Global dynamics of stationary solutions of the extended Fisher–Kolmogorov equation Jaume Llibre, Marcelo Messias, and Paulo R. da Silva Citation: J. Math. Phys. 52, 112701 (2011); doi: 10.1063/1.3657425 View online: http://dx.doi.org/10.1063/1.3657425 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v52/i11 Published by the AIP Publishing LLC. Additional information on J. Math. Phys. Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions http://jmp.aip.org/?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/726139175/x01/AIP-PT/HC_JMPCoverPg_0713/chp_books_banner1640x440.jpg/6c527a6a7131454a5049734141754f37?x http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Jaume Llibre&possible1zone=author&alias=&displayid=AIP&ver=pdfcov http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Marcelo Messias&possible1zone=author&alias=&displayid=AIP&ver=pdfcov http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Paulo R. da Silva&possible1zone=author&alias=&displayid=AIP&ver=pdfcov http://jmp.aip.org/?ver=pdfcov http://link.aip.org/link/doi/10.1063/1.3657425?ver=pdfcov http://jmp.aip.org/resource/1/JMAPAQ/v52/i11?ver=pdfcov http://www.aip.org/?ver=pdfcov http://jmp.aip.org/?ver=pdfcov http://jmp.aip.org/about/about_the_journal?ver=pdfcov http://jmp.aip.org/features/most_downloaded?ver=pdfcov http://jmp.aip.org/authors?ver=pdfcov JOURNAL OF MATHEMATICAL PHYSICS 52, 112701 (2011) Global dynamics of stationary solutions of the extended Fisher–Kolmogorov equation Jaume Llibre,1,a) Marcelo Messias,2,b) and Paulo R. da Silva3,c) 1Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain 2Departamento de Matemática Estatı́stica e Computação – FCT–UNESP, Rua Roberto Simonsen, 305, CEP 19060–900 P. Prudente, São Paulo, Brazil 3Departamento de Matemática – IBILCE–UNESP, Rua C. Colombo, 2265, CEP 15054–000 S. J. Rio Preto, São Paulo, Brazil (Received 8 June 2011; accepted 6 October 2011; published online 1 November 2011) In this paper we study the fourth order differential equation d4u dt4 + q d2u dt2 + u3 − u = 0, which arises from the study of stationary solutions of the Extended Fisher– Kolmogorov equation. Denoting x = u, y = du dt , z = d2u dt2 , v = d3u dt3 this equation be- comes equivalent to the polynomial system ẋ = y, ẏ = z, ż = v, v̇ = x − qz − x3 with (x, y, z, v) ∈ R4 and q ∈ R. As usual, the dot denotes the derivative with re- spect to the time t. Since the system has a first integral we can reduce our analysis to a family of systems on R3. We provide the global phase portrait of these systems in the Poincaré ball (i.e., in the compactification of R3 with the sphere S2 of the infinity). C© 2011 American Institute of Physics. [doi:10.1063/1.3657425] I. 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 ∂u ∂t = ∂2u ∂x2 + u − u3 was originally proposed in 1937 to model the interaction of dispersal and fitness in biological populations. The EFK-equation or more precisely the extended Fisher–Kolmogorov equation, ∂u ∂t = −γ ∂4u ∂x4 + ∂2u ∂x2 + u − u3, γ > 0 was proposed in 1988 as a higher order model equation for physical systems that are bistable (i.e., the EFK-equation has two uniform states u(x) = ± 1 which are stable, separated by a third uniform state u(x) = 0). For stationary solutions, that is, the solutions which do not depend on the time t, the EFK-equation reduces to the ordinary differential equation −γ d4u dx4 + d2u dx2 + u − u3 = 0, γ > 0. By the transformation x = 4 √ γ x̄, q = − 1√ γ , a)Electronic mail: jllibre@mat.uab.cat. b)Electronic mail: marcelo@fct.unesp.br. c)Electronic mail: prs@ibilce.unesp.br. 0022-2488/2011/52(11)/112701/12/$30.00 C©2011 American Institute of Physics52, 112701-1 Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions http://dx.doi.org/10.1063/1.3657425 http://dx.doi.org/10.1063/1.3657425 http://dx.doi.org/10.1063/1.3657425 