PERIODIC SOLUTIONS FOR A CLASS OF SECOND ORDER ODES WITH A NAGUMO CUBIC TYPE NONLINEARITY

. We prove the existence of multiple periodic solutions as well as the presence of complex proﬁles (for a certain range of the parameters) for the steady–state solutions of a class of reaction–diﬀusion equations with a FitzHugh–Nagumo cubic type nonlinearity. An application is given to a second order ODE related to a myelinated nerve axon model.

1. Introduction and main results.

Motivation about the model investigated.
The study of the so called Nagumo (or FitzHugh-Nagumo) type equations has proven to be relevant not only for its significance from the point of view of the investigations of nerve fiber models but also from the theoretical point of view, since the peculiar nonlinearity which appears in these equations, as well as the various different features shown by the solutions, have stimulated the development of new theoretical tools from nonlinear functional analysis and the theory of dynamical systems. Usually, according to [18], by a Nagumo equation, we mean an autonomous system of the form where f is a cubic nonlinearity like In some works the condition 1 0 f (s) ds > 0 is added. This corresponds to the case 0 < a < 1 2 for a function f as in (2). Nonlinear ODEs or PDEs in which a Nagumo type cubic function plays a central role in the equation arise not only in mathematical models for neurobiology [11,12] but 4046 CHIARA ZANINI AND FABIO ZANOLIN they appear in population genetics, chemical reaction theory, combustion theory and in a wide range of physical systems where the presence of excitable media are taken into account (see, for instance, the Introduction and the References in [16,22]). Generalizations of these equations to PDEs systems of the form have been considered by various authors as well (see [8] and the references therein). The search of the steady states or the travelling wave solutions for equation (1) leads to the study of some autonomous systems (corresponding, typically, to some third order ODEs). Some authors, however, with the aim of developing suitable variants of the FitzHugh-Nagumo model, have modified the original system, in order to take into account some special physical (or physiological) features of the model under consideration. With this respect, we recall the contributions by Grindrod and Sleeman [9] for the study of a myelinated nerve axon and by Chen and Bell [7] for the case of a nerve fiber model with spines (see also [4]). In both of the above recalled models, the equation for the stationary solutions is a nonautonomous second order nonlinear ODE with periodic coefficients which are stepwise functions. The associated dynamics in the phase-plane is that of two superimposed first order autonomous systems which switch from one to the other in a periodic fashion. Both the models present a typical threshold behavior. In [9,7] the authors investigated the range of the parameters for which the trivial solution is the only periodic one and also proved the stability of such equilibrium solutions. Reaction-diffusion equations in a one-dimensional non-homogeneous environment (medium) have been investigated by several authors in the last decade. See, for instance, [2,3,16,22] and the references therein. In the above quoted papers, a typical model equation takes the form where the function g(u, x) is defined piecewise as follows: . , x 2n < x < +∞ g 0 (u, x), otherwise.
In our study, we consider a medium having an inhomogeneous structure which repeats periodically. A natural problem for this kind of models is the search of stationary periodic solutions. Such kind of researches have raised some interest and have been performed not only in the case of neural models, but also in different areas of investigation, for instance, in connection with problems in nonlinear optics dealing with nonlinear Schrödinger equations with periodic inhomogeneous terms [5,14]. Periodic structures have been also considered in [6,19,21] in a different context, namely, the study of front propagation.
In some recent papers [23,24] we studied the existence and multiplicity of nontrivial periodic steady states for the Chen and Bell equation where g > 0 is a constant such that 1/g represents a Ohmic resistance, f (v) is a Nagumo type nonlinearity and n(x) is a periodic stepwise positive coefficient. We subsequently also proved the presence of complex (i.e. chaotic-like) solutions for the same model [25], by assuming (for simplicity) f negative and convex on ]0, a[ and positive and concave on ]a, 1[ , with  (2), as well as to more general kind of functions f (v) whose graph is a \/\-shaped curve satisfying 1 0 f (s) ds > 0, as in Figure 1.
