DYNAMICAL SYSTEMS
Volume33, Number10, October2013 pp.4595–4611
CANARD TRAJECTORIES IN 3D PIECEWISE LINEAR SYSTEMS
Rafel Prohens and Antonio E. Teruel
Dep. Ci`encies Matem`atiques i Inform`atica 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
˙ u=du
dτ =g(u,v, ε), εv˙ =εdv
dτ =f(u,v, ε), (1) 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 ofuare called slow variables, while the coordinates ofvare called fast variables. The variable τ is referred to as the slow time. Changing the time τ to the fast timet=τ /ε, system (1) is written as
u′= du
dt =εg(u,v, ε), v′= dv
dt =f(u,v, ε). (2) 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 per- turbed system (1) by combining the behaviour of the singular orbits, corresponding to the limiting cases given by ε = 0. In particular, by settingε = 0 in (1) and in (2), we get respectively the reduced problem
˙
u=g(u,v,0), 0=f(u,v,0), (3)
2010Mathematics 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 MTM2011- 22751.
1
and the layer problem
u′=0, v′ =f(u,v,0). (4) The reduced problem is ans-dimensional vector field defined on the critical man- ifoldS={(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 matrixDvf(u0,v0,0) have nonzero real part.
ConsiderS0⊂ S a compact set such that every point inS0is a normally hyper- bolic singular point. From Fenichel’s Theorems [13], the submanifoldS0persists as a locally invariant slow manifold, Sε, of the perturbed system (1) for every small enoughε. 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 prob- lem (3). Fenichel also proved that there exist a stable and an unstable invariant foliation with baseSεwith 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 matrixDvf(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 Benoˆıt 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 oscil- lation 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 manifoldS and then following a re- pelling 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 andq = 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].
The model of the self-coupled FitzHugh-Nagumo system, [9], the 3D Hodgkin- Huxley 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 man- ifold S. A question that arises in this setting is; when smoothness is no longer present, what does remain from previous dynamic behaviour? Under suitable as- sumptions, in [22] the authors prove the existence of canard cycles in singularly perturbed piecewise differential systems with s= 2 andq= 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
u′1=ε(a11u1+a12u2+a13v+b1), u′2=ε(a21u1+a22u2+a23v+b2), v′=u1+|v|,
(5)
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 lineF ={(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} andS−∩ {v <0} persist as locally invariant manifolds.
As a first result, we claim that these perturbed manifolds are in fact half–planes, denoted bySε+ andSε−, 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 overSε and the flow surrounding it. Before presenting the first result we introduce some preliminary notation. Let t1, d1, d2
andd3be
t1=a11+a22, d1=a11a22−a12a21,
d2=a12a23−a13a22, d3=b1a22−b2a12. (7)
(a)
u2
u1
S v
F
S+
S−
(b)
u2
u1
Sε v
Sε+
Sε−
Figure 1. (a) The critical manifold S made from the union of the half–planesS+ andS−,which intersect along the fold lineF.
(b) The slow manifold Sε for ε > 0, which is the union of the half–planesSε+and Sε−.
GivenA, B two subsets ofRn withAcompact, we denote d(A, B) = max
x∈A
y∈Binf kx−yk
,
where k kstands 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 conditionp∈R3 att= 0.
Theorem 1.1. For ε > 0 there exist two real values λ+1 = 1 +a13ε+O(ε2) and λ−1 =−1−a13ε+O(ε2), and two half–planes
Sε+=n
(u1, u2, v)∈R3 :v≥0, λ+1 −εa22
u1+εa12u2
+ (λ+1)2−εt1λ+1 +ε2d1
v=−b1ε+ d3
λ+1 ε2
, Sε−=n
(u1, u2, v)∈R3 :v≤0, λ−1 −εa22
u1+εa12u2
+ (λ−1)2−εt1λ−1 +ε2d1
v=−b1ε+ d3
λ−1 ε2
such that the manifoldSε=Sε+∪ Sε− satisfies the following properties:
a) The manifold Sεis 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)andd(S0−,Sε−) =O(ε).
c) If εis 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>0such that for eacht∈(−t0, t0) d(ϕ(t;p),Sε) =d(p,Sε)eλ+1t.
e) If p∈ {v <0},then there exists t0>0such that for eacht∈(−t0, t0) d(ϕ(t;p),Sε) =d(p,Sε)eλ−1t.
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 fromSε+ 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 intersectionSε+∩ 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)whereε >0.
a) If a126= 0,then the set Sε+∩ Sε−∩ {v= 0}contains a unique point, which is pc=
− d3
λ+1λ−1 ε2,−b1
a12 + d3
λ+1λ−1a12
(λ+1 +λ−1 −εa22)ε,0
.
