This is a preprint of: “On the ”traveling pulses” of the limit of the FitzHugh-Nagumo equations when  → 0”, Jaume Llibre, Clàudia Valls, Nonlinear Anal. Real World Appl., vol. 73, 103891, 2023. DOI: [10.1016/j.nonrwa.2023.103891] ON THE TRAVELING PULSES OF THE FITZHUGH-NAGUMO EQUATIONS JAUME LLIBRE1 AND CLAUDIA VALLS2 Abstract. In this paper we characterize the phase portraits of the polynomial differential systems ẋ = y, ẏ = −cy + x(x − a)(x − 1) + w, with a, c, w ∈ R. We give the complete description of their phase portraits in the Poincaré disc (i.e. in the compactification of R2 adding the circle S1 of the infinity) modulo topological equivalence. When w = 0 the homoclinic orbits of the origin of coordinates are related with the travelling pulses of the FitzHugh-Nagumo equations. 1. Introduction and statement of the main results Numerous problems of applied mathematics, or in physics, chemistry, economics, ... are modeled by polynomial differential systems. Excluding linear differential systems the quadratic polynomial differential systems are the ones with the lowest degree of complexity, and the large bibliography on them proves their relevance, see the books [1, 12, 13] and the surveys [2, 3]. After the quadratic polynomial differential systems come the cubic ones, which also have many applications. This paper deals with the travelling waves of the FitzHugh-Nagumo equations (1) ut = uxx + f (u) − w, wt = ε(u − γw), being f (u) = u(u − a)(1 − u), where a < 1/2, ε and γ are positive real numbers. Here we are interested in the case ε << 1 and γ << 1. The FitzHugh-Nagumo equations are a simplification of the Hodgking-Hexley equations for nerve conduction, see FitzHugh [5] and Nagumo et al. [10]. A solution of system (1) of the form (u(s), w(s)) with s = x − ct with c ̸= 0 is a travelling wave and it satisfies the differential system (2) −cu′ = u′′ + f (u) − w, −cw′ = ε(u − γw), where the prime denotes derivative with respect to the variable s. A travelling way satisfying (u, w) → (0, 0) when s → ±∞ is a travelling pulse. We write the differential system (2) as the differential system of first order (3) u′ = v, v ′ = −cv − f (u) + w, w′ = −(ε/c)(u − γw). 2010 Mathematics Subject Classification. Primary 34A05. Secondary 34C05, 37C10. Key words and phrases. FitzHugh-Nagumo system, Poincaré compactification, dynamics at infin- ity, traveling pulse. 1 2 J. LLIBRE AND C. VALLS System (3) has R3 as its phase space and it has a origin (0, 0, 0) as a singular point, and consequently a travelling pulse is a homoclinic orbit to the origin. If ε = 0 it follows that each plane w = constant is invariant, i.e. if an orbit of system (3) has a point in a such plane the whole orbit is contained in that plane. The objective of this paper is to study the phase portraits of system (3) with ε = 0 that we shall write as (4) x′ = y, y ′ = −cy − x(x − a)(1 − x) + w with a, c, w ∈ R and the prime means derivative with respect to s ∈ R. We are specially interested in the homoclinic orbits at the origin when c > 0, a < 1/2 and w = 0. For more details in the travelling pulses see [6, 7, 8, 9]. Doing the change of variables (x, y, t, c) → (x, −y, −t, c) in what follows we assume that c ≥ 0. (a) (b) (c) (d) (e) (f) (g) Figure 1. The seven phase portraits in the Poincaré disc of Theorem 1. More precisely, first we classify the phase portraits of the polynomial differential systems (4) in the Poincaré disc modulo topological equivalence. As any polynomial differential system, system (4) can be extended to an analytic system on a closed disc of radius one, whose interior is diffeomorphic to R2 and its boundary, the circle S1 , plays the role of the infinity. This closed disc is denoted by D2 and FITZBURGH NAGUMO POLYNOMIAL DIFFERENTIAL SYSTEMS 3 called the Poincaré disc, because the technique for doing such an extension is the Poincaré compactification for a polynomial differential system in R2 , which is described in details in Chapter 5 of [4]. In this paper we shall use the notation of that chapter. By using this compactification technique the dynamics of system (4) in a neighborhood of the infinity can be studied. Theorem 1. The phase portraits of the polynomial differential system (4) in the Poincaré disc are topologically equivalent to one of the seven phase portraits pre- sented in Figure 1. 