Canard trajectories in 3D piecewise linear systems

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(1)DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 33, Number 10, October 2013. doi:10.3934/dcds.2013.33.4595 pp. 4595–4611. CANARD TRAJECTORIES IN 3D PIECEWISE LINEAR SYSTEMS. Rafel Prohens and Antonio E. Teruel Dep. Ciències Matemàtiques i Informàtica Universitat de les Illes Balears 07122 Palma de Mallorca, Illes Balears, Spain. (Communicated by Hinke Osinga) Abstract. We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold Sε . As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle.. 1. Introduction and main results. Singularly perturbed systems of ordinary differential equations are written in standard form as du dv u̇ = = g(u, v, ε), εv̇ = ε = f (u, v, ε), (1) dτ dτ where (u, v) ∈ Rs × Rq are the state variables, f and g are sufficiently smooth functions and 0 < ε ≪ 1 is a small parameter. From the expression above, the coordinates of u are called slow variables, while the coordinates of v are called fast variables. The variable τ is referred to as the slow time. Changing the time τ to the fast time t = τ /ε, system (1) is written as dv du = εg(u, v, ε), v′ = = f (u, v, ε). (2) u′ = dt dt Systems (1) and (2) are differentiably equivalent and their phase portraits are the same. It can be understood that the dynamics of both systems exhibit an slow-fast explicit splitting. In this setting, system (1) and (2) are called slow-fast systems. Usually, system (1) is referred to as the slow system whereas system (2) is called the fast system. Fenichel’s geometric theory [13] allows us to analyse the dynamics of the perturbed system (1) by combining the behaviour of the singular orbits, corresponding to the limiting cases given by ε = 0. In particular, by setting ε = 0 in (1) and in (2), we get respectively the reduced problem u̇ = g(u, v, 0),. 0 = f (u, v, 0),. (3). 2010 Mathematics Subject Classification. Primary: 34E15; Secondary: 34E17, 37G05, 34D15. Key words and phrases. Singular perturbation, slow–fast system, canard solutions, piecewise linear differential systems, slow manifold. R. Prohens and A. E. Teruel are supported by a MCYT/FEDER grant number MTM201122751.. 1.

(2) 2. RAFEL PROHENS AND ANTONIO E. TERUEL. and the layer problem u′ = 0,. v′ = f (u, v, 0).. (4). The reduced problem is an s-dimensional vector field defined on the critical manifold S = {(u, v) ∈ Rs+q | f (u, v, 0) = 0}, which is assumed to be an s-dimensional manifold. Regarding to the layer problem, the critical manifold S is fulfilled by singular points. A singular point (u0 , v0 ) ∈ S is said to be normally hyperbolic if the eigenvalues of the Jacobian matrix Dv f (u0 , v0 , 0) have nonzero real part. Consider S0 ⊂ S a compact set such that every point in S0 is a normally hyperbolic singular point. From Fenichel’s Theorems [13], the submanifold S0 persists as a locally invariant slow manifold, Sε , of the perturbed system (1) for every small enough ε. Moreover, the restriction of the flow of the perturbed system (1) to the slow manifold Sε is a small smooth perturbation of the flow of the reduced problem (3). Fenichel also proved that there exist a stable and an unstable invariant foliation with base Sε with the dynamics along each foliation being a small smooth perturbation of the flow of the layer problem. See also Jones, [18], for a survey of a geometric approach to singular perturbation theory. Roughly speaking, orbits of the perturbed system (1) are composed by slow and fast segments. The former ones close to the flow of the reduced problem while the latter ones are close to the flow of the layer problem. A general question is; what does remain of this dynamic behaviour when normal hyperbolicity is lost? e.g., at the points (u0 , v0 ) ∈ S in which the critical manifold is folded, that is, in which the determinant of the Jacobian matrix Dv f (u0 , v0 , 0) is equal to zero. Several articles have addressed this subject and different tools and approaches have been used. For instance, we refer the reader to the works of Benoı̂t et al., [2], Dumortier and Roussarie, [11], Krupa and Szmolyan, [19, 20], and Desroches and Jeffrey, [7, 8]. Related to the loss of normal hyperbolicity is the appearance of relaxation oscillation and canard orbits. A canard orbit is a solution of the singularly perturbed system following an attracting branch of the slow manifold, Sε , passing close to a non-normally hyperbolic point of the critical manifold S and then following a repelling branch of the slow manifold. The analysis of canard orbits can be achieved by the study of the linearized system in a neighbourhood of the so-called folded singular points [10, 26, 27, 28]. Since the systems that we deal with are piecewise linear, we would emphasize that the presence of folded points is not necessary here to ensure the appearance of canard segments for 0 < ε ≪ 1. All needed information for such analysis is obtained from the eigenvalues of the involved matrices. Then, some of the solutions of the slow-fast systems consist of a mixture of long periods of small changes interspersed by short periods of sudden changes. This mixed dynamic behaviour appears quite naturally in many applications. We refer the reader to the introduction of the works [6, 10] for some additional references. In particular, in neuroscience this phenomenon can be found related to some models of neuronal activity, see [12] for instance. One of these models reproduces accurately the bursting activity of spiking neurons. For a detailed exposition on this subject see [16, §9] and the references therein. Singularly perturbed three dimensional differential systems (1) where the slow and fast dynamics have dimension s = 1 and q = 2, respectively, often appear in applications. See, for instance, the Hindmarsh-Rose model of bursting neurons [15], the three dimensional Volterra-Gause model of predator-prey model type [14], and [21] for a physical model. See also the works [24, 25]..