In this article we consider as a sample model the one discussed by Grindrod and Sleeman in [9]. The analysis of such model "inspired" some arguments in the proofs of Chen and Bell's paper [7, p. 394]. Grindrod and Sleeman in [9] studied the case of a myelinated nerve axon, thought as an infinitely long cylindrical membrane, covered by a sheath of lipoproteins (the myelin) which insulates the axon from the external ionic fluid. It is known that gaps in the sheath (named as nodes of Ranvier) occur at evenly-spaced intervals. This is taken into account in Grindrod and Sleeman's model. In fact, the authors consider the resulting dynamics by superimposing two different equations (one for the myelinated part of the fiber, the other one for the node of Ranvier). The search of the stationary solutions leads to the following system.
where θ, µ, L, R > 0 are given constants. The function v(x) = u(x, t) = constant with respect to t in Grindrod and Sleeman's model represents the transmembrane potential, that is the difference between the longitudinal axoplasmic and the external fluid potentials. The nonlinearity f is a sufficiently smooth function having a \/\-shaped graph. For instance, the choice of a cubic nonlinearity, as the function defined in (2) and with 0 < a < 1 2 , is considered as appropriate (see [9, p. 121]).
In the present work we can relax (at a certain extent) such kind of assumptions. As a general hypothesis for f we suppose (H 1 ) f : R → R is locally Lipschitz, with f (0) = 0 and such that f (s) > 0, for s < 0 and f (s) < 0, for s > 1.
In some results we will add more specific hypotheses which require a Nagumo type shape for the graph of f. In such a case we will replace condition (H 1 ) with the following and, moreover, f (s) < 0 for 0 < s < a and s > 1, f (s) > 0 for s < 0 and a < s < 1. (6) We also introduce the primitive With respect to the myelinated nerve axon model in [9], the parameter θ ∈ ]0, L[ (the case biologically relevant requires θ << L) represents the width of each of the Ranvier nodes, which are assumed to be periodically distributed along the nerve fiber and located at distance L − θ apart. By a scaling, one could assume L = 1 like in Grindrod and Sleeman's model [9, p. 123]. In [9] the authors also assumed µ = 1. For technical reasons, we prefer, however, to let the parameters µ and L free. Like in [9] we are interested only in those stationary states v(x) which satisfy the condition since it has been proved in [9] that [0, 1] is a positively invariant set for the solutions of the associated reaction-diffusion equation. In particular, we shall focus our attention to the search of periodic solutions as well as to the detection of solutions presenting some kind of more complex structure which, in any case, will have range in [0, 1].
With the aim of analyzing (4) as a single equation, we introduce two auxiliary coefficients α(x) and β(x) and consider the new equation For the weights α, β we assume (H 2 ) α, β : R → R + := [0, +∞) are L-periodic measurable functions, with α, β ∈ L 1 ([0, L]), such that We recover system (4) from (7), for the particular choice Our goal, however, is to study equation (7), possibly for a more general choice of α and β. We observe that equation (7) is general enough to include other Nagumo type equations considered in the literature. For instance the equation (3) from the Chen and Bell model [7] turns out to be a special case of (7) with the choice 1.2. A dynamical system approach. In our proofs, we follow a dynamical system approach and therefore we consider the spatial variable x as a time. With the position x ←→ t, equation (7) becomes where the dot denotes the differentiation with respect to t. From this perspective, the number L, representing the basic length structure pattern of the fiber which repeats periodically, from now on, will be indicated as the new constant T. In particular, in condition (H 2 ) , as well as in (4), the substitution L ←→ T is tacitly assumed. Dealing with (7) we always suppose f to be a locally Lipschitz function and α, β : . We also write equation (10) in the phase-plane (q, p) = (v,v), as the first order system The fundamental theory of ODEs ensures that, for every t 0 ∈ R and z 0 = (q 0 , p 0 ) ∈ R 2 , there exists a unique (noncontinuable) solution ζ(·) = ζ(·; t 0 , z 0 ) of (11) satisfying the initial condition ζ(t 0 ) = z 0 . When t 0 = 0, we usually take the simplified notation ζ(t, z 0 ) := ζ(t; 0, z 0 ). According to a classical approach [15], the search of periodic (harmonic and subharmonic) solutions for system (11) can be performed by looking for the fixed points and periodic points of the Poincaré's map where dom(Ψ) is an open set. By the continuous dependence of the solutions with respect to the initial data, Ψ turns out to be a homeomorphism of its domain onto its image. Since we are looking for solutions v(·) to (10) such that we notice that such solutions are the same also for any other equation where the function f (s) coincides with the given one in [0, 1], but is possibly different elsewhere.