Ifd3>0,then the orbit throughpc is a primary canard and ifd3<0,then the orbit throughpc is a faux–canard. Ifd3= 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 Theorem1.1is a direct consequence of these lemmas. The end of the section is devoted to the proof of Theorem1.2. 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 settingε= 0 in system (5), we get the layer system
u′1= 0, u′2= 0, v′=u1+|v|.
(8) 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 manifoldS = S+∪ S−,defined in (6), see Figure1(a).
Since the layer vector field is locally linear, the spectrum of the Jacobian matrix at any singular pointp∈ S ∩{v6= 0},is{0,0,1}or{0,0,−1}according top∈ S+∩ {v >0}orp∈ S−∩{v <0},respectively. ThenS+∩{v >0}is a repelling normally hyperbolic manifold andS−∩{v <0}is an attracting normally hyperbolic manifold.
Singular points on the fold lineF 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 thatv′ >0 over the plane{u1= 0}.Then, the straight lineF attracts orbits in{v <0}and repels orbits in{v >0}(see Figure2).
u2
u1
v S
F
S+
S−
Figure 2. Representation of the flow of the layer equation, at- tracting and repelling normally hyperbolic half–planesS− andS+, and the fold lineF.
Now, we continue by considering the reduced system
˙
u1=a11u1+a12u2+a13v+b1,
˙
u2=a21u1+a22u2+a23v+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 whenv6= 0 we have
˙
v=−|v|
v u˙1.
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) =
F+(u1, u2, v) ifv >0, F−(u1, u2, v) ifv <0, where
F+(u1, u2, v) =
a11u1+a12u2+a13v+b1
a21u1+a22u2+a23v+b2
−a11u1−a12u2−a13v−b1
and
F−(u1, u2, v) =
a11u1+a12u2+a13v+b1
a21u1+a22u2+a23v+b2
a11u1+a12u2+a13v+b1
.
The projection mapπ(u1, u2, v) = (u2, v) induces on R2\ {(u2,0) :u2∈R}the planar piecewise linear differential system
u˙2
˙ v
=
A+
u2
v
+b+ ifv >0,
A− u2
v
+b− ifv <0,
(10)
where
A+=
a22 a23−a21
−a12 a11−a13
, b+= b2
−b1
,
(11) A− =
a22 a23+a21
a12 a11+a13
, b−= b2
b1
.
The flows defined by the systems (9) and (10) are differentially conjugated inS \ F and inR2\ {(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 ˙u2 =a22u2+b2, v˙ = 0. Whena22 6= 0 a singular pointe= (−b2/a22,0) appears inF.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 matricesA+ andA−.
Direct computations show that the eigenvaluesβ+andγ+of the matrixA+,and the eigenvaluesβ− andγ− of the matrixA− 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 Theorems1.1and1.2.
The flow of (5) is defined by the composition of the two linear flows which are associated with the differential system
x′ =
A+εx+bε ifv≥0, A−εx+bε ifv≤0,
(13) wherex= (u1, u2, v)T and
A±ε =
εa11 εa12 εa13
εa21 εa22 εa23
1 0 ±1
, bε=
εb1
εb2
0
.
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. Forε >0the eigenvalues of the matrixA+ε expand in power series in εasλ+1 = 1 +a13ε+O(ε2), λ+2 =β+ε+O(ε2)andλ+3 =γ+ε+O(ε2),whereβ+and γ+ are the eigenvalues of the matrix A+ in (11). Moreover, the eigenvalues of the matrixA−ε expand in power series inεasλ−1 =−1−a13ε+O(ε2), λ−2 =β−ε+O(ε2) andλ−3 =γ−ε+O(ε2), whereβ− andγ− 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 A−ε follows in an analogous way. Since the characteristic polynomial ofA+ε
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 matrixA+ε can be expanded in power series inεas
λ+1 = 1 +αε+O(ε2),
λ+2 =βε+O(ε2), (15)
λ+3 =γε+O(ε2), whereαis a real number andβ,γ are real or complex.
From equality (14) we obtain that
λ+1 +λ+2 +λ+3 = 1 +εt1,
λ+1λ+2 +λ+1λ+3 +λ+2λ+3 =ε(t1−a13+εd1), (16) λ+1λ+2λ+3 =ε2(d1+d2).
Then, from (15) and (16) its follows that α=a13,
β+γ=t1−a13, (17)
βγ=d1+d2.
Note thatβandγsatisfy the same equations thatβ+andγ+in (12). Thenβ=β+ andγ=γ+.