mailto: jllibre@mat.uab.cat mailto: marcelo@fct.unesp.br mailto: prs@ibilce.unesp.br 112701-2 Llibre, Messias, and da Silva J. Math. Phys. 52, 112701 (2011) we brought into the form of the canonical equation d4u dt4 + q d2u dt2 + u3 − u = 0. (1) Denoting x = u, y = du dt , z = d2u dt2 , v = d3u dt3 we get the polynomial differential system ẋ = y, ẏ = z, ż = v, v̇ = x − qz − x3 (2) with (x, y, z, v) ∈ R4 and q ∈ R being negative. Besides the large amount of papers concerning the Fisher–Kolmogorov and extended Fisher– Kolmogorov equations existing in the literature (see, for instance, Refs. 1, 7, and 8), there are few works describing their dynamics. The aim of this paper 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 ẋ = F(x), x ∈ Rn (3) be a smooth differential system and S : Rn → Rn, S(x) = y be a linear map satisfying S ◦ S = Id. We say that (3) is symmetric with respect to S if ẏ(t) = F(y(t)). We say that (3) is reversible with respect to S if ẏ(t) = −F(y(t)). We point out some properties of symmetric and reversible systems. (a) Their phase portraits are symmetric with respect to Fix(S) = {x ∈ Rn : S(x) = x}. (b) If x(t) 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) implies 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). (a) It is reversible with respect to the involution R(x, y, z, v) = (x,−y, z,−v). (b) It has the first integral H (x, y, z, v) = q 2 y2 − x2 2 − z2 2 + vy + x4 4 . For h �= 0 and h �= − 1/4 the set H− 1(h) is a smooth three-dimensional manifold. (c) The flow of system (2) on H−1(h) ∩ (R4 \ {y = 0}) is determined by the constrained three- dimensional differential system ẋ = y, ẏ = z, 4yż = 4h + 2x2 − x4 − 2qy2 + 2z2. (4) Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-3 Stationary solutions of EFK-equation J. Math. Phys. 52, 112701 (2011) FIG. 1. Phase portrait of system (5) at the infinity of the Poincaré ball: there is a circle of equilibrium points containing the endpoints of the yz-plane; the positive (negative) endpoints of the z-axis behave like unstable (stable) nonhyperbolic improper nodes. (d) The equilibrium points of system (2) are (0, 0, 0, 0) ∈ H− 1(0) and ( ± 1, 0, 0, 0) ∈ H− 1( − 1/4). The plane y = 0 is an impasse surface for system (4) according to the terminology used in Refs. 6 and 9. Under the rescaling dt = 4ydτ we transform system (4) into the regularized vector field given by ẋ = 4y2, ẏ = 4yz, ż = 4h + 2x2 − x4 − 2qy2 + 2z2. (5) As any polynomial differential system, Eq. (5) can be extended to an analytic system on a closed ball of radius one, whose interior is diffeomorphic toR3 and its boundary, the two-dimensional sphere S2, plays the role of the infinity. This closed ball is denoted by D3 and is called the Poincaré ball, because the technique for doing such an extension is precisely the Poincaré compactification for a polynomial differential system in R3, which is described in detail in Ref. 2 and a summary of it is given in Sec. III 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 ⊆ D3 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 ∈ D3, defined on its maximal interval Ip = R, because D3 is compact. Then the α-limit set of p is the invariant set α(p) = {q ∈ D3 : ∃ {tn} such that tn → −∞ and ϕ(tn) → q as n → ∞}. In a similar way, the ω-limit set of p is the invariant set ω(p) = {q ∈ D3 : ∃ {tn} such that tn → ∞ and ϕ(tn) → q as n → ∞}. We also study the phase portrait of system (5) on the Poincaré ball. In order to state our next results, we introduce the notation: • D3 ++ = {(x, y, z) ∈ R3 : x2 + y2 + z2 < 1, y > 0, z > 0} and D3 +− = {(x, y, z) ∈ R3 : x2 + y2 + z2 < 1, y > 0, z < 0}; Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-4 Llibre, Messias, and da Silva J. Math. Phys. 52, 112701 (2011) • A = {(x, y, z) ∈ D3 : x2 + y2 + z2 < 1, y > 0}; • LN = {x = 0, 0 ≤ y ≤ 1, y2 + z2 = 1} and LS = {x = 0, − 1 ≤ y ≤ 0, y2 + z2 = 1}. We denote by Shq the closure of the surface ż = 4h + 2x2 − x4 − 2qy2 + 2z2 = 0 in the Poincaré ball. We also denote by Sh the closure of Shq ∩ {y = 0} in the Poincaré ball. We remember that if C ⊆ D3, then ∂C denotes its boundary and C denotes its closure in the Poincaré ball. Theorem 3: The polynomial differential system (5) in the Poincaré 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). (b) The plane y = 0 is invariant by the flow. The set of all finite equilibrium points of system (5) is the finite part of the curve Sh . The phase portrait on y = 0 is as it is shown in Figure 2. (c) The eigenvalues of the linear part of system (5) at the point (x, 0, z) are 0, 4z, and 4z. (d) System (5) has no periodic orbits. (e) If p ∈ A, then α(p) and ω(p) are contained in the set of equilibrium points contained in ∂A. (f) If p ∈ A, then – α(p) ∩ (LN\{(0, 0, 1), (0, 1, 0)}) = ∅, – α(p) ∩ (Sh ∩ {−1 < z < 0}) = ∅, – ω(p) ∩ (LS\{(0, 0, − 1), (0, 1, 0)}) = ∅, and – ω(p) ∩ (Sh ∩ {0 < z < 1}) = ∅. (g) If α(p) (resp. ω(p)) is contained in (Sh ∩ {−1 < z < 1, z �= 0}), then α(p) (resp. ω(p)) is formed by only one equilibrium point. Remark: According to the statement (a) of Theorem 3 it is enough to describe the phase portrait of system (5) only on D3++. But since D3++ is not invariant by the flow of (5) and the minimal compact invariant set containing D3++ 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. (a) If p ∈ ∂A, then α(p) and ω(p) are completely characterized in Figures 1 and 2. (b) The surface Shq is the boundary separating two open regions Z+ = {(x, y, z) ∈ A : ż > 0} and Z− = {(x, y, z) ∈ A : ż < 0}. If p ∈ Z+ is such that the whole orbit passing through p is contained in Z+, then ω(p) ⊂ LN. If p ∈ Z− is such that the whole orbit passing through p is contained in Z−, then α(p) ⊂ Sh ∩ {z ≥ 0} and ω(p) ⊂ Sh ∩ {z ≤ 0}. (c) The boundary of the surface Shq at infinity is the great circle x = 0. If the orbit through p ∈ A intersects Shq a more detailed analysis is necessary. Remark: We emphasize that the conclusions of Theorems 2–4 are concerned to the dynam- ics 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 y2 − x2 2 − z2 2 + vy + x4 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 to 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. The paper is organized as follows. In Sec. II we prove Theorem 1 and study the linear part of the differential system (2) at the equilibrium points. In Sec. III we give a summary of the formulas Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-5 Stationary solutions of EFK-equation J. Math. Phys. 52, 112701 (2011) (a) (b) (c) (d) (e) FIG. 2. (Color online) Global phase portrait of system (5) on the disk {y = 0} ∩ D3 : (a) h < − 1 4 , (b) h = − 1 4 , (c) − 1 4 < h < 0, (d) h = 0, (e) h > 0. The bold line is formed by equilibrium points. Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-6 Llibre, Messias, and da Silva J. Math. Phys. 52, 112701 (2011) related with the Poincaré compactification of a polynomial vector field in R3, 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 Sec. IV we prove Theorem 2, and in Sec. V we prove Theorem 3. In Sec. VI we prove Theorem 4 and we study the α- and ω-limit sets when the parameter h varies. II. PROOF OF THEOREM 1 (a) Denote X(x, y, z, v) = (y, z, v, x − qz − x3). If (y1, y2, y3, y4) = R(x, y, z, v) and (ẋ, ẏ, ż, v̇) = X (x, y, z, v), then (ẏ1, ẏ2, ẏ3, ẏ4) = −X (y1, y2, y3, y4). Thus, (a) is proved. (b) Since d H dt = Hx ẋ + Hy ẏ + Hzż + Hvv̇ = 0, it follows that H is a first integral of system (2). The gradient of H is given by ∇H (x, y, z, v) = (−x + x3, qy + v,−z, y), which is equal to (0, 0, 0, 0) if and only if x = 0, 1, − 1 and y = z = v = 0. Since H(0, 0, 0, 0) = 0 and H (±1, 0, 0, 0) = − 