To make this claim more precise, we state the following lemma which also improves a similar result in [25, Lemma A.1]. Lemma 1.1. Let α, β satisfy (H 2 ) and suppose that F : R → R is a locally Lipschitz function such that F(0) = 0 and F(s) > 0 for s < 0, F(s) < 0 for s > 1.
Proof. The result is an easy variant of the maximum principle. We give the details for the reader convenience. Equation (13) where h : R × R → R, is a Carathéodory function, that is, h(·, s) is measurable for all s ∈ R, h(t, ·) is continuous for almost every t ∈ R and, for every r > 0 and and, for almost every t ∈ R, Let v(·) be a solution of (14) and using the first sign condition in (16) With a similar argument we can prove, from the second sign condition in (16) This gives the first assertion of the Lemma. Assume now that v(·) : R → R is a periodic solution (14) Using the sign condition (16) and choosing, respectively, c = 0 and c = 1, we find that v < 0 and v > 1. Hence, there existt,t, We notice that until now, only the sign condition (16) was assumed.
To conclude the proof, consider again a solution v(·) of (14) defined on [t 0 , t 1 ] such that v(t 0 ), v(t 1 ) > 0. From the first part of the Lemma, we already know we also havev(t * ) = 0 and hence, by (15) and the uniqueness of the solutions for the initial value problems we obtain v = 0 on [t 0 , t 1 ], a contradiction.
with α, β satisfying (H 2 ). (a 1 ) Every initial value problem associated to (17) has a unique solution which is defined for every t ∈ R. (a 2 ) The sets of periodic solutions (of any period) of (10) and (17) coincide. (a 3 ) Let v(·) be a solution of (17) and assume that there exists a two-sided se- is a solution of (10). In view of the above properties, in the sequel, when dealing with equation (10), besides condition (H 1 ) we also assume (which implies that all the solutions of (10) are defined for every t ∈ R) and lim inf Such extra hypotheses, which are satisfied by f ε , are harmless with respect to the search of periodic solutions or solutions v(·) with v(t i ) ∈ [0, 1] for a sequence t i which is unbounded from above and from below (see (a 2 ) and (a 3 ), respectively). Note that, according to (a 1 ), when we replace f with f ε we can assume that all the solutions of system (11) are globally defined and thus the associated Poincaré's map turns out to be a global homeomorphism of the plane onto itself. After these preliminary remarks, we are now in position to start the study of equation (10). More precisely, we are interested in finding nontrivial periodic solutions, as well as we want to prove the existence of solutions which have some kind of complex behavior. With this respect, our paper is organized as follows. In Section 2 we prove a result of existence and multiplicity of T -periodic solutions under general assumptions on the coefficients. Our main results (Proposition 1 and Proposition 2) when applied to the particular case of Grindrod and Sleeman's model (4) require a suitable lower bound for θµ. This is consistent with the existence of only the trivial solution proved in [9] for µ = 1 and θ << T. In Section 3 we confine our attention to (4) which is analyzed by means of phase-plane techniques. In Theorem 3.1 our argument follows closely an analysis performed in [9, pp.124-130], with the substantial difference that in [9] the goal was that of proving the nonexistence of nontrivial solutions, whence here we are addressed to the opposite case (i.e., the existence of nontrivial solutions) for a different range of parameters. In Theorem 3.2 we consider also the problem of multiplicity of periodic solutions which are characterized by their oscillatory properties on [0, θ]. Finally, in Section 4 we prove the existence of infinitely many subharmonic solutions and chaotic behavior for (4). To this aim we apply recent results from the theory of topological horseshoes combined with some technical estimates for the time-mappings associated to (4).