2. Infinite singular points The following lemma summarizes the information at the infinite singular points of system (4). Lemma 2. There is a pair of infinite singular points located at the origin of the local charts U2 ∪ V2 and are formed by two elliptic sectors separated by parabolic sectors, and the infinity straight line crosses the two parabolic sectors leaving in each side an elliptic sector, see Figure 2. (a) (b) (c) (d) (e) Figure 2. The local phase portraits of the blow ups of Lemma 2. Proof. First we determine the local phase portrait of the infinite singular points in the local chart U1 . The expression of system (4) in this chart is u̇ = 1 − (a + 1)v + av 2 − cuv 2 + wv 3 − u2 v 2 , v̇ = −uv 3 . There are no infinite singular points in the local chart U1 . 4 J. LLIBRE AND C. VALLS Now we analyze the phase portrait in the local chart U2 , we only need to study the origin of U2 , the other infinite singular points have been studied in the local chart U1 . The expression of the system in this chart is u̇ = v 2 + cuv 2 − u4 + (a + 1)u3 v − au2 v 2 − wuv 3 , (5) v̇ = −v(−cv 2 + u3 − (a + 1)u2 v + auv 2 + wv 3 ). Note that the origin of the local chart U2 is a singular point whose linear part is identically zero. We need to do a blow-up procedure to study its local phase portrait. We first introduce the coordinates (u, v1 ) where v1 = v/u. In these new coordinates we obtain u̇ = −u2 (u2 − v12 − (a + 1)u2 v1 − cuv12 + au2 v12 + wu2 v13 ), v̇1 = −uv13 . We introduce the reparameterization of time ds = udt and we get u̇ = −u(u2 − v12 − (a + 1)u2 v1 − cuv12 + au2 v12 + wu2 v13 ), (6) v̇1 = −v13 , where now the dot means derivative in the new time s. Only the origin is on the straight line u = 0 and again its linear part is identically zero. We apply a second blow-up introducing the new coordinates (u, v2 ) where v2 = v1 /u. In these new coordinates we obtain the system u̇ = −u3 (1 − (a + 1)uv2 − v22 − cuv22 + au2 v22 + wu3 v23 ), v̇2 = u2 v2 (1 − (a + 1)uv2 − 2v22 − cuv22 + au2 v22 + wu3 v23 ). We introduce the reparameterization of time dr = u2 ds and we get u̇ = −u(1 − (a + 1)uv2 − v22 − cuv22 + au2 v22 + wu3 v23 ), v̇2 = v2 (1 − (a + 1)uv2 − 2v22 − cuv22 + au2 v22 + wu3 v23 ). where now the dot means derivative in √ the new time r.√ There are three singular points in u = 0 which are (0, 0), (0, −1/ 2) and (0, 1/ 2). Computing the eigen- values of the Jacobian matrix at these singular points we get that√ (0, 0) is a saddle (the two eigenvalues are −1, 1) and the singular points (0, ±1/ 2) are both stable nodes (the two eigenvalues are in both cases −2, −1/2). Therefore the local phase portrait near the straight line u = 0 is topologically equivalent to the one of Figure 2(a). Undoing the rescaling dr = u2 ds we get the phase portrait of Figure 2(b). Going back through the changes of variables from the phase portrait of Figure 2(b), we obtain the local phase portrait at the origin of system (6) which is topologically equivalent to the one of Figure 2(c). Again undoing the rescaling ds = udt we obtain the phase portrait of Figure 2(d), and going back through the changes of variables from the phase portrait of Figure 2(d), we obtain the local phase portrait at the origin of system (5) which is topologically equivalent to the one of Figure 2(e). Hence the origin of the local chart U2 is formed by two elliptic sectors separated by parabolic