(3) CANARDS IN 3D PWLS. 3. The model of the self-coupled FitzHugh-Nagumo system, [9], the 3D HodgkinHuxley model, [23, 27] and the stellate cell model [28], for instance, are applications of singularly perturbed three dimensional systems where the slow dynamics is two dimensional whereas the fast dynamic is one dimensional, i.e. s = 2 and q = 1. It is worth to observe that most results assume smoothness on the critical manifold S. A question that arises in this setting is; when smoothness is no longer present, what does remain from previous dynamic behaviour? Under suitable assumptions, in [22] the authors prove the existence of canard cycles in singularly perturbed piecewise differential systems with s = 2 and q = 1. This fact suggests that canards are not exclusively a differential phenomenon, but rather a geometric one. In this paper we consider singularly perturbed 3–dimensional piecewise linear differential systems. We use this approach because, there are many works in which versions of piecewise linear differential systems are able to reproduce the dynamic behaviour exhibited by general nonlinear systems. For example, a piecewise linear version of the Michelson system reproduces global dynamic behaviours, [3, 4], as well as bifurcations, [5], that are characteristic of the Michelson system. In particular, we deal with the singularly perturbed piecewise linear differential system  ′  u1 = ε(a11 u1 + a12 u2 + a13 v + b1 ), u′ = ε(a21 u1 + a22 u2 + a23 v + b2 ), (5)  ′2 v = u1 + |v|,. where 0 < ε ≪ 1. The flow of the system (5) is formed by the composition of two linear flows, each defined into a half–space, {v ≥ 0} or {v ≤ 0}. In spite of the fact that the vector field is not differentiable, the flow defined by (5) is smooth, even when the orbits cross the common boundary {v = 0}. The critical manifold S = {(u1 , u2 , v) : u1 + |v| = 0} is made from the union of the two half–planes S + = {(u1 , u2 , v) : v ≥ 0, u1 + v = 0}, S − = {(u1 , u2 , v) : v ≤ 0, u1 − v = 0},. (6). which intersect along the fold line F = {(0, u2 , 0) : u2 ∈ R}, see Figure 1(a). As we see in Section 2, points in the manifold S, except those contained in the fold line F , are normally hyperbolic singular points of the layer problem. Since the vector field defined by (5) is smooth in the half–spaces {v > 0} and {v < 0}, indeed it is linear, the Fenichel’s theory applies to this system in each one of the previous open regions. Therefore, under the flow of system (5), the open half–planes S + ∩ {v > 0} and S − ∩ {v < 0} persist as locally invariant manifolds. As a first result, we claim that these perturbed manifolds are in fact half–planes, denoted by Sε+ and Sε− , also defined on {v = 0}. Therefore, the slow manifold Sε is the union of these two half–planes, see Figure 1(b). In this result we also give a description of both the flow defined over Sε and the flow surrounding it. Before presenting the first result we introduce some preliminary notation. Let t1 , d1 , d2 and d3 be t1 = a11 + a22 ,. d1 = a11 a22 − a12 a21 ,. d2 = a12 a23 − a13 a22 ,. d3 = b1 a22 − b2 a12 .. (7).

(4) 4. RAFEL PROHENS AND ANTONIO E. TERUEL. S. S+. v. Sε. v. Sε+. u2. u2. F. S−. u1. u1. Sε− (b). (a). Figure 1. (a) The critical manifold S made from the union of the half–planes S + and S − , which intersect along the fold line F . (b) The slow manifold Sε for ε > 0, which is the union of the half–planes Sε+ and Sε− . Given A, B two subsets of Rn with A compact, we denote d(A, B) = max inf kx − yk , x∈A. y∈B. where k k stands for the euclidean norm. As usual, ϕ(t; p) denotes the solution of the initial value problem given by the differential system (5) through the initial condition p ∈ R3 at t = 0. 2 Theorem 1.1. For ε > 0 there exist two real values λ+ 1 = 1 + a13 ε + O(ε ) and − 2 λ1 = −1 − a13 ε + O(ε ), and two half–planes n Sε+ = (u1 , u2 , v) ∈ R3 : v ≥ 0, λ+ 1 − εa22 u1 + εa12 u2 d3 2 + 2 2 + (λ+ ) − εt λ + ε d v = −b ε + , ε 1 1 1 1 1 λ+ 1 n Sε− = (u1 , u2 , v) ∈ R3 : v ≤ 0, λ− 1 − εa22 u1 + εa12 u2 d3 2 − 2 − 2 + (λ1 ) − εt1 λ1 + ε d1 v = −b1 ε + − ε λ1. such that the manifold Sε = Sε+ ∪ Sε− satisfies the following properties: a) The manifold Sε is locally invariant under the flow of (5). b) Let S0 be a compact subset of S. If S0+ = S0 ∩ S + and S0− = S0 ∩ S − , then d(S0+ , Sε+ ) = O(ε) and d(S0− , Sε− ) = O(ε). c) If ε is small enough, the flow over Sε defined by (5) is a regular perturbation of the reduced flow over S. d) If p ∈ {v > 0}, then there exists t0 > 0 such that for each t ∈ (−t0 , t0 ) +. d(ϕ(t; p), Sε ) = d(p, Sε )eλ1 t . e) If p ∈ {v < 0}, then there exists t0 > 0 such that for each t ∈ (−t0 , t0 ) −. d(ϕ(t; p), Sε ) = d(p, Sε )eλ1 t ..