2. Existence of nontrivial periodic solutions. In this section, following a variational approach like in [23], we prove the existence of nontrivial periodic solutions for (10) under general assumptions on α, β and f. First of all, we define r(t, s) := β(t)s − α(t)f (s) and We look for the existence of nontrivial critical points of the functional on the Hilbert space H 1 T . As usual, by H 1 T we mean the Sobolev space of functions u ∈ L 2 ([0, T ]) having a weak derivativeu ∈ L 2 ([0, T ]). For our purposes in the definition of weak derivative, as in [17], we assume that v is the weak derivative of T is the space of indefinitely differentiable T -periodic functions. Note that, according to this convention, every map u ∈ H 1 T is an absolutely continuous function [17, p.6-7]). The norm || · || for H 1 T is the standard one ||u|| := |u| 2 2 + |u| 2 2 1/2 associated to the inner product The functional I : and, moreover, if v(·) is a critical point of I (that is a solution of the corresponding Euler equation), thenv has a weak derivativev which is a solution of the periodic problemv  For R(t, s) := s 0 r(t, ξ) dξ, we have that which implies that the functional I is bounded from below, since where c > 0 is a suitable constant independent of u. Let (u j ) j be a Palais-Smale sequence in H 1 T , that is we suppose that I(u j ) is bounded with I (u j ) → 0. The boundedness of I(u j ) implies that there exists a constant K > 0 such that By standard properties of Sobolev spaces, we find that (u j ) j is bounded in H 1 T and therefore, there exists a point u ∈ H 1 T such that, passing if necessary to a subsequence, u j u in H 1 T and u j → u in C([0, T ], R). Hence, On the other hand, we have that Then, problem (21) has a nontrivial solution.
Proof. Consider the functional I defined in (20) where we assume that the map r(t, s), besides (r 1 ) and (r 2 ), satisfies also (r 3 ) (thanks to the previously discussed harmless modification of f outside a neighborhood of [0, 1]). By the assumption (22) we have that This implies that the minimum of I on H 1 T is achieved at a nontrivial function. Proof. We define η 0 (t) := α(t)f (0)−β(t). The assumption (8) implies that η 0 (t) < 0 for a.e. t ∈ [0, T ]. Hence, the quadratic form is positive definite on H 1 T . Using the fact that we find that u = 0 is a strict local minimum for the functional I. This implies that there are ρ > 0 and η > 0 (and we can also take ρ < ||e|| ) such that On the other hand, by (22) We already know that I achieves its absolute minimum at some level d ≤ I(e) < 0 < η ≤ c. In this manner, we can prove the existence of at least two nontrivial critical points which are positive T -periodic solutions of (10) with range on ]0, 1].
Example 1. If we consider equation (4), with f satisfying (H 1 ), we enter in the setting of Proposition 1 and have that condition (22) is satisfied if there is a constant e ∈ ]0, 1] such that It may be worth to observe that in order to have condition (24) fulfilled, it is sufficient to assume 1 0 f (s) ds > 0 and choose a number e close to 1, provided that the quotient µRθ T − θ is sufficiently large. For µ and R fixed constants, the condition for the existence of nontrivial solutions will be satisfied for θ not too close to zero, a result which is consistent with that of Grindrod and Sleeman [9] who proved the existence of only the trivial periodic solution for θ small.
