Dynamics of the FitzHugh–Nagumo system having invariant algebraic surfaces

In this paper, we study the dynamics of the FitzHugh–Nagumo system x˙=z,y˙=bx-dy,z˙=xx-1x-a+y+cz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=z,\;\dot{y}=b\left( x-dy\right) ,\;\dot{z}=x\left( x-1\right) \left( x-a\right) +y+cz$$\end{document} having invariant algebraic surfaces. This system has four different types of invariant algebraic surfaces. The dynamics of the FitzHugh–Nagumo system having two of these classes of invariant algebraic surfaces have been characterized in Valls (J Nonlinear Math Phys 26:569–578, 2019). Using the quasi-homogeneous directional blow-up and the Poincaré compactification, we describe the dynamics of the FitzHugh–Nagumo system having the two remaining classes of invariant algebraic surfaces. Moreover, for these FitzHugh–Nagumo systems we prove that they do not have limit cycles.


Introduction
The FitzHugh-Nagumo system is given by the partial differential system where f (u) = u (u − 1) (u − a) and 0 < a < 1/2, ε > 0, γ > 0 are parameters. We say that a bounded solution (u, v) (x, t) of the FitzHugh-Nagumo system (1) with x, t ∈ R is a traveling wave if (u, v) (x, t) = (u, v) (ξ), where ξ = x+ct and c is the constant denoting the wave speed. Substituting u = u (ξ) , v = v (ξ) into (1), we obtain the ordinary differential systeṁ x = z = P (x, y, z) , Here, the dot denotes derivative with respect to ξ, x = u, y = v, z =u, b = ε/c and d = γ; see for more details [11]. The FitzHugh-Nagumo system (1) is a classical differential system introduced independently by FitzHugh [8] and Nagumo et al. [19]. It is an important model for describing the excitation of neural membranes and the propagation of nerve impulses along an axon. Besides its biological interest, the FitzHugh-Nagumo system has gained wide investigation from the mathematical point of view, such as the existence, uniqueness and stability of its traveling wave solutions; see, for instance, [2,9,[12][13][14]19], etc.
In recent years, the FitzHugh-Nagumo system (2) has been investigated from the points of view of its dynamics and integrability. The analytical integrability of the FitzHugh-Nagumo system (2) has been studied by Llibre and Valls in [17]. The Liouvillian integrability of the planar FitzHugh-Nagumo  Table 2.
Darboux invariants and first integrals of system (2) Parameters Darboux invariants The paper is organized as follows. We present some preliminary results in Sect. 2. The proofs of Theorems 3 and 4 will be given in Sects. 3 and 4, respectively.

Preliminary results
In this section, we recall some results that we shall need for proving our theorems.
Lemma 5. Let f (x, y, z) = 0 be an invariant algebraic surface of the differential system (2) with the constant cofactor k = 0, and let I = f (x, y, z)e −kt be its associated Darboux invariant. Assume that φ(t) = (x(t), y(t), z(t)) is a solution of system (2) not contained in the invariant surface f (x, y, z) = 0.
(a) If k > 0, the ω-limit of φ(t) inside the Poincaré ball B 3 is contained in the sphere of Poincaré S 2 intersection with the closure of the surface f (x, y, z) = 0, and the α-limit of φ(t) inside the Poincaré ball B 3 is contained in the closure of the surface f (x, y, z) = 0. (b) If k < 0, then statement (a) holds interchanging the ω-limit by the α-limit.
The proof of Lemma 5 is given in Proposition 5 of [16]. In fact, there it is proved for polynomial differential systems in R 2 , but that proof works for polynomial differential systems in R n . To describe the dynamics of the FitzHugh-Nagumo system (2) having an invariant algebraic surface, by Lemma 5, we only need to investigate it on the invariant surface.
A proof of Lemma 6 can be found in Lemma 2.1 of [15]. For studying the local phase portrait of the singular points of a two-dimensional differential systems whose linear part is identically zero, we shall use the quasi-homogeneous directional blow-up (or (α, β)blow-up) technique; see Chapter 3 of [7] or [1,6] for more details. The quasi-homogeneous directional blow-up coordinate change can be written as where (α, β) ∈ N + × N + . Using the Newton diagram, we can determine the values of α and β; see [3,4] for more details.
A phase portrait of a differential system defined in an open set U of R n is the decomposition of U as the union of all the orbits of the differential system.
We say that two phase portraits one defined on the open set U and the other defined in the open set V are topologically equivalent if there exists a homeomorphism h : U → V which send the orbits on U onto the orbits of V , preserving or reversing the orientation of all the orbits.
The separatrices of a differential system or vector field on a surface S are the singular points, the limit cycles and the separatrices of all its hyperbolic sectors. The set Σ of all separatrices of a vector field on S is a closed set. The open connected components of S \ Σ are called canonical regions. A separatrix configuration Σ * of a vector field on S is the union of Σ with one orbit in each canonical region.
Let Σ * 1 and Σ * 2 be two separatrices configurations of two vector fields on the surface S. We say that Σ * 1 and Σ * 2 are topologically equivalent if there exists a homeomorphism h : Σ * 1 → Σ * 2 which send the orbits on Σ * 1 onto the orbits of Σ * 2 , preserving or reversing the orientation of all the orbits. Neumann provided the following theorem in [20]. This theorem shows that the phase portrait of a differential system on a surface is determined by its separatrix configurations if all the singular points of the differential system are isolated.
The following result provides necessary and sufficient conditions in order that all the roots of a polynomial g (z) ∈ R[z] have negative real parts; see p. 231 of [10].
Theorem 8. (Routh-Hurwitz Criterion) All roots of the real polynomial g (z) = a 0 z n + a 1 z n−1 + · · · + a n−1 z + a n (a 0 > 0) have negative real parts if and only if is the Hurwitz determinant of order i (i = 1, 2, . . . , n).