(5) CANARDS IN 3D PWLS. 5. We remark that the manifold Sε = Sε+ ∪ Sε− defined in Theorem 1.1 satisfies the same properties as those of Fenichel’s theory for smooth vector fields [13, 18]. Moreover, from Theorem 1.1(d) the locally invariant half–plane Sε+ ∩ {v > 0} is asymptotically unstable. In fact, while the orbit through a point p ∈ {v > 0} remains in the positive half-space, it moves away from Sε+ with an exponential rate. Similarly, from Theorem 1.1(e) the locally invariant half–plane Sε− ∩ {v < 0} is exponentially stable. Thus points contained in the intersection Sε+ ∩ Sε− ∩ {v = 0} correspond to orbits passing from the stable branch to the unstable branch of the slow manifold Sε , or vice versa. In the first case the orbit is called a primary canard and in the second case it is called a faux–canard. The next theorem establishes necessary and sufficient conditions on singularly perturbed piecewise linear systems (5) for the existence and location of primary canards. Theorem 1.2. Consider the system (5) where ε > 0. a) If a12 6= 0, then the set Sε+ ∩ Sε− ∩ {v = 0} contains a unique point, which is d3 b1 d3 2 + − + + − (λ + λ1 − εa22 )ε, 0 . pc = − + − ε , − a12 λ1 λ1 λ1 λ1 a12 1. If d3 > 0, then the orbit through pc is a primary canard and if d3 < 0, then the orbit through pc is a faux–canard. If d3 = 0 then no primary canards exist. b) If a12 = 0 then no primary canards exist. More concretely: if b1 6= 0 then Sε+ ∩ Sε− ∩ {v = 0} = ∅; if b1 = 0 then Sε+ ∩ Sε− ∩ {v = 0} is the invariant straight line {(0, u2 , 0) : u2 ∈ R}.. The rest of the paper is organized in three sections. In Section 2 we deal with the dynamic behaviour of the unperturbed systems associated to (5), that is, we describe both the layer system and the reduced one. In Section 3, and through the Lemmas 3.1–3.3, we analyse the singularly perturbed systems (5) for ε > 0. The proof of Theorem 1.1 is a direct consequence of these lemmas. The end of the section is devoted to the proof of Theorem 1.2. In Section 4 we show an example of a canard orbit in a piecewise linear differential system. Furthermore, by changing the vector field away from {v = 0} (just by adding two new linear pieces), we make canard orbit close to form a periodic orbit. 2. Unperturbed systems. The section is organized in two parts. In the first one we analyse the flow of the layer problem associated with the singularly perturbed system (5). In the second part we address the reduced problem. By setting ε = 0 in system (5), we get the layer system  ′  u1 = 0, u′ = 0, (8)  ′2 v = u1 + |v|.. The flow defined by this system is very simple. In fact, orbits are contained in vertical lines and singular points completely fill the piecewise linear manifold S = S + ∪ S − , defined in (6), see Figure 1(a). Since the layer vector field is locally linear, the spectrum of the Jacobian matrix at any singular point p ∈ S ∩{v 6= 0}, is {0, 0, 1} or {0, 0, −1} according to p ∈ S + ∩ {v > 0} or p ∈ S − ∩{v < 0}, respectively. Then S + ∩{v > 0} is a repelling normally hyperbolic manifold and S − ∩{v < 0} is an attracting normally hyperbolic manifold..

(6) 6. RAFEL PROHENS AND ANTONIO E. TERUEL. Singular points on the fold line F do not have a well–defined Jacobian matrix, so they are not normally hyperbolic singular points. The local flow surrounding F follows from (8) by noting that v ′ > 0 over the plane {u1 = 0}. Then, the straight line F attracts orbits in {v < 0} and repels orbits in {v > 0} (see Figure 2).. S+. v. S. u2. F. S−. u1. Figure 2. Representation of the flow of the layer equation, attracting and repelling normally hyperbolic half–planes S − and S + , and the fold line F . Now, we continue by considering the reduced system   u̇1 = a11 u1 + a12 u2 + a13 v + b1 , u̇2 = a21 u1 + a22 u2 + a23 v + b2 ,  0 = u1 + |v|,. (9). which is defined on the manifold S = S + ∪ S − . From the last equation in (9) and by taking the derivative when v 6= 0 we have v̇ = −. |v| u̇1 . v. Thus, the vector field defined by the reduced system on the submanifold S \ F is given by the piecewise linear function + F (u1 , u2 , v) if v > 0, F(u1 , u2 , v) = F− (u1 , u2 , v) if v < 0, where  a11 u1 + a12 u2 + a13 v + b1 F+ (u1 , u2 , v) =  a21 u1 + a22 u2 + a23 v + b2  −a11 u1 − a12 u2 − a13 v − b1 . and.  a11 u1 + a12 u2 + a13 v + b1 F− (u1 , u2 , v) =  a21 u1 + a22 u2 + a23 v + b2  . a11 u1 + a12 u2 + a13 v + b1 .