From Lemma3.1, we emphasize that the spectrum of the matrixA+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 eigenvaluesλ+2 andλ+3,which tend to zero asεtends to zero. In Lemma3.2we see that these eigenvalues are responsible for the slow dynamic in{v≥0}.
Letw+be the eigenvector associated to the eigenvalueλ+1 of the matrix (A+ε)T, where superscriptT stands for the transpose. Letw−be the eigenvector associated to the eigenvalueλ−1 of the matrix (A−ε)T.Then it follows that
(w+)TA+ε =λ+1(w+)T, (w−)TA−ε =λ−1(w−)T. (18) Let us now consider the following setsPε+={p∈R3: (w+)T(A+εp+bε) = 0}and Pε− ={p∈ R3 : (w−)T(A−εp+bε) = 0}, and their restriction to the half–spaces {v ≥ 0} and {v ≤ 0} given by Sε+ = Pε+∩ {v ≥ 0} and Sε− = Pε−∩ {v ≤ 0}, respectively, and let us takeSε=Sε+∪ Sε−.The lemma below is satisfied.
Lemma 3.2. a) The set Pε+ (resp. Pε−) is an invariant plane under the flow of the linear system x′=A+εx+bε(resp. x′ =A−εx+bε).
b) The half–planesSε+andSε−are locally invariant under the flow of system(13).
Moreover, the points(u1, u2, v)∈ Sε+ satisfy thatv≥0 and (λ+1 −εa22)u1+εa12u2+ (λ+1)2−εt1λ+1 +ε2d1
v=−εb1+ε2d3
λ+1 , whereas the points(u1, u2, v)∈ Sε− satisfy thatv≤0 and
(λ−1 −εa22)u1+εa12u2+ (λ−1)2−εt1λ−1 +ε2d1
v=−εb1+ε2d3
λ−1 . c) IfS0 is a compact subset of S and we takeS0+=S0∩ S+ andS0−=S0∩ S−;
thend(S0+,Sε+) =O(ε)andd(S0−,Sε−) =O(ε).
Proof. We are going to prove the statements for the setsPε+ and Sε+. The corre- sponding proofs forPε− andSε− follow in a similar way.
From the equality (18) it follows that Pε+=
p∈R3: (w+)Tp=−(w+)Tbε
λ+1
. (19)
Hence Pε+ is an orthogonal plane to the vectorw+.Since for any given pointpin Pε+the vector field atp(i.e. A+εp+bε) is orthogonal tow+,we conclude thatPε+
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 inSε+. By solving w+ from (18), we have
w+= λ+1 −εa22, εa12,(λ+1)2−εt1λ+1 +ε2d1T . Hence, statement (b) follows from (19) by taking into account that
(w+)Tbε=εb1λ+1 −ε2d3.
LetS0 be a compact subset ofS and consider the compact setS0+ =S0∩ S+. From statement (b) of this lemma and by considering thatλ+1 = 1+O(ε),if|u1|,|u2| and|v|are bounded, then the points in the half–planeSε+satisfy thatu1+v=O(ε).
Moreover, the points on S+ satisfy that u1+v= 0.From these facts we conclude thatd(S0+,Sε+) =O(ε).
We note that from the former lemma, the half–planeSε+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–planesSε+ andSε− 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 eacht∈(−t0, t0) d(ϕε(t;p),Sε+) =d(p,Sε+)eλ+1t.
c) If p∈R3∩ {v <0},then there exists t0>0 such that for eacht∈(−t0, t0) d(ϕε(t;p),Sε−) =d(p,Sε−)eλ−1t.
Proof. Consider the projectionπε : 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 thatπεis a diffeomorphism for each small enoughε >0. Moreover
π−ε1(u2, v) =
− εa12
λ±1 −εa22
u2−(λ±1)2−εt1λ±1 +ε2d1
λ±1 −εa22
v
− b1ε λ±1 −εa22
+ d3ε2
λ±1(λ±1 −εa22), u2, v
,
whereλ±1 stands forλ+1 orλ−1 depending on{v >0}or{v <0}, respectively.
Hence, πε induces on {(u2, v) ∈ R2 : v 6= 0} the piecewise linear differential system
u′2 v′
=
Bε+
u2
v
+c+ε ifv >0,
Bε− u2
v
+c−ε ifv <0,
(20)
where
B±ε =
ε
a22− a21a12ε
λ±1−εa22
ε
a23−a21(λ±1)2−εt1λ±1+ε2d1
λ±1−εa22
−ελ±a12
1−εa22 ±1−(λ
±