If we further suppose that f (s) is continuously differentiable in a neighborhood of s = 0 with f (0) < 0, then, according to Proposition 2, we conclude that (4)  3. Phase-plane analysis of the Grindrod -Sleeman model. In the preceding section we have considered the general equation (10). From now on, we focus our attention on a more specific equation which is strictly related to the Grindrod -Sleeman model. Accordingly, we assume condition (H * 1 ) and also If we suppose that the coefficients α, β are like in (9), we can describe the dynamics associated to (10) in the phase-plane as the superposition of the two planar systemṡ When dealing with the two autonomous systems separately, we consider the dynamical systems associated to them. Using the language of dynamical systems, for any point P ∈ R 2 , we denote by γ + (P ), γ − (P ) and γ(P ), the positive, negative semi-orbit and the orbit of the system passing through P.
We follow the orbits of the first system for t ∈ [0, θ] and those of the second one for t ∈ [θ, T ] and then repeat the switching from one system to the other by T -periodicity. Hence, the associated Poincaré's map Ψ can be represented like where Φ is the map that associates to any initial point (q(0), p(0)) ∈ R 2 the solution of (26) evaluated at the time t = θ and where Y (·) is the fundamental matrix of the linear system (27). This remark suggests, like in [9, pp.124-130], to analyze separately the phaseportraits of the two systems. The geometry associated to (26) is that of a center at (a, 0) surrounded by a homoclinic orbit at zero that we denote by Γ 0 . The phase-portrait of system (27) is that of a standard saddle with p = −q/ √ R and p = q/ √ R as stable and unstable manifolds, respectively. Both systems (26) and (27) are conservative with energies respectively. Taking a constant c with F (a) < c ≤ F (1), we define the set For c < 0, we see that Γ c is a periodic orbit surrounding (a, 0). Larger values of µ produce the effect of decreasing the fundamental period of such a periodic orbit and, correspondingly, to increase the number of turns (rotation number) around the equilibrium point (a, 0) during the time interval [0, θ]. The set Γ 0 \ {(0, 0)} is an orbit which is homoclinic to (0, 0). For 0 < c < F (1), the set Γ c is an orbit path which intersects the vertical axis q = 0 at the points (0, ± √ 2µc) and the horizontal axis p = 0 at a point (q c , 0) with a < q c < 1 and F (q c ) = c. Finally, Γ c for c = F (1) consists of the equilibrium point (1, 0) and the intersection of its stable and unstable manifolds with the strip [0, 1] × R.
For any c with F (a) < c ≤ F (1), and (y 1 , z 1 ), (y 2 , z 2 ) ∈ Γ c with y 1 < y 2 < 1 and z 1 , z 2 ≥ 0 (z i := √ µ 2(c − F (y i )), i = 1, 2), we can compute the time σ c (y 1 , y 2 ) needed to move from (y 1 , z 1 ) to (y 2 , z 2 ) along Γ c in the upper half-plane, using the time-mapping formula where y + c is the solution of the equation F (y) = c, with a < y + c ≤ 1, so that (y + c , 0) is the intersection of the set Γ c with the abscissa. Actually, for 0 < c < F (1) we have y + c = q c (according to a notation introduced above). On the other hand, for F (a) < c < 0, the closed orbit Γ c intersects the abscissa at another point (y − c , 0) with 0 < y − c < a such that F (y − c ) = c. Notice that by (H * 1 ), we know that F (s) is strictly decreasing on [0, a] and strictly increasing on [a, 1]. Therefore, the constants y ± c are well defined. The number σ c (y 1 , y 2 ) is also equal to the time needed to move from (y 2 , −z 2 ) to (y 1 , −z 1 ) along Γ c in the lower half-plane. In particular, for every c ∈ ]F (a), 0[ , we have that τ (c) := 2σ c (y − c , y + c ) is the fundamental period of the periodic orbit Γ c . Observe that lim c→0 − τ (c) = +∞.