(7) CANARDS IN 3D PWLS. 7. The projection map π(u1 , u2 , v) = (u2 , v) induces on R2 \ {(u2 , 0) : u2 ∈ R} the planar piecewise linear differential system  u2  +  A + b+ if v > 0,   v  u̇2 = (10) v̇   u2  − −  +b if v < 0,  A v where. A+ =. . a22 −a12. A− =. . a22 a12. a23 − a21 a11 − a13. . a23 + a21 a11 + a13. b+ =. , . ,. . b− =. b2 −b1. . . . b2 b1. , (11) .. The flows defined by the systems (9) and (10) are differentially conjugated in S \ F and in R2 \ {(u2 , 0) : u2 ∈ R}, respectively. Since at both sides of the line {(u2 , 0) : u2 ∈ R} and close to it, the second coordinate of the vector field (10) takes opposite signs, system (10) can not be continuously extended to that line. The flow on {(u2 , 0) : u2 ∈ R} can be obtained by the Filippov extension of the system (10) to the boundary {v = 0} (see [17]), that is u̇2 = a22 u2 + b2 , v̇ = 0. When a22 6= 0 a singular point e = (−b2 /a22 , 0) appears in F . This singular point is called a folded singular point. Since the linearisation of the reduced system at folded singular points gives the local behaviour of the reduced flow in differential singularly perturbed systems, folded singular points play an important role in the study of canard trajectories, see [19, 20, 26]. However, for singularly perturbed piecewise linear differential systems this behaviour can be obtained from the eigenvalues of the matrices A+ and A− . Direct computations show that the eigenvalues β + and γ + of the matrix A+ , and the eigenvalues β − and γ − of the matrix A− can be written in terms of the values defined in (7) as β + + γ + = t1 − a13 , β + γ + = d1 + d2 , β − + γ − = t1 + a13 , β − γ − = d1 − d2 .. (12). 3. Perturbed system. In this section we give some lemmas to analyse the flow of the perturbed system (5). Moreover, we relate it to the flow of the layer problem (8) and to the flow of the reduced problem (9). At the end of the section we will use this relationship to prove Theorems 1.1 and 1.2. The flow of (5) is defined by the composition of the two linear flows which are associated with the differential system  +  Aε x + bε if v ≥ 0, x′ = (13)  − Aε x + bε if v ≤ 0, where x = (u1 , u2 , v)T and  εa11  εa21 A± = ε 1. εa12 εa22 0.  εa13 εa23  , ±1. .  εb1 bε =  εb2  . 0.

(8) 8. RAFEL PROHENS AND ANTONIO E. TERUEL. Hence, locally the flow of (13) can be derived from the analysis of the eigenvalues and the eigenvectors of both linear systems. Lemma 3.1. For ε > 0 the eigenvalues of the matrix A+ ε expand in power series in + + 2 + 2 ε as λ+ = 1 + a ε + O(ε ), λ = β ε + O(ε ) and λ = γ + ε + O(ε2 ), where β + and 13 1 2 3 + + γ are the eigenvalues of the matrix A in (11). Moreover, the eigenvalues of the − − 2 − 2 matrix A− ε expand in power series in ε as λ1 = −1−a13 ε+O(ε ), λ2 = β ε+O(ε ) − − 2 − − − and λ3 = γ ε + O(ε ), where β and γ are the eigenvalues of the matrix A in (11). − Proof. Let us prove the lemma for the matrix A+ ε . The result for the matrix Aε + follows in an analogous way. Since the characteristic polynomial of Aε. λ3 − (1 + εt1 )λ2 + ε(t1 − a13 + εd1 )λ − ε2 (d1 + d2 ) = 0,. (14). tends to λ3 − λ2 = 0 as ε tends to zero, we conclude that the eigenvalues of the matrix A+ ε can be expanded in power series in ε as 2 λ+ 1 = 1 + αε + O(ε ), 2 λ+ 2 = βε + O(ε ),. λ+ 3. (15). = γε + O(ε2 ),. where α is a real number and β, γ are real or complex. From equality (14) we obtain that + + λ+ 1 + λ2 + λ3 = 1 + εt1 , + + + + + λ+ 1 λ2 + λ1 λ3 + λ2 λ3 = ε(t1 − a13 + εd1 ), + + λ+ 1 λ2 λ3. (16). 2. = ε (d1 + d2 ).. Then, from (15) and (16) its follows that α = a13 , β + γ = t1 − a13 ,. (17). βγ = d1 + d2 . Note that β and γ satisfy the same equations that β + and γ + in (12). Then β = β + and γ = γ + . From Lemma 3.1, we emphasize that the spectrum of the matrix A+ ε decomposes into two parts. One consisting on the eigenvalue λ+ 1 , which is responsible for the fast dynamic in {v ≥ 0} when ε tends to zero. The other one is formed by the + eigenvalues λ+ 2 and λ3 , which tend to zero as ε tends to zero. In Lemma 3.2 we see that these eigenvalues are responsible for the slow dynamic in {v ≥ 0}. + T Let w+ be the eigenvector associated to the eigenvalue λ+ 1 of the matrix (Aε ) , − where superscript T stands for the transpose. Let w be the eigenvector associated − T to the eigenvalue λ− 1 of the matrix (Aε ) . Then it follows that + + T (w+ )T A+ ε = λ1 (w ) ,. Pε+. − − T (w− )T A− ε = λ1 (w ) . 3. + T. (A+ ε p + bε ). (18). Let us now consider the following sets = {p ∈ R : (w ) = 0} and Pε− = {p ∈ R3 : (w− )T (A− p + b ) = 0}, and their restriction to the half–spaces ε ε {v ≥ 0} and {v ≤ 0} given by Sε+ = Pε+ ∩ {v ≥ 0} and Sε− = Pε− ∩ {v ≤ 0}, respectively, and let us take Sε = Sε+ ∪ Sε− . The lemma below is satisfied..

