount of papers concerning the Fisher-Kolmogorov and Extended Fisher-Kolmogorov equations existing on the literature (see for instance [1,7,8]), there are a few works describing their dynamics. The aim of this note is to describe the global dynamics of stationary solutions of the EFK-equation, more precisely to characterize all the α-and ω-limit sets of all orbits of this equation. Before doing it let us remember some basic results about symmetric and reversible systems, which shall be used later on in the study of system (2). Let be a smooth differential system and S : R n → R n , S(x) = y be a linear map satisfying S • S = Id. We say that (3) is symmetric with respect to S ifẏ(t) = F (y(t)). We say that (3) is reversible with respect to S iḟ y(t) = −F (y(t)).
We point out some properties of symmetric and reversible systems. (a) Their phase portraits are symmetric with respect to is a solution of system (3), then S(x(t)) = y(t) is a solution of (3) for the symmetric case, and S(x(t)) = y(−t) is a solution of (3) for the reversible case. For the reversible case: (c1) Any orbit meeting Fix(S) at two different points is a periodic orbit. (c2) Any equilibrium point or periodic orbit on Fix(S) cannot be an attractor or a repeller.
(c3) Intersection of (un)-stable manifolds with Fix(S) imply the existence of heteroclinic or homoclinic orbits. Our first result about system (2) is the following. Theorem 1. The following statements hold for system (2).
The plane y = 0 is an impasse surface for system (4) according to the terminology used in [6,9]. Under the rescaling dt = 4ydτ we transform system (4) into the regularized vector field given by As any polynomial differential system, equations (5) can be extended to an analytic system on a closed ball of radius one, whose interior is diffeomorphic to R 3 and its boundary, the 2-dimensional sphere S 2 , plays the role of the infinity. This closed ball is denoted by D 3 and called the Poincaré ball, because the technique for doing such an extension is precisely the Poincaré compactification for a polynomial differential system in R 3 , which is described in details in [2] and a summary of it is given in section 3 ahead. By using this compactification technique the dynamics of system (5) at infinity is studied and we have the following result.

Theorem 2.
For all values of the parameters h, q ∈ R the phase portrait of system (5) on the sphere of infinity is as shown in figure 1.
We say that a set V ⊆ D 3 is invariant by the flow of (5) if for any p ∈ V the whole orbit passing through p is contained in V . The sphere of the infinity is always an invariant set.
Let ϕ(t) = ϕ(t, p) be the solution of the compactified system (5) passing through the point p ∈ D 3 , defined on its maximal interval I p = R, because D 3 is compact. Then the α-limit set of p is the invariant set α(p) = {q ∈ D 3 : ∃ {t n } such that t n → −∞ and ϕ(t n ) → q as n → ∞}. In a similar way, the ω-limit set of p is the invariant set We also study the phase portrait of system (5) on the Poincar ball. In order to state our next results, we introduce the notation: We denote by S hq the closure of the surfaceż = 4h+2x 2 −x 4 −2qy 2 +2z 2 = 0 in the Poincar ball. We also denote by S h the closure of S hq ∩ {y = 0} in the Poincar ball. We remember that if C ⊆ D 3 then ∂C denotes its boundary and C denotes its closure in the Poincar ball.
Theorem 3. The polynomial differential system (5) in the Poincar ball satisfies the following statements.
(a) It is symmetric with respect to the involution S(x, y, z) = (x, −y, z), and reversible with respect to the involution R(x, y, z) = (−x, −y, −z). (e) If p ∈ A then α(p) and ω(p) are contained in the set of equilibrium points contained in ∂A.
is formed by only one equilibrium point.
Remark. According to statement (a) of Theorem 3 it is enough to describe the phase portrait of system (5) only on D 3 ++ . But since D 3 ++ is not invariant by the flow of (5) and the minimal compact invariant set containing D 3 ++ is A, we shall describe all the α-and ω-limit sets of the orbits contained in A.
Theorem 4. The α-and ω-limit sets of the solutions of the compactified system (5) satisfy the following statements.
The boundary of the surface S hq at infinity is the great circle x = 0.
If the orbit through p ∈ A intersects S hq a more detailed analysis is necessary.
Remark. We emphasize that the conclusions of Theorems 2, 3 and 4 are concerned to the dynamics of the differential system (5) and its compactification. The orbits of the original differential system, i.e. system (2), are contained in the hyper surfaces H −1 (h) where H = q 2 y 2 − x 2 2 − z 2 2 + vy + x 4 4 . System (2) on H −1 (h) ∩ {y = 0} is topologically equivalent to system (5) on {y = 0} that is, removing from its orbits the impasse points, and inverting the orientation of the orbits contained on {y < 0}. No additional information is given for the orbits of system (2) passing through {y = 0}. Due to this fact, in spite of system (5) has no periodic orbits, our analysis is not sufficient to detect periodic orbits of the original system if they cross {y = 0}. A way to study the complete phase portrait is to solve H = 0 on the variable z and study the corresponding two differential systems according the sign of the square root which appears after the substitution of the variable z. But to do this work is equivalent to write another paper longer than this one.