sectors, and the infinite straight line crosses the two parabolic sectors leaving in each side an elliptic sector. □ FITZBURGH NAGUMO POLYNOMIAL DIFFERENTIAL SYSTEMS 5 3. Finite singular points We first obtain the following result concerning the non-existence of limit cycles. Proposition 3. System (4) has no limit cycles. Proof. The divergence of system (4) is constant equal to −c. Therefore if c > 0 it follows from the Bendixson Theorem (see [4, Theorem 7.10]) that there are no limit cycles. On the other hand if c = 0 system (4) has the polynomial first integral y2 x4 (a + 1)x3 ax2 (7) H(x, y) = − + − − wx, 2 4 3 2 and so when c = 0 there are no limit cycles. □ Proposition 4. The vector field (P (x, y), Q(x, y)) = (y, −cy − x(x − a)(1 − x) + w), associated to system (4) is a generalized rotated vector field with respect the param- eter c. Proof. Since ! P Q ∂P ∂Q = −y 2 ≤ 0, ∂c ∂c the proposition follows. See for more details [4, Chapter 7], [11] and [14, Chapter 4]. □ The finite singular points of system (4) are the real solutions of ẋ = ẏ = 0. Computing such solutions we obtain y = 0 and x = x∗ where x∗ is any real solution of F (x) = x(x − a)(x − 1) + w = 0. We have several possibilities. 3.1. Three real roots for F (x). In this case we have that F (x) is of the form (x − r1 )(x − r2 )(x − r3 ) with r1 , r2 , r3 ∈ R and r1 < r2 < r3 if (i) the parameters of F (x) when r3 ̸= 1 − r2 . are (r2 − 1)r2 (r3 − 1)r3 −r2 + r22 − r3 + r2 r3 + r32 w= , a= , r2 + r3 − 1 r2 + r3 − 1 with either r2 < 0, r2 + r3 < 1 and (r2 − 1)2 + 2r2 r3 < r3 ; or 0 < 2r2 < 1, r2 + r3 > 1 and (r2 − 1)2 + 2r2 r3 > r3 ; or 2r2 ≥ 1 and r3 > r2 ; (ii) or the parameters of F (x) when r3 = 1 − r2 , are w = 0, a = r1 , r2 = 0, and so r3 = 1. Now we study the local behavior of each singular point (ri , 0) with i = 1, 2, 3. The Jacobian matrix at each singular point (ri , 0) is of the form   Q 0 1 , j∈{1,2,3},j̸=i (ri − rj ) −c 6 J. LLIBRE AND C. VALLS whose eigenvalues are q Q −c ± c2 + 4 j∈{1,2,3},j̸=i (ri − rj ) λ= . 2 Therefore, since r1 < r2 < r3 we get that (r1 , 0) is a saddle. Moreover, (r2 , 0) is a stable node if c > 0 and c2 + 4(r2 − r1 )(r2 − r3 ) ≥ 0, it is a stable focus if c > 0 and c2 + 4(r2 − r1 )(r2 − r3 ) < 0 and it is a center if c = 0 (by the proof of Proposition 3 it has a polynomial first integral). Finally, (r3 , 0) is a saddle. 3.2. Two real roots for F (x). In this case we can write F (x) = (x − r1 )2 (x − r2 ) with r1 , r2 ∈ R and r1 ≠ r2 whenever 2 2 (r1 − 1) r1 r1 (3r1 − 2) p (8) w = , a= , 2r1 −1 ̸= 0, r1 ̸= 1−r2 ± r2 (r2 − 1). 2r1 − 1 2r1 − 1 Now we study the local behavior of each singular point (r1 , 0) and (r2 , 0). We first study the local behavior of (r2 , 0). The Jacobian matrix at the singular point (r2 , 0) is   0 1 , (r2 − r1 )2 −c whose eigenvalues are p −c ± c2 + 4(r2 − r1 )2 λ= 2 and so (r2 , 0) is a saddle. The Jacobian matrix at each singular point (r1 , 0) is of the form   0 1 0 −c and so it is semihyperbolic if c ̸= 0 and nilpotent if c = 0. In the first case using [4, Proposition 2.19] we get that it is a saddle-node and in the second case, using [4, Proposition 3.5] we get that it is a cusp. 3.3. One real root for F (x). In this case we have that either F (x) = (x − r1 )3 with r1 ∈ R, or F (x) = (x − r1 )(x2 − 2αx + (α2 + β 2 )) with r1 , α, β ∈ R. The first case is not possible because the equation (x − r1 )3 = x(x − a)(x − 1) + w has no real solutions. The second case is achieved whenever α2 − 2α3 + α4 + β 2 − 2αβ 2 + 2α2 β 2 + β 4 3α2 − β 2 − 2α w= , a= , 2α − 1 ̸= 0. 