(9) CANARDS IN 3D PWLS. 9. Lemma 3.2. a) The set Pε+ (resp. Pε− ) is an invariant plane under the flow ′ − of the linear system x′ = A+ ε x + bε (resp. x = Aε x + bε ). + − b) The half–planes Sε and Sε are locally invariant under the flow of system (13). Moreover, the points (u1 , u2 , v) ∈ Sε+ satisfy that v ≥ 0 and + 2 + 2 2 d3 (λ+ , 1 − εa22 )u1 + εa12 u2 + (λ1 ) − εt1 λ1 + ε d1 v = −εb1 + ε λ+ 1 whereas the points (u1 , u2 , v) ∈ Sε− satisfy that v ≤ 0 and − 2 − 2 d3 2 . (λ− 1 − εa22 )u1 + εa12 u2 + (λ1 ) − εt1 λ1 + ε d1 v = −εb1 + ε λ− 1. c) If S0 is a compact subset of S and we take S0+ = S0 ∩ S + and S0− = S0 ∩ S − ; then d(S0+ , Sε+ ) = O(ε) and d(S0− , Sε− ) = O(ε). Proof. We are going to prove the statements for the sets Pε+ and Sε+ . The corresponding proofs for Pε− and Sε− follow in a similar way. From the equality (18) it follows that (w+ )T bε Pε+ = p ∈ R3 : (w+ )T p = − . (19) λ+ 1 Hence Pε+ is an orthogonal plane to the vector w+ . Since for any given point p in + + Pε+ the vector field at p (i.e. A+ ε p + bε ) is orthogonal to w , we conclude that Pε is invariant under the flow of the linear system. This proves statement (a). From statement (a), the set Sε+ is a locally invariant half–plane. Now we deal with the equation satisfied by the points in Sε+ . By solving w+ from (18), we have T + 2 + 2 . w+ = λ+ 1 − εa22 , εa12 , (λ1 ) − εt1 λ1 + ε d1. Hence, statement (b) follows from (19) by taking into account that 2 (w+ )T bε = εb1 λ+ 1 − ε d3 .. Let S0 be a compact subset of S and consider the compact set S0+ = S0 ∩ S + . From statement (b) of this lemma and by considering that λ+ 1 = 1+O(ε), if |u1 |, |u2 | and |v| are bounded, then the points in the half–plane Sε+ satisfy that u1 +v = O(ε). Moreover, the points on S + satisfy that u1 + v = 0. From these facts we conclude that d(S0+ , Sε+ ) = O(ε). We note that from the former lemma, the half–plane Sε+ is locally invariant under the flow of the perturbed system (13). In particular, orbits in Sε+ remain in Sε+ until they reach the boundary of the half–plane at {v = 0}. In the next result we discuss the behaviour of the flow of the perturbed system (13) defined over the locally invariant half–planes Sε+ and Sε− and in its neighbourhood. Lemma 3.3. Let ϕε : R × R3 → R3 be the flow defined by the system (13). a) For each ε > 0 small enough, the restriction of ϕε to the locally invariant half–plane Sε+ (resp. Sε− ) is a regular perturbation of the restriction to S + (resp. S − ) of the flow defined by the reduced system (9). b) If p ∈ R3 ∩ {v > 0}, then there exists t0 > 0 such that for each t ∈ (−t0 , t0 ) +. d(ϕε (t; p), Sε+ ) = d(p, Sε+ )eλ1 t . c) If p ∈ R3 ∩ {v < 0}, then there exists t0 > 0 such that for each t ∈ (−t0 , t0 ) −. d(ϕε (t; p), Sε− ) = d(p, Sε− )eλ1 t ..

(10) 10. RAFEL PROHENS AND ANTONIO E. TERUEL. Proof. Consider the projection πε : Sε ∩ {v 6= 0} → R2 , given by πε (u1 , u2 , v) = (u2 , v). From the expression of the half–planes Sε+ and Sε− appearing in Lemma 3.2(b), it is clear that πε is a diffeomorphism for each small enough ε > 0. Moreover 2 εa12 (λ± )2 − εt1 λ± 1 + ε d1 πε−1 (u2 , v) = − ± u2 − 1 v ± λ1 − εa22 λ1 − εa22 d3 ε2 b1 ε + ± ± , u2 , v , − ± λ1 − εa22 λ1 (λ1 − εa22 ) + − where λ± 1 stands for λ1 or λ1 depending on {v > 0} or {v < 0}, respectively. Hence, πε induces on {(u2 , v) ∈ R2 : v 6= 0} the piecewise linear differential system   + u2  + c+ if v > 0, B  ε ′  v  ε u2 (20) = v′    − u2 −  + cε if v < 0,  Bε v. where. . and.  Bε± =  . ε a22 −. a21 a12 ε λ± 1 −εa22. 12 −ε λ±a−εa 1. 22. .  2 (λ± )2 −εt1 λ± 1 +ε d1 ε a23 − a21 1 λ± −εa 22  1   ± 2 2 (λ± 1 ) −εt1 λ1 +ε d1 ±1 − ± λ −εa 1. 22.  d3 a21 + ε2 λ± (λa±21−εa ε b2 − ε λ±b1−εa 22 22 1 1 1 )  .  c± ε = b1 d3 ε − λ± −εa + ε λ± (λ± −εa ) . 1. 22. 1. 1. 22. We remark that, the meaning of ±1 in the matrix Bε± is that it stands for 1 in the matrix Bε+ and for −1 in the matrix Bε− . To prove statement (a) of the lemma it is enough to show that the flow associated to system (20) is a regular perturbation of the flow defined by the system (10). First, we change the time t in system (20) by the slow time τ = εt. We note that − from the expressions of λ+ 1 and λ1 appearing in Lemma 3.1 it follows that ! 2 2 λ+ − εt1 λ+ 1 1 1 + ε d1 1− = a11 − a13 + O(ε), ε λ+ 1 − εa22 and 1 ε. −1 −. λ− 1. 2. 2 − εt1 λ− 1 + ε d1 λ− 1 − εa22. !. = a11 + a13 + O(ε).. Therefore, the matrices and the vectors defining the piecewise linear system in slow time, tend to the matrices and the vectors defining system (10) as ε tends to zero. We conclude the statement from the continuous dependence on the parameters of the solutions of the differential equations. Now we analyse the behaviour of the flow surrounding the locally invariant half– plane Sε+ . Let p be a point in R3 ∩ {v > 0} and let v1+ be the eigenvector associated + + to the eigenvalue λ+ 1 . Since v1 is not parallel to Sε , the point p can be expressed + as a sum of a point q in Sε and a point q1 in the straight line {rv1+ : r ∈ R},.