1 4 , each level H−1(h), h �= 0,− 1 4 is a three-dimensional invariant manifold of system (2) on R4. This proves statements (b) and (d). (c) The dynamics on each level H− 1(h) except on the surface H− 1(h) ∩ {y = 0} is determined by the constrained system (4). In fact, on H−1(h) ∩ (R4 \ {y = 0}), v = 4h + 2x2 − x4 − 2qy2 + 2z2 4y . Thus, system (2) becomes system (4). � The orbits of system (4) are defined only outside the impasse hyper-surface 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. Proposition 5: Consider system (2). (a) For any q ∈ R, there exist λ > 0, μ > 0 such that the eigenvalues at (0, 0, 0, 0) are ±λ ∈ R and ± μi. (b) The eigenvalues at ( ± 1, 0, 0, 0) are (b1) ±λ ∈ R and ±ν ∈ R, with λν �= 0, if q ∈ (−∞,−√ 8); (b2) ±λ ∈ R \ {0} with algebraic multiplicities equal to 2, if q = −√ 8; (b3) ± a ± bi, with ab �= 0, if q ∈ (−√ 8, √ 8); (b4) ± μi with algebraic multiplicities equal to 2 and μ �= 0, if q = √ 8; (b5) ± μi and ± νi with μν �= 0, if q ∈ ( √ 8,∞). Proof: Denote by X(x, y, z, v) = (y, z, v, x − qz − x3). We have DX (0, 0, 0, 0) = ⎛ ⎜⎜⎝ 0 1 0 0 0 0 1 0 0 0 0 1 1 0 −q 0 ⎞ ⎟⎟⎠ . Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-7 Stationary solutions of EFK-equation J. Math. Phys. 52, 112701 (2011) The eigenvalues of DX(0, 0, 0, 0) are ± √ −2q + 2 √ q2 + 4 2 , ± √ −2q − 2 √ q2 + 4 2 . Since for any q ∈ R we have that −2q + 2 √ q2 + 4 > 0 and −2q − 2 √ q2 + 4 < 0 statement (a) is verified. We also have DX (±1, 0, 0, 0) = ⎛ ⎜⎜⎝ 0 1 0 0 0 0 1 0 0 0 0 1 −2 0 −q 0 ⎞ ⎟⎟⎠ . The eigenvalues of DX( ± 1, 0, 0, 0) are ± √ −2q + 2 √ q2 − 8 2 , ± √ −2q − 2 √ q2 − 8 2 . If q ∈ (−∞,−√ 8), then both −2q ± 2 √ q2 − 8 are positive. If q = −√ 8, then −2q ± 2 √ q2 − 8 = −2q > 0. If q ∈ (−√ 8, √ 8), then both −2q ± 2 √ q2 − 8 are non-real. If q = √ 8, then −2q ± 2 √ q2 − 8 = −2q < 0. If q ∈ ( √ 8,∞), then both −2q ± 2 √ q2 − 8 are negative. � III. THE POINCARÉ COMPACTIFICATION IN R3 A polynomial vector field X in Rn can be extended to a unique analytic vector field on the sphere Sn . 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 Sn−1 of the sphere Sn . Poincaré introduced this compactification for polynomial vector fields in R2. Its extension to Rn for n > 2 can be found in Ref. 2 and some applications in Refs. 4 and 5. In this section we describe the Poincaré compactification for polynomial vector fields in R3 following closely what is made in Ref. 2. In R3 we consider the polynomial differential system ẋ = P1(x, y, z), ẏ = P2(x, y, z), ż = P3(x, y, z), or equivalently its associated polynomial vector field X = (P1, P2, P3). The degree n of X is defined as n = max{deg(Pi ) : i = 1, 2, 3}. Let S3 = {y = (y1, y2, y3, y4) ∈ R4 : ‖y‖ = 1} be the unit sphere in R4, and S+ = {y ∈ S3 : y4 > 0} and S− = {y ∈ S3 : y4 < 0} be the northern and southern hemispheres of S4, respectively. The tangent space to S3 at the point y is denoted by TyS3. Then the tangent plane T(0,0,0,1)S 3 = {(x1, x2, x3, 1) ∈ R4 : (x1, x2, x3) ∈ R3} is identified with R3. We consider the central projections f+ : R3 = T(0,0,0,1)S3 −→ S+ and f− : R3 = T(0,0,0,1) S3 −→ S− defined by f± (x) = ± (x1, x2, x3, 1)/ x, where x = ( 1 + ∑3 i=1 x2 i )1/2 . Through these central projections R3 is identified with the northern and southern hemispheres. The equator of S3 is S2 = {y ∈ S3 : y4 = 0}. Clearly, S2 can be identified with the infinity of R3. The maps f+ and f− define two copies of X on S3, one Df+ ◦ X in the northern hemisphere and the other Df− ◦ X in the southern one. Denote by X the vector field on S3 \ S2 = 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 X (y) = y4 ⎛ ⎜⎜⎝ 1 − y2 1 −y2 y1 −y3 y1 −y1 y2 1 − y2 2 −y3 y2 −y1 y3 −y2 y3 1 − y2 3 −y1 y4 −y2 y4 −y3 y4 ⎞ ⎟⎟⎠ ⎛ ⎝ P1 P2 P3 ⎞ ⎠ , Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-8 Llibre, Messias, and da Silva