2α − 1 2α − 1 The Jacobian matrix at the singular point (r1 , 0) is   0 1 , β 2 + (r1 − α)2 −c whose eigenvalues are p −c ± c2 + 4(β 2 + (r1 − α)2 ) λ= 2 Therefore, (r1 , 0) is a saddle because −β 2 − (r1 − α)2 < 0. FITZBURGH NAGUMO POLYNOMIAL DIFFERENTIAL SYSTEMS 7 4. Phase portraits in the Poincaré disc Here we shall study the phase portraits in the Poincaré disc using the local phase portraits of the finite and infinite singular points, together with Proposition 3, which states that system (4) has no limit cycles. For all values of the parameters a, c and w system (4) at infinity has a unique pair of singular points, the origins of U2 and V2 , which in the Poincaré disc has one elliptic sector and two parabolic sectors in its sides, see Lemma 2. 4.1. One real root for F (x). In this case system (4) has a unique finite singular point which is a saddle, see subsection 3.3. Taking into account the singular points at infinity the unique possible global phase portrait in the Poincaré disc is given in Figure 1(a). 4.2. Two real roots for F (x). Case c = 0. Then system (4) is a Hamiltonian system with the Hamiltonian H(x, y) given in (7). This system has two finite singular points, a saddle and a cusp, see subsection 3.2. The unique separatrices in the interior of the Poincaré disc are the six separa- trices, the four of the saddle and the two of the cusp. We claim that there is no connections between the saddle (r2 , 0) and the cusp (r1 , 0). Indeed, if a connection exists we must have H(r1 , 0) = H(r2 , 0), and taking into account (8) this equality, after some computations, is equivalent to 3(r1 − 1)r1 + 1 = 0. 2r1 − 1 Since this equation has no real roots, the claim is proved. Taking into account the local phase portraits at the finite and infinite singu- lar points, and that there are no connections between the separatrices the unique possible phase portrait in the Poincaré disc is 1(b). Case c > 0. Now system (4) has two finite singular points, a saddle at (r2 , 0) and a saddle-node at (r1 , 0). We consider that r1 < r2 , the case r1 > r2 can be studied in a similar way. Computing the eigenvalues of the linear part of system (4) at the singular points (r1 , 0) and (r2 , 0) we obtain that the local separatrices near these two singular points have the orientation with respect to the straight line y = 0 shown in Figure 3. Note that system (4) has seven separatrices in the interior of the Poincaré disc, we must determine where they born and where they die. Now taking into account the flow of system (4) on y = 0 and the local phase portraits of the finite and infinite singular points we get that the α- and ω-limit of the separatrices drawn in Figure 3 are determined. It remains to determine the α-limit of the separatrix s1 of (r1 , 0) and the ω-limit of the separatrix s2 of (r2 , 0). The ω-limit of the unstable separatrix of (r2 , 0) that we must determine has three possibilities: first to be the origin of the local chart V2 , second to connect with the free stable separatrix of (r1 , 0), and third to reach the saddle-node (r1 , 0) through its parabolic sector. The first case can be realized for the value c = 1, r1 = 0 and 8 J. LLIBRE AND C. VALLS Figure 3. The phase portrait with a finite saddle and a finite saddle- node. r2 = 1 in Figure 1(c), the third case is realized by c = 2, r1 = 0, r2 = 1 in Figure 1(e). So by continuity moving c between the values 1 and 2 we obtain the phase portrait of Figure 1(d). 4.3. Three real roots for F (x). Case c = 0. Then system (4) is a Hamiltonian system with the Hamiltonian H(x, y) given in (7). This system has three finite sin- gular points, two saddles at (r1 , 0) and (r3 , 0), and a center at (r2 , 0), see subsection 3.3. The unique separatrices in the interior of the Poincaré disc are the eight sep- aratrices of the two saddles. We claim that there are two connections between the saddles (r1 , 0) and (r3 , 0) forming a heteroclinic loop. Indeed, if such con- nections exist we must have H(r1 , 0) = H(r3 , 0), and this is possible whenever r1 − 2r2 + r3 = 0. Taking into account the local phase portraits of the finite and infinite singular points, and that there is a unique heteroclinic loop