(11) CANARDS IN 3D PWLS. 11. i.e. p = q + q1 , with q1 = rv1+ . As far as ϕ(t; p) remains in {v ≥ 0} and ϕ(t; q) remains in Sε+ it can be expressed as Z t + + + + ϕ(t; p) = eAε t q + eAε (t−s) bε ds + reAε t v1+ = ϕ(t; q) + reλ1 t v1+ . 0. Hence, we conclude that. +. d(ϕ(t; p), Sε+ ) = d(p, Sε+ )eλ1 t , which ends the proof of the statement (b). Statement (c) follows in a similar way. We note that, in this case, the points p ∈ R3 ∩ {v < 0} have to be expressed as the sum p = q + rv1− , where q ∈ Sε− , r ∈ R and v1− is the eigenvector associated to the eigenvalue λ− 1. We end this section with the proofs of Theorems 1.1 and 1.2. Proof of Theorem 1.1. The statement (a) of the theorem is a straightforward consequence of Lemma 3.2(b). Theorem 1.1(b) is proved as Lemma 3.2(c). Finally, statements (c), (d) and (e) of the theorem are proved as Lemmas 3.3(a), (b) and (c), respectively. Proof of Theorem 1.2. Consider the intersection of the locally invariant half–planes Sε+ and Sε− and the plane {v = 0}. From Lemma 3.2(b), the intersection points satisfy the following system of linear equations d3 2 (λ+ 1 − εa22 )u1 + εa12 u2 = −b1 ε + + ε λ1 (21) d3 2 − (λ1 − εa22 )u1 + εa12 u2 = −b1 ε + − ε λ1 + − whose determinant is λ1 − λ1 εa12 . If a12 6= 0 then the system (21) has a unique solution which is the point pc in the statement (a) of this theorem. Moreover, the flow passes from Sε+ to Sε− , or vice versa, through this point. The direction of the flow can be obtained from the sign of the third component of the vector field on pc . Since it depends on the sign of the first component of pc , see (13), we conclude that if d3 > 0, then the orbit γpc through pc goes from Sε− to Sε+ . Since Sε− is the stable branch of the slow manifold Sε and Sε+ is the unstable branch of Sε , the orbit γpc is a canard. If d3 < 0, then the orbit goes in the opposite direction, i.e. from the unstable branch Sε+ to the stable branch Sε− . Hence γpc is a faux–canard. Finally, if d3 = 0 then pc is a singular point of the differential system. Hence, no orbit crosses through pc from one branch to the other of Sε . Therefore no primary canards exist. If a12 = 0 then the system (21), for ε small enough, has no solution unless b1 = 0. If it is the case, then d3 = 0 and the set of solutions is given by the straight line {(0, u2 , 0) : u2 ∈ R}. We are going to see that this straight line is made up with orbits of the perturbed system (13) and, hence, from the uniqueness of solutions Theorem, there is no orbit passing from one branch to the other of the slow manifold. Assume that a22 6= 0, then the straight line is made up by the critical point (0, −b2 /a22 , 0) plus the solutions (0, −b2 /a22 ± eεa22 t , 0). Otherwise, when b2 6= 0 the straight line contains the orbit {(0, εb2 t, 0) : t ∈ R}; and when b2 = 0 each point of the straight line is a critical point..

(12) 12. RAFEL PROHENS AND ANTONIO E. TERUEL. 4. Example of canard cycle. In this section we apply Theorem 1.1 and Theorem 1.2 to a particular family of singularly perturbed piecewise linear differential system, (5), given by a11 = a13 = b1 = 0, a12 < 0, a21 = a22 = a23 = 0 and b2 = 1, i.e.  ′  u1 = εa12 u2 , u′ = ε, (22)  ′2 v = u1 + |v|. We note that system (22) is a piecewise linear version of the differential system x′ = −2ε y, y ′ = ε, z ′ = x + z 2 , which is considered in [26]. The matrices of the linear problems associated to (22) are     0 εa12 0 0 εa12 0  0  0 0 0  , A− 0 0 , A+ ε = ε = 1 0 1 1 0 −1. + + − − − and their eigenvalues are λ+ 1 = 1, λ2 = λ3 = 0 and λ1 = −1, λ2 = λ3 = 0, respectively. According to Theorem 1.1, the slow manifold Sε of (22) is the union of the unstable half–plane o n Sε+ = (u1 , u2 , v) ∈ R3 : v ≥ 0, u1 + εa12 u2 + v = −ε2 a12. and the stable half–plane o n Sε− = (u1 , u2 , v) ∈ R3 : v ≤ 0, −u1 + εa12 u2 + v = ε2 a12 .. These half–planes intersect at a unique point pc = (−ε2 a12 , 0, 0). Since d3 = −a12 > 0, from Theorem 1.2, the orbit γpc through this point is a primary canard. Let ϕ(t; p) denote the flow of system (22). The expression of ϕ(t; p) can be obtained by integrating the two linear systems associated to (22) and by combining the results conveniently. Hence, the first coordinates of ϕ(t; p) can be written as ϕ1 (t; p) = p1 + εta12 p2 +. ε 2 t2 a12 , 2. ϕ2 (t; p) = p2 + εt, where p = (p1 , p2 , p3 )T . Since the plane {v = 0} separates the two half–spaces where the vector field is linear, no expression can be obtained to describe the third component of flow defined all the time. Next, we present a local expression of ϕ3 (t; p) which is defined in a neighbourhood of the initial time t = 0. Indeed, the expression of ϕ3 (t; p) depends on the sign of the third coordinate of the initial condition p3 . Moreover, when p3 = 0 this expression depends on the direction of the flow at p. This direction is upward when p1 > 0, and downward when p1 < 0 (see the third equation in (22)). Therefore, setting R+ = {p ∈ R3 : p3 > 0, or p3 = 0 and p1 > 0} and R− = {p ∈ R3 : p3 < 0, or p3 = 0 and p1 < 0}, it follows that: if p ∈ R+ , then t2 ϕ3 (t; p) = et − 1 p1 + εa12 et − 1 − t p2 + et p3 + ε2 a12 et − 1 − t − , 2 for t ∈ (t−p , tp ); and if p ∈ R− , then ϕ3 (t; p) = 1 − e. −t. . p1 + εa12 e. −t. . −t. 2. . − 1 + t p2 + e p3 − ε a12 e. −t. t2 −1+t− 2. . ,.