STATIONARY SOLUTIONS OF EXTENDED FISHER-KOLMOGOROV EQUATION 7
The paper is organized as follows. In section 2 we prove Theorem 1 and study the linear part of the differential system (2) at the equilibrium points. In section 3 we give a summary of the formulas related with the Poincaré compactification of a polynomial vector field in R 3 , because they will be used along this paper. We also study how the invariant algebraic surfaces H −1 (h) extend to infinity in the Poincaré ball (see Lemma 6). In section 4 we prove Theorem 2, and in section 5 we prove Theorem 3. In section 6 we prove Theorem 4 and we study the α-and ω-limit sets when the parameter h varies.
it follows that H is a first integral of system (2). The gradient of H is given by Thus system (2) becomes system (4). The orbits of system (4) are defined only outside the impasse hypersurface by the corresponding similar elements of system (5). Note that the phase portrait of system (4) is the same as of system (5) by removing from its orbits the impasse points and inverting the orientation of the orbits contained in y < 0. Now we study the linear part of the differential system (2) at the equilibrium points.

The Poincaré compactification in R 3
A polynomial vector field X in R n can be extended to a unique analytic vector field on the sphere S n . The technique for making such an extension is called the Poincaré compactification and allows us to study a polynomial vector field in a neighborhood of infinity, which corresponds to the equator S n−1 of the sphere S n . Poincaré introduced this compactification for polynomial vector fields in R 2 . Its extension to R n for n > 2 can be found in [2] and some applications in [4,5]. In this section we describe the Poincaré STATIONARY SOLUTIONS OF EXTENDED FISHER-KOLMOGOROV EQUATION 9 compactification for polynomial vector fields in R 3 following closely what is made in [2].
In R 3 we consider the polynomial differential systeṁ x = P 1 (x, y, z),ẏ = P 2 (x, y, z),ż = P 3 (x, y, z), or equivalently its associated polynomial vector field X = (P 1 , P 2 , P 3 ). The degree n of X is defined as n = max{deg(P i ) : i = 1, 2, 3}. Let S 3 = {y = (y 1 , y 2 , y 3 , y 4 ) ∈ R 4 : y = 1} be the unit sphere in R 4 , and S + = {y ∈ S 3 : y 4 > 0} and S − = {y ∈ S 3 : y 4 < 0} be the northern and southern hemispheres of S 4 respectively. The tangent space to S 3 at the point y is denoted by T y S 3 . Then the tangent plane We consider the central projections f + : R 3 = T (0,0,0,1) S 3 −→ S + and f − : . Through these central projections R 3 is identified with the northern and southern hemispheres. The equator of S 3 is S 2 = {y ∈ S 3 : y 4 = 0}. Clearly S 2 can be identified with the infinity of R 3 . The maps f + and f − define two copies of X on S 3 , one Df + • X in the northern hemisphere and the other Df − • X in the southern one. Denote by X the vector field on S 3 \ S 2 = S + ∪ S − which, restricted to S + coincides with Df + • X and restricted to S − coincides with Df − • X.
The expression for X(y) on S + ∪ S − is where P i = P i (y 1 /|y 4 |, y 2 /|y 4 |, y 3 /|y 4 |). Written in this way X(y) is a vector field in R 4 tangent to the sphere S 3 . Now we can extend analytically the vector field X(y) to the whole sphere S 3 by p(X)(y) = y n−1 4 X(y). This extended vector field p(X) is called the Poincaré compactification of X on S 3 .
When we work with the expression of the compactified vector field p(X) in the local charts we usually omit the factor 1/(∆z) n−1 . We can do that through a rescaling of the time variable.
In what follows we shall work with the orthogonal projection of p(X) from the closed northern hemisphere to y 4 = 0, we continue denoting this projected vector field by p(X). Note that the projection of the closed northern hemisphere is a closed ball B of radius one, whose interior is diffeomorphic to R 3 and whose boundary S 2 corresponds to the infinity of R 3 . Of course p(X) is defined in the whole closed ball D 3 in such a way that the flow on the boundary is invariant. This new vector field on D 3 will be called the Poincaré compactification of X, and D 3 will be called the Poincaré ball. Remark. All the points on the invariant sphere S 2 at infinity in the coordinates of any local chart U i and V i have z 3 = 0. Also, the points in the interior of the Poincaré ball, which is diffeomorphic to R 3 , are given in the local charts U 1 , U 2 and U 3 by z 3 > 0 and in the local charts V 1 , V 2 and V 3 by z 3 < 0. (e) Curve L S : contains the α and ω-limit sets of its own points. (f) South hemisphere (0, 0, 1): It is the α and ω-limit of itself; it is the ω-limit of any regular point on S 2 and it is the α-limit of some orbits on the plane y = 0 (see figure 2).

Conclusions
We describe the global dynamics of a polynomial differential system in R 4 which corresponds to the stationary solutions of the EFK-equation. We find a first integral and thus we reduce our analysis to a family of polynomial differential systems in R 3 . We provide the global phase portraits of these systems in the Poincaré ball. Moreover we characterize all the α-and ωlimit sets of all orbits of this system.