J. Math. Phys. 52, 112701 (2011) where Pi = Pi(y1/|y4|, y2/|y4|, y3/|y4|). Written in this way X (y) is a vector field in R4 tangent to the sphere S3. Now we can extend analytically the vector field X (y) to the whole sphere S3 by p(X )(y) = yn−1 4 X (y). This extended vector field p(X) is called the Poincaré compactification of X on S3. As S3 is a differentiable manifold in order to compute the expression for p(X) we can consider the eight local charts (Ui, Fi) , (Vi, Gi), where Ui = {y ∈ S3 : yi > 0} and Vi = {y ∈ S3 : yi < 0} for i = 1, 2, 3, 4; the diffeomorphisms Fi : Ui → R3 and Gi : Vi → R3 for i = 1, 2, 3, 4 are the inverses of the central projections from the origin to the tangent planes at the points ( ± 1, 0, 0, 0), (0, ± 1, 0, 0), (0, 0, ± 1, 0), and (0, 0, 0, ± 1), respectively. Now we do the computations on U1. Suppose that the origin (0, 0, 0, 0), the point (y1, y2, y3, y4) ∈ S3, and the point (1, z1, z2, z3) in the tangent plane to S3 at (1, 0, 0, 0) are collinear. Then we have 1/y1 = z1/y2 = z2/y3 = z3/y4, and consequently, F1(y) = (y2/y1, y3/y1, y4/y1) = (z1, z2, z3) defines the coordinates on U1. As DF1(y) = ⎛ ⎜⎜⎝ −y2/y2 1 1/y1 0 0 −y3/y2 1 0 1/y1 0 −y4/y2 1 0 0 1/y1 ⎞ ⎟⎟⎠ and yn−1 4 = (z3/ z)n−1, the analytical vector field p(X) becomes zn 3 ( z)n−1 (−z1 P1 + P2,−z2 P1 + P3,−z3 P1 ) , where Pi = Pi(1/z3, z1/z3, z2/z3). In a similar way we can deduce the expressions of p(X) in U2 and U3. These are zn 3 ( z)n−1 (−z1 P2 + P1,−z2 P2 + P3,−z3 P2 ) , where Pi = Pi(z1/z3, 1/z3, z2/z3) in U2, and zn 3 ( z)n−1 (−z1 P3 + P1,−z2 P3 + P2,−z3 P3 ) , where Pi = Pi(z1/z3, z2/z3, 1/z3) in U3. The expression for p(X) in U4 is zn+1 3 ( P1, P2, P3 ) , now denoting Pi = Pi(z1, z2, z3). The expression for p(X) in the local chart Vi is the same as in Ui multiplied by ( − 1)n − 1. 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 y4 = 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 R3 and whose boundary S2 corresponds to the infinity of R3. Of course, p(X) is defined in the whole closed ball D3 in such a way that the flow on the boundary is invariant. This new vector field on D3 will be called the Poincaré compactification of X, and D3 will be called the Poincaré ball. Remark: All the points on the invariant sphere S2 at infinity in the coordinates of any local chart Ui and Vi have z3 = 0. Also, the points in the interior of the Poincaré ball, which is diffeomorphic to R3, are given in the local charts U1, U2, and U3 by z3 > 0 and in the local charts V1, V2, and V3 by z3 < 0. Lemma 6: Let f(x1, x2, x3) = 0 be an algebraic surface of R3 = T(0,0,0,1)S3 of degree m. The extension of this surface to the boundary of the Poincaré ball is obtained solving the system ym 4 f ( x1 y4 , x2 y4 , x3 y4 ) = 0, y4 = 0. Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-9 Stationary solutions of EFK-equation J. Math. Phys. 52, 112701 (2011) Proof: We project the northern hemisphere y4 > 0 of the sphere S3 on y4 = 0, i.e., on the Poincaré ball using the equations x1 = y1 y4 , x2 = y2 y4 , x3 = y3 y4 . Thus, the points on the infinity correspond to the points on the equator y4 = 0 of S3. � IV. PROOF OF THEOREM 2 In this section we shall make an analysis of the flow of system (5) near and at infinity. In order to do it in Subsections 4A–4C we shall analyze the Poincaré compactification of system (5) in the local charts Ui and Vi, i = 1, 2, 3 as described in Sec. III and in Subsection 4D we put together the results obtained to obtain the proof of Theorem 2. A. In the local charts U1 and V1 From the results of Sec. III the expression of the Poincaré compactification p(X) of system (5) in the local chart U1 is given by ż1 = −4z1z2 3(z2 1 − z2), ż2 = −1 − 4z2 1z2z2 3 + 4hz4 3 + 2z2 3 − 2qz2 1z2 3 + 2z2 2z2 3, (6) ż3 = −4z2 1z2 3. For z3 = 0 (which corresponds to the points on the