(13) CANARDS IN 3D PWLS. 13. for t ∈ (t−p , tp ). The endpoints of the intervals of definition t−p ≤ 0 ≤ tp correspond to the time in which the solution passes through the separation plane. Assuming that one of these values does not exist, then the corresponding endpoint is infinity. Since pc ∈ R+ , the next proposition is a direct consequence of the expression of the flow shown above. Proposition 1. The canard orbit γpc is given by t2 |t| 2 2 ϕ(t; pc ) = −ε a12 1 − , εt, −ε a12 t 1 + 2 2. for t ∈ R.. From Proposition 1 and the expression of the slow manifold Sε = Sε+ ∪ Sε− , it is easy to check that the canard orbit γpc remains in Sε+ for t > 0 and in Sε− for t < 0. In Figure 3 we represent the canard orbit γpc for the parameters a12 = −1.3 and ε = 1e − 1. 1.5. Sε+ γp c. 1. 0.5. –1.5. –1.4 –0.8 –1 –1.2 –0.6. –1 0.5. 1. 1.5. –0.5. Sε−. –1. –1.5. Figure 3. Canard orbit γpc crossing from the stable half–plane Sε− through the unstable half–plane Sε+ of the piecewise linear differential system (22). Hence, for γpc to be a periodic orbit, it is necessary that γpc leaves Sε− and Sε+ in negative and positive time, respectively. This will be achieved by adding two new linear pieces to the system (22). So that, for each arbitrary but fixed positive number η, we consider the four–pieces linear differential system  u  F (x) if v ≥ η, F o (x) if |v| ≤ η, (23) x′ =  l F (x) if v ≤ −η, where x = (u1 , u2 , v)T ,   εa12 u2 + a1 (v − η) , F u (x) =  ε − a22 (v − η) 2 u1 + η + a3 (v − η). .  εa12 u2 + a1 (v + η) , F l (x) =  ε + a22 (v + η) 2 u1 + η − a3 (v + η). a1 , a2 , a3 ∈ R, and F o (x) is the piecewise linear vector field defined by the differential system (22)..

(14) 14. RAFEL PROHENS AND ANTONIO E. TERUEL. Let ϕ̃(t; p) be the flow defined by the piecewise linear differential system (23). It is easy to conclude that ϕ̃ coincides with ϕ when we restrict ϕ̃ to the central region {(u1 , u2 , v) : |v| ≤ η}. Therefore, the slow manifold S̃ε of the system (23) is the union of the unstable branch n o S̃ε+ = (u1 , u2 , v) ∈ R3 : 0 ≤ v ≤ η, u1 + εa12 u2 + v = −ε2 a12 and the stable branch o n S̃ε− = (u1 , u2 , v) ∈ R3 : −η ≤ v ≤ 0, −u1 + εa12 u2 + v = ε2 a12 ,. which are locally invariants under the flow ϕ̃. In fact, the slow manifold has boundaries at {v = 0} and at {v = ±η} (border planes) through which the flow leaves the manifold, see Figure 4. Moreover, we emphasize that the orbit γpc is also a canard in this new setting, and its expression, given in Proposition 1, is correct while the orbit remains in the central region {|v| ≤ η}. That is, for r 2η 1 ∗ ε2 − |t| ≤ t = − 1. (24) ε a12 v C(p2 ) p+ c. {v = η}. S̃ε+. {v = 0}. qc pc. u1. u2. γp c. {v = −η} Figure 4. Representation of the canard cycle γpc , slow manifolds S˜ε ∪S̃ε and the border planes {v = η}, {v = 0} and {v = −η}, which separate the regions where the system is linear. We highlight the points of intersection of γpc with the border planes. An important thing that allows us to get a canard cycle is the fact that the system (23) is time–reversible with respect to the involution R(u1 , u2 , v) = (u1 , −u2 , −v). This means that the flow ϕ̃(t; p) is reversible in the sense that R (ϕ̃(t; p)) = ϕ̃ (−t; R(p)) . We note that the set of fixed points of involution R corresponds to the u1 –axis. Therefore, since pc is on the u1 –axis, a sufficient condition for the orbit γpc to be periodic is that it intersects the u1 –axis at a new point, which we denote by qc , see Figure 4. Let us seek now the initial conditions p = (p1 , p2 , η) on the plane {v = η} such that orbits through them, reach the u1 –axis. That is, ϕ2 (t; p) = 0 and ϕ3 (t; p) = 0,.