with the two saddles, it follows the existence of a unique topologically equivalent phase portrait in the Poincaré disc given in Figure 1(f). Case c > 0. Now system (4) has three finite singular points, a saddle at (r1 , 0) a stable node (or a stable focus) at (r2 , 0) and a saddle at (r3 , 0). We consider fixed r1 < r2 < r3 and we vary c from 0 to ∞. Note that system (4) has eight separatrices in the interior of the Poincaré disc and we must determine where they born and where they die. Now taking into account the flow of system (4) on y = 0 and the local phase portraits of the finite and infinite singular points we get that the α- and ω-limit of the separatrices drawn in Figure 4 are determined. It remains to determine the α-limit and ω-limit of the four separatrices which for c = 0 form the heteroclinic loop. It follows from Proposition 4 that the vector field associated to system (4) is a generalized rotated vector field with respect to the parameter c. So, the heteroclinic loop that exists for c = 0 breaks down (see [14, Chapter 4]) and since for c > 0 sufficiently small the singular point (r2 , 0) is a stable focus, the ω-limit of the separatrices s1 and s2 of Figure 4 is the stable focus (r2 , 0) and consequently the α- limit of the separatrices s3 and s4 of Figure 4 are determined provinding the phase portrait 1(g). Note that when c increases the stability of the stable focus, and after FITZBURGH NAGUMO POLYNOMIAL DIFFERENTIAL SYSTEMS 9 Figure 4. The phase portrait with two finite saddles and a finite anti- saddle, i.e. either a center, or a focus, or a node. of the stable node, increases and consequently the ω-limit of the separatrices s1 and s2 is always the stable singular point (r2 , 0). Acknowledgements The first author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is supported through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020. References [1] J.C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe, Geometric configurations of singulari- ties of planar polynomial differential systems. A global classification in the quadratic case , Birkhäuser, 2020 [2] C. Chicone and T. Jinghuang, On General Properties of Quadratic Systems, The American Mathematical Monthly, 89, No. 3 (1982), 167–178. [3] W.A. Coppel, A Survey of Quadratic Systems, J. Differential Equations 2 (1966), 293–304. [4] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer, 2006. [5] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membranes, J. Biophys 1 (1961), 445–466. [6] S.P. Hastings, Single and multiple pulse waves for the FitzHugh-Nagumo equations, SIAM J. Appl. Math. 42 (1982), 247–260. [7] C. Jones, Stability of the travelling wave solution of the FitzHugh–Nagumo system, Trans. Amer. Math. Soc. 286 (1984), 431–469. [8] C. Jones, N. Kopell, and R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, Patterns and dynamics in reactive media (Minneapolis, MN, 1989), 101– 115, IMA Vol. Math. Appl., 37, Springer, New York, 1991. [9] R.B. Langer, Existence and uniqueness of pulse solutions to the FitzHugh-Nagumo equations, Thesis (Ph.D.)–Northeastern University. 1980. 51 pp. [10] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axons, Proc. IRL 50 (1960), 2061–2070. [11] L.M. Perko, Rotated vector fields, J. Differential Equations 103 (1993), 127–145. [12] J.W. Reyn, Phase portraits of planar quadratic systems, Mathematics and its Applications, 583, Springer, 2007. [13] Ye Yanqian el al., Theory of limit cycles, Trans. of Mathematical Monographs 66, Amer. Math. Soc., Providence, RI, 2 edition, 1984. [14] Z. Zhang et al., Qualitative theory of differential equations, Translations of Mathematical Monographs, Vol. 101, Amer. Math. Soc., 1992. 10 J. LLIBRE AND C. VALLS 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 2 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: cvalls@math.tecnico.ulisboa.pt