In fact, for any fixed µ > 0, as c tends to 0 − , the closed orbit Γ c approaches the homoclinic Γ 0 . We also notice that the period τ (c) depends upon the parameter µ. When we want to put in evidence this fact, we'll denote it as τ µ (c). From the time-mapping formula it is clear that, for any fixed c ∈ ]F (a), 0[ , it follows that τ µ (c) → 0 as µ → +∞.
Consider now the linear system (27). Taking a constant d with we consider the set The set Υ d is the part of the trajectory of (27) passing through the point and contained in the strip [0, 1] × R. For any (w, z) ∈ Υ d , with w ∈ ]y d , 1] and z = √ 2d + R −1 w 2 , we can compute the time ς(y d , w) needed to move in the upper half-plane along γ + (P d ) from P d to the point (w, z), which is the point on which the semi-orbit γ + (P d ) intersects the vertical line q = w. The same time is needed to move from (w, −z) to P d along γ − (P d ) contained in the lower half-plane. Using a time-mapping formula, we find the following integral expression for ς(y d , w).
As described in [9], the strategy to find a periodic solution is based on the following steps: We choose a point P := (w, z) in the region and consider the orbit path of (27) from P := (w, −z) to P. Such orbit path is contained in Υ d for d := E 2 (w, z) and crosses the abscissa at the point P d := (y d , 0) for y d := 2R|E 2 (w, z)|. Next, we consider the orbit path of (26) from P to P contained in Γ c for c := E 1 (w, z)/µ . If 0 ≤ c < F (1) such orbit path belongs to the strip {(q, p) : q ≥ w}, while, if c < 0, we have different possibilities to connect P to P along Γ c because, in this situation, Γ c is a closed orbit. More precisely, we can go from P to P along Γ c remaining (as above) in the strip {(q, p) : q ≥ w}, or we can start in P and stop at P after a certain number of turns around (a, 0). In [9], the authors ruled out this latter possibility and focused their attention only upon the T -periodic solutions (q(t), p(t)) of (4) which satisfy In order to obtain a T -periodic trajectory of (4) passing through P and following the procedure explained above, we shall require that T − θ 2 = ς(y d , w), y d := 2R|E 2 (w, z)| and also, simultaneously, for some nonnegative integer j. More in detail, we assume j = 0 if 0 ≤ E 1 (w, z) < µF (1) and whenever Condition A is imposed. In [9, Theorem 1.3] and for µ = 1, T = 1, Grindrod and Sleeman proved that for each fixed R > 0 there exists θ 0 > 0 such that if 0 < θ < θ 0 then there are no periodic solutions of (4) satisfying Condition A. As a complementary result, we prove the existence of periodic solutions for θ close to T.
we can write (28), as and therefore (28) is satisfied provided that the point P = (w, z) is chosen with (a relation already given in [9, Equation (1.8)]). Observe that Since we look for solutions satisfying Condition A, we take P = (w, z) with w ∈ [a, 1[ and study (29) with j = 0. First of all, we introduce the constant F (a)).
By the definition of m * it follows that, for every slope m with 0 < m < m * , the line p = mq intersects the stable manifold of (1, 0) at a point  Figure 2 for a graphical explanation). According to the above positions, the time σ c (w, y + c ) is given by the function ϕ(w, m, µ) is well defined. By the properties of the time-mapping and structure of the orbits of (26) in the phase-plane we easily prove that lim µ→+∞ ϕ * (m, µ) = lim m→0 + ϕ * (m, µ) = 0, with ϕ * (m, µ) strictly increasing as a function of m.