Proof of Theorem 3
Poincaré introduced his compactification for polynomial vector fields in R 2 . For the extension of the Poincaré compactification to polynomial vector fields in R n , see [5]. Using the results of [5] for the Poincaré compactification in R 3 , we get that in the local chart U 1 the Poincaré compactified system (2) becomesu On the invariant plane w = 0 (the infinity in U 1 ), system (6) reduces tȯ This system has no singular points. In the local chart U 2 , the Poincaré compactified system (2) writeṡ On the invariant plane w = 0 (again the infinity in U 2 ), system (8) becomeṡ The singular points of system (9) fill up the straight line u = 0. Once we have studied the singular points on the local charts U 1 and U 2 , the unique additional singular point at infinity can be the origin of the local chart U 3 . Of course, at infinity all the diametrically points with respect to the origin of B 3 of the singular points of the charts U k for k = 1, 2, 3 are also infinite singular points.
In the local chart U 3 , system (2) is given bẏ So, the origin of U 3 is also a singular point. In summary, taking into account the definition of the local charts U k for k = 1, 2, 3 and their symmetric charts V k for k = 1, 2, 3 (see [5], or in particular [15]) all the infinite singular points of the FitzHugh-Nagumo system (2) fill up the boundary of the plane x = 0 at infinity in the Poincaré sphere.
This and the flows on the local charts U 1 and V 1 complete the proof of Theorem 3.
Applying (1, 1)-type quasi-homogeneous directional blow-up, we get that the origin of system (19) has the local phase portrait given in Fig. 3.
In summary on the invariant surface f 3 = 0, we have the phase portrait in the Poincaré disc of system (11) in Fig. 4.
The singular points of system (21) are characterized by equation (22). The parameters a, b, c, d satisfy Table 1). Equation (22) can be reduced to The solutions of this equation are .
Let e 0 = (0, 0), e 1 = (dy 1 , y 1 ) and e 2 = (dy 2 , y 2 ), that is, Doing the adequate computations, we have Consider the subsystems of system (21) From equation (23), system (24) has two finite singular points e 0 and e 1 . The Jacobian matrix of system (24) is Since det (J (e 0 )) = −(a + 1) 2 /9 < 0, e 0 is a saddle. The Jacobian matrix J (x, y) at the singular point e 1 is not defined at e 1 because of η (e 1 ) = 0. This means that system (24) is not analytic at e 1 . In order to get local phase portrait of system (24) at e 1 , we go back to investigate system (2). From Lemma 10, it follows that the restriction of the singular point E 1 to the surface f 4 = 0 becomes the singular point e 1 of system (24), which is repeller.
This completes the proof of Theorem 4.