sphere S2 of the infinity) (6) reduces to ż1 = 0, ż2 = −1, (7) from which follows that system (5) has no equilibrium point nor periodic orbits in the portion of the Poincaré sphere parametrized by the local chart U1, which contains the positive endpoint of the x-axis. It implies that there are no trajectories of system (5) which tend to or come from infinity through this part of the sphere, where the dynamics is given by system (7), which is trivial. The flow in the local chart V1 is the same as the flow in the local chart U1 reversing appropriately the time, since the compactified vector field p(X) in V1 coincides with the vector field p(X) in U1 multiplied by ( − 1)n − 1, where n = 4 is the degree of system (5) (for details see Sec. III). Hence, system (5) also has trivial dynamics on the portion of the infinite sphere parametrized by the local chart U2, which contains negative endpoint of the x-axis. Actually, this dynamics is given by the system ż1 = 0, ż2 = 1. See Figure 1 which shows the dynamics of system (5) on the Poincaré sphere for a view of the dynamics on the portions of this sphere, containing the endpoints of the x-axis, described above. B. In the local charts U2 and V2 Again using the results of Sec. III we have the expression of the Poincaré compactification p(X) of system (5) in the local chart U2, which is given by ż1 = −4z2 3(z1z2 − 1), ż2 = −2z2 2z2 3 + 4hz4 3 + 2z2 1z2 3 − z4 1 − 2qz2 3, (8) ż3 = −4z2z2 3. For z3 = 0 (which corresponds to the points on the sphere S2 of the infinity) system (8) has a line of equilibria given by the z2-axis and the linear part of the system at these equilibria has three null eigenvalues. Let us study the flow near these line of equilibria. From the compactification procedure described in Sec. III follows that the z1z2-plane is invariant under the flow of (8), so we Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-10 Llibre, Messias, and da Silva J. Math. Phys. 52, 112701 (2011) (a) U2 (b) V2 FIG. 3. (Color online) Dynamics of system (5) on the sphere of the infinity in the local charts U2 (a) and V2 (b). There is a line of equilibria in the z2-axis. can completely describe the dynamics on the sphere at infinity. In fact, if z3 = 0 system (8) restricted to the z1z2-plane is given by ż1 = 0, ż2 = −z4 1. (9) Hence, the phase portrait of system (8) restricted to this plane is as shown in Figure 3(a). See also Figure 1 which shows the global phase portrait of system (5) on the Poincaré sphere. The flow in the local chart V2 is the same as the flow in the local chart U2 reversing the time (see Figure 3(b)), because the compactified vector field p(X) in V2 coincides with the vector field p(X) in U2 multiplied by ( − 1)n − 1, where n = 4 is the degree of system (5). C. In the local charts U3 and V3 The expression of the Poincaré compactification p(X) of system (5) in the local chart U3 is ż1 = −4hz1z4 3 − 2z3 1z2 3 + z5 1 + 2qz1z2 2z2 3 − 2z1z2 3 + 4z2 2z2 3, ż2 = −4hz2z4 3 − 2z2 1z2z2 3 + z4 1z2 + 2qz3 2z2 3 + 2z2z2 3, (10) ż3 = −4hz5 3 − 2z2 1z3 3 + z4 1z3 + 2qz2 2z3 3 − 2z3 3. For z3 = 0, system (10) restricted to the invariant z1z2-plane reduces to ż1 = z5 1, ż2 = z2z4 1. The solutions of this system behave like shown in Figure 4(a), which corresponds to the dynamics of system (5) at infinity in the local chart U3. The dynamics at infinity in the chart V3 is as shown in Figure 4(b). Indeed, for z1 �= 0 the system is equivalent to ż1 = z1, ż2 = z2, whose origin is an improper node. The set {z1 = 0} determines a line of equilibria. See also Figure 1. (a) U3 (b) V3 FIG. 4. (Color online) Dynamics of system (5) on the sphere of the infinity in the local charts U3 (a) and V3 (b). There is a line of equilibria in the z2-axis. Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-11 Stationary solutions of EFK-equation J. Math. Phys. 52, 112701 (2011) D. Dynamics of system (5) on the Poincaré sphere of the infinity Considering the analysis made in Subsections 4A–4C we have a global picture of the dynamical behavior of system (5) on the sphere at infinity. The system has a line of (nonhyperbolic) equilibria containing the endpoints of the yz-plane and there are no more equilibrium points on the sphere. The equilibria at the endpoints of the z-axis behave like improper nodes, even being nonhyperbolic. The global dynamics on the sphere of the infinity, constructed based on the calculations in the local charts Ui and Vi, i = 1, 2, 3, is shown in Figure 1. We observe that the description of the complete phase portrait of system (5) on the sphere at infinity was possible because of the invariance of this set under the flow of the compactified system, since the dynamics near the line of equilibria is highly degenerate. V. PROOF OF THEOREM 3 We denote Y(x, y, z) = (4y2, 4yz, 4h + 2x2 − x4 − 2qy2 + 2z2). Proof of Theorem 3: (a) If (y1, y2, y3) = S(x, y, z) and (ẋ, ẏ, ż) = Y (x, y, z), then (ẏ1, ẏ2, ẏ3) = Y (y1, y2, y3). If (y1, y2, y3) = R(x, y, z), then (ẏ1, ẏ2, ẏ3) = −Y (y1, y2, y3). (b) Since ẏ = 4yz, the plane y = 0 is invariant by the flow. The regularized system (5) has equilibrium points given by (x, 0, z) with 4h + 2x2 − x4 + 2z2 = 0. Applying Lemma 6, the extension of this curve to the boundary of the Poincaré ball is obtained by solving the system ω4 ( 4h + 2 ( x ω )2 − ( x ω )4 + 2 ( z ω )2 ) = 0, ω = 0. It means that the boundary at infinity of this curve is the union of the north and south hemispheres (0, 0, ± 1). (c) The linearization of Y at (x, 0, z) has the matrix ⎛ ⎝ 0 0 0 0 4z 0 4x − 4x3 0 4z ⎞ ⎠. It is immediate that the eigenvalues are 0 and 4z. (d) Since ẋ > 0 for y �= 0 it is impossible to have a periodic orbit. (e) If p ∈ A, then α(p) and ω(p) are contained in ∂A. In fact, since ϕ(t, p) ∈ D3, and D3 is bounded, α(p) and ω(p) are not empty. Moreover, since x′ > 0 in A, y′ > 0 in D3 ++, and y′ < 0 in D3 +−, it follows that α(p), ω(p) ⊂ ∂A. Due to the fact that ∂A is invariant, the Poincaré–Bendixson Theorem can be applied. Since there are neither periodic orbits, nor graphics on ∂A (see Theorem 2) we conclude that all the α(p) and ω(p) are formed by equilibrium points. (f) It is a direct consequence of the sign of the eigenvalues of the linearization of Y at (x, 0, z), and the signs of ẋ, ẏ, and ż in A. (g) Statement (c) implies that all equilibrium points in Sh ∩ {−1 < z < 1, z �= 0} are normally hyperbolic. Thus, the invariant manifold theory can be applied, see Ref. 3. � VI. THE α- AND ω-LIMIT SETS OF SYSTEM (5) SOLUTIONS Proof of Theorem 4: (a) Since ∂A is an invariant set and the phase portrait of the differential system (5) is sketched in Figures 1 and 2, the item (a) is proved. (b) The sign of ż determines the region where the flow goes up (ż > 0), and the region where the flow goes down (ż < 0). So the surface Shq is the boundary separating Z+ and Z−. The openness follows of the continuity of ż. If the whole orbit passing through p is contained in Z+, then Theorem 3 guarantees that ω(p) is contained in the set of equilibrium points on ∂A. Moreover, since ż > 0 through the orbit, ω(p) is contained in LN. Analogously, if the whole orbit passing through p is contained in Z−, then ż < 0 implies the statements. Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions 112701-12 Llibre, Messias, and da Silva J. Math. Phys. 52, 112701 (2011) (c) The boundary of the surface ż = 4h + 2x2 − x4 − 2qy2 + 2z2 = 0 at infinity is the great circle x = 0. In fact, according to Lemma 6 it follows solving the system ω4 ( 4h + 2 ( x ω )2 − ( x ω )4 − 2q ( y ω )2 + 2 ( z ω )2 ) = 0, ω = 0. The region on the Poincaré ball where ż > 0 (the inner one) ends at the circle {x = 0} of the boundary of the Poincaré ball. � In short, we have the following informations about the equilibrium points on ∂A. (a) Sh ∩ {z ≥ 0}: contains the α- and ω-limit sets of its own points; contains the α-limit of any point p at A which has the whole orbit passing through p contained in the region where z′ < 0; and depending on the parameter h; contains the α- or the ω-limit of some orbits on the plane y = 0 (see Figure 2). (b) North hemisphere (0, 0, 1): It is the α- and ω-limit of itself; it is α-limit of any regular point on S2; and it is the ω-limit of some orbits on the plane y = 0 (see Figure 2). (c) Curve LN: contains the α- and ω-limit sets of its own points; contains the ω-limit of any point at A which has the whole orbit passing through it contained in the region where z′ > 0. (d) Sh ∩ {z ≤ 0}: contains the α- and ω-limit sets of its own points; contains the ω-limit of any point p at A which has the whole orbit passing through p contained in the region where z′ < 0 and depending on the parameter h; contains the α- or the ω-limit of some orbits on the plane y = 0 (see Figure 2). (e) Curve LS: 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 S2; and it is the α-limit of some orbits on the plane y = 0 (see Figure 2). VII. CONCLUSIONS We describe the global dynamics of a polynomial differential system in R4 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 R3. We provide the global phase portraits of these systems in the Poincaré ball. Moreover, we characterize all the α- and ω-limit sets of all orbits of this system. ACKNOWLEDGMENTS J.L. is partially supported by the MICIIN/FEDER (Grant No. MTM2008–03437), the Generalitat de Catalunya (Grant No. 2009SGR-410), and ICREA Academia. M.M. is supported by CNPq-Brazil under the Project No. 305204/2009-2. P.R.S. is partially supported by CNPq and FAPESP. All the authors are supported by the Int. Coop. Proj. CAPES/MECD–TQED II and PHB-2009-0025. 1 Bonheure, D., “Multitransition kinks and pulses for fourth order equations with a bistable nonlinearity,” Ann. Inst. Henri Poincareé 21, 319–340 (2004). 2 Cima, A. and Llibre, J., “Bounded polynomial vector fields,” Trans. Am. Math. Soc. 318, 557–579 (1990). 3 Hirsch, M. W., Pugh, C. C., and Shub, M., “Invariant manifolds,” Bull. Am. Math. Soc. 76, 1015–1019 (1970). 4 Llibre, J., Messias, M., and da Silva, P. R., “On the global dynamics of the Rabinovich system,” J. Phys. A: Math. Theor. 41(27), 275210 (21pp.) (2008). 5 Llibre, J., Messias, M., and da Silva, P. R., “Global dynamics of the Lorenz system with invariant algebraic surface,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 20, 3137–3155 (2010). 6 Llibre, J., Sotomayor, J., and Zhitomirskii, M., “Impasse bifurcations of constrained systems Differential Equations and Dynamical Systems, Fields Institute Communications,” Am. Mat. Soc. A(31), 235–255 (2002). 7 Peletier, L. A. and Troy, W. C., Spatial Patterns, Higher Order Models in Physics and Mechanics (Birkhäuser, Boston - Basel - Berlin, 2001). 8 Peletier, L. A. and Troy, W. C., “Spatial patterns described by the extended Fisher–Kolmogorov (EFK) equation: kinks,” Diff. Integral Eq. 8, 1279–1304 (1995). 9 Sotomayor, J. and Zhitomirskii, M., “Impasse Singularities of Differential Systems of the Form A(x)x′ = F(x),” J. Differ. Equations 169, 567–587 (2001). Downloaded 18 Jul 2013 to 200.130.19.215. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions http://dx.doi.org/10.1016/S0294-1449(03)00037-4 http://dx.doi.org/10.1016/S0294-1449(03)00037-4 http://dx.doi.org/10.2307/2001320 http://dx.doi.org/10.1090/S0002-9904-1970-12537-X http://dx.doi.org/10.1088/1751-8113/41/27/275210 http://dx.doi.org/10.1142/S0218127410027593 http://dx.doi.org/10.1142/S0218127410027593 http://dx.doi.org/10.1006/jdeq.2000.3908 http://dx.doi.org/10.1006/jdeq.2000.3908