We take now m such that 0 < m < min{m # , m * µ } and observe that, in order to have (28) and (29) satisfied we need to find a suitable m such that Such inequality is always satisfied for m > 0 sufficiently small (that is for m in a right neighborhood of 0) as the left-hand side of (30) tends to T while the righthand side tends to 0 as m → 0 + . If we denote by m 1 the infimum of the m > 0 such that (30) does not hold, we obtain, on the contrary, that (30) will be satisfied for every m ∈ ]0, m 1 [ . To conclude the proof, we define

PERIODIC SOLUTIONS FOR SECOND ORDER ODES 4059
For every θ ∈ ]θ * 0 , T [ , we determine a line p = m(θ)q with slope m = m(θ) such that m < m * µ . For all the points (w, z) on the segment [B m Q m ] we have equation (28) satisfied. On the other hand, by the choice of m(θ) < m 1 we also have that θ > φ * (m, µ). Therefore, there exists at least one point (w, z) ∈ [B m Q m ] \ {Q m } such that (29) holds (with j = 0). This gives the existence of at least one solution for system (4) satisfying Condition A. To conclude this section, we now look for solutions of (4) which are not necessarily concave on [0, θ], in the sense that they present a certain number of oscillations in such interval. To this aim, we look for the solvability of system (28)-(29) in the case when (29) holds for some integer j ≥ 1. Accordingly, we have: and (25). Let T, R > 0, be fixed. For every θ ∈ ]0, T [ and each integer j ≥ 1, there exists µ * > 0 such that for all µ > µ * system (4) has a T -periodic solution v(·) with v(t) − a having exactly 2j-zeros in ]0, θ[ .
Proof. Along the proof, we keep unaltered all the notation introduced for the proof of Theorem 3.1. First of all, we take θ ∈ ]0, T [ and, consequently, we have the slope m(θ) ∈ ]0, m # [ fixed. By this choice, we have condition (28) satisfied for all the points (w, z) lying on the segment L θ := {(w, z) : z = m(θ)w, 0 < w < 1, z > 0}. Next, we choose so that the point B := (a, m(θ)a) lies in the interior of the region bounded by the homoclinic Γ 0 , because the energy of the level line of (26) passing through B satisfies E 1 (B) < 0. The fact that F is strictly increasing on [a, 1], guarantees that there exists a unique point Q µ := (q µ , p µ ) ∈ Γ 0 ∩ L θ for a < q µ < y + 0 < 1. Every point (w, z) on the segment [BQ µ [⊂ L θ belongs to a closed orbit Γ c of (26), for c = E 1 (w, z)/µ , having a fundamental period that we have already denoted by τ µ (c).
Suppose now that we have fixed an integer j ≥ 1 which represents the number of turns around the center (a, 0). As we already observed, τ µ (c) → 0 as µ → +∞, hence we can find µ * = µ * (θ, j) ≥ µ # (θ) such that τ µ (c) < θ/2(j + 1) for all µ > µ * , with c = E 1 (B)/µ. In this manner, we obtain On the other hand, we already know that for any µ > µ # (θ), it holds that τ µ (c) → +∞ as c → 0 − . The continuity of the maps σ and τ with respect to their arguments, allows us to claim that there exists a pointP = (w,z) = (w, m(θ)w) ∈ [B, Q µ [ for which (29) is satisfied (see also Figure 3 for a graphical explanation). For the solutionṽ(·) of (4) departing fromP at t = 0, there exists 0 < t + 1 < t 0 is strictly concave on [0, t 0 1 ] and each of the intervals [s 0 i , t 0 i+1 ], achieving its maximum for t = t + 1 and for t = t + i+1 , respectively. The same solution is strictly convex in each of the intervals [t 0 i , s 0 i ], achieving its minimum for t = t − i . For t ∈ [θ, T ], the solution satisfies the linear equation v − R −1 v = 0 and therefore it is convex.