(15) CANARDS IN 3D PWLS. 15. for a convenient time t. Hence, from the expression of the flow ϕ in the central region 0 ≤ v ≤ η we obtain the system 0 = p2 + εt, 0 = (et − 1) p1 + εa12 (et − 1 − t) p2 + et η + ε2 a12 et − 1 − t −. t2 2. . From this, we get the relation p1 = C(p2 ), where −1 p2 p 1 − ε 2 2 − ε2 e (η + εa12 (p2 + ε)) + a12 p2 − ε a12 . C(p2 ) = 1 − e 2. .. (25). ∗ ∗ Let p+ c = ϕ̃(t , pc ), where t is the value of the time defined in (24). Hence, is the point of intersection of the canard orbit γpc with the plane {v = η}, see Figure 4. Then r r 2η 2η 2− 2− ε ε p+ = −η − εa , −ε + , η . 12 c a12 a12. p+ c. Therefore, a sufficient condition for γpc to be a periodic orbit is the existence of a + value of the time t+ c > 0 such that the orbit through pc intersect the plane {v = η} just at the graph of the function C(p2 ), see Figure 4, that is + − , (26) ϕ̃(t+ c , pc ) ∈ (C(p2 ), p2 , η) : p2 ∈ R + and ϕ̃3 (t, p+ c ) > η for t ∈ (0, tc ). Condition (26) leads us to a system of two equations. + + + ϕ̃1 (t+ c ; pc ) = C (ϕ̃2 (tc ; pc )) , + ϕ̃3 (t+ c ; pc ) = η,. (27). with seven unknowns ε, η, a12 , a1 , a2 , a3 and t+ c . Now we proceed by fixing five of these unknowns to compute the remaining ones. Although the flow is explicitly known in the half–space {v > η}, solving this systems in general is not possible analytically. One of the difficulties comes from that one of the unknowns, t+ c , appears involved in exponential and trigonometric functions. The existence and location of solutions often entails an important analytical study. This study is the key point of some references, see for example [3, 4, 5]. Since this analysis goes beyond the objective of this work, we limit ourselves to present a numerical solution of system (27). Setting ε = 0.1, a12 = −1.3, η = 1, a1 = −1 and a2 = 3.4, and solving (27) for + a3 and t+ c , we obtain the values a3 ≈ 0.583695486652 and tc ≈ 2.3372454. Figure 5 contains different views of the canard cycle γpc obtained by using the software package Dynamic Solver [1]. Pictures (a), (b) and (c), represent the projection of γpc on the planes (u2 , v), (u1 , v) and (u1 , u2 ), respectively. Pictures (d) and (e) represent a three dimensional view of the canard cycle and a graph of the variable v versus the time t, respectively. We recall that fast dynamics takes place on perturbed vertical straight lines, while the slow dynamics follows the slow manifold Sε = Sε+ ∪Sε− . In the (u1 , v) projection, one can observe the portion of the canard orbit following the fast manifold and the portion which follows the slow manifold. As shown in that drawing, the growth of the variable v during the slow phase is lost during the fast phase. We turn now to the drawing which represents the variable v versus time. As it can be observed, along a period, the time interval in which the variable v increases (the slow phase).

(16) 16. RAFEL PROHENS AND ANTONIO E. TERUEL. ◆◆. ❨. ✒ ❄ ❄ ◆◆. ✒. ✯. (a). (b) ❲❲. (c). ✒. ❲❲. ✻ (d). (e). Figure 5. Different views of the canard cycle γpc exhibited by system (23) with ε = 0.1, a12 = −1.3, η = 1, a1 = −1, a2 = 3.4 and a3 ≈ 0.583695486652. (a) Projection of γpc on the plane (u2 , v). (b) Projection of γpc on the plane (u1 , v). (c) Projection of γpc on the plane (u1 , u2 ). (d) Three dimensional view of the canard γpc . (e) Representation of the variable v versus the time. is, approximately, one tenth of the time interval in which the variable v decreases (the fast phase). This is because ε = 0.1. 4.1. Conclusions. In this paper we have presented a way to prove the existence of canard orbits in three–dimensional slow-fast systems. The approach is based in getting the explicit expression of the slow manifold. The expression of this manifold allows us to obtain the set of intersection points between their attracting and repelling branches. The calculations can be performed since the family of systems we deal with are piecewise linear differential systems. We apply the former results to prove the existence of a canard trajectory. By using numerical arguments, we show that this canard orbit closed to form a canard cycle. Acknowledgments. We thank the referees for their comments and suggestions which have been very helpful for improving our manuscript. REFERENCES [1] J. M. Aguirregabiria, “Dynamics Solver v. 1.91,” 2012, http://tp.lc.ehu.es/jma.html. [2] E. Benoı̂t, J. L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math., 32, (1981), 37–119. [3] V. Carmona, F. Fernández-Sanchez and A. E. Teruel, Existence of a reversible T-point heteroclinic cycle in a piecewise linear version of the Michelson system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1032–1048. [4] V. Carmona, F. Fernández-Sanchez, E. Garcı́a-Medina and A. E. Teruel, Existence of homoclinic connections in continuous piecewise linear systems, Chaos, 20 (2010), pp.8..

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