We conclude this section with a short comment about the main differences between the results obtained here and those of Section 2. In Propositions 1-2, by means of functional analytic tools, we have obtained results for the existence of at least one (respectively two) nontrivial periodic solutions for a general equation including Grindrod and Sleeman model as a special case. On the contrary, in Theorems 3.1-3.2, by means of a more direct phase-plane analysis approach which concerns only the case of system (4), we are able to achieve more information about the oscillatory behavior of the solutions and also to improve the multiplicity result (at least for certain values of the parameters).
Theorem 3.2 suggests the fact that the dynamical properties of the solutions become more and more complex as the parameter µ increases. The presence of some kind of chaotic behaviour is investigated in the next section, with the aid of some topological tools.

4.
Solutions with a complex behavior. Throughout this section we continue to use, without any change, the notation of Section 3.  (25) and let T, R and θ ∈ ]0, T [ be fixed. Then there is a constant ω * > 0 such that, for every µ > ω * , there exists a compact invariant set Λ ⊆ ]a, 1[× ]0, +∞) for which Ψ| Λ is semiconjugate via a continuous surjection h to a two-sided Bernoulli shift on two symbols. Moreover, for each periodic sequence of two symbols (i k ) k∈Z there is a point z ∈ Λ which is periodic and such that h(z) = (i k ) k∈Z .
We remark that the assertion of Theorem 4.1 about the semiconjugation of Ψ| Λ has the following interpretation in terms of the solutions to (4): Consider a twosided sequence of two symbols ξ := (i k ) k∈Z with i k ∈ {1, 2}, ∀ k ∈ Z. Then, there is at least one solution v ξ = v ξ (x) of (4) which is defined for every x ∈ R and satisfies 0 < v ξ (x) < 1, ∀ x ∈ R as well as (v ξ (0), d dx v ξ (0)) ∈ Λ. Moreover, the symbol i k = 1 means that v ξ (x) has precisely two strict maximum points separated by one strict minimum point along the time interval [(k − 1)T, (k − 1)T + θ], while, the symbol i k = 2 means that v ξ (x) has precisely three strict maximum points separated by two strict minimum points along the time interval and v ξ (kT ) > 0. If the sequence (i k ) k∈Z is periodic, that is i k = i k+ for some ≥ 1, then we can take v ξ (·) as a T -periodic solution as well. A variant of Theorem 4.1 is the following. be fixed. Then there exists a constant θ * = θ * (µ) > 0 such that for every θ > θ * and T = θ + δ, the same conclusion of Theorem 4.1 holds.
In [9] the authors focused their attention to the case of a "healthy" nerve fiber with θ << T and proved the nonexistence of nontrivial periodic solutions.Theorem 4.2 is, in some sense, complementary as we give more relevance to the nonlinear part of the equation which gives account of the unmyelinated part of the fiber.
We present only the proof of Theorem 4.1. A similar argument can be adapted to get a proof of Theorem 4.2. For the full details how to apply the abstract results about chaotic-like dynamics to our setting, we refer to [25] where the Chen and Bell model (instead of the Grindrod-Sleeman one) was discussed.
Proof. By (H * 1 ), as we already observed, we know that F is negative and strictly decreasing on ]0, a] and strictly increasing on [a, 1]. Hence, (25) implies that there exists a unique point in ]a, 1[ where F vanishes. Let b ∈ ]a, 1[ be such that Consider for a moment the trajectories of the subsystem (27) and we choose We also fix w − ∈ ]0, a[ such that  The set Γ c0 is a closed curve contained in the region bounded by the homoclinic which is strictly star-shaped around (a, 0). It represents a periodic orbit of system (26) with fundamental period Γ c0 intersects the vertical line q = a at the points (a, ± √ µ 2(F (w 0 − ) − F (a)) ) . For our geometry, we need to have the intersection of Γ c0 with the positive vertical half-line through (a, 0) above the intersection of the same vertical line with the unstable manifold p = q √ R of (27). Accordingly, we take µ > ω 1 := a 2 2R(F (w 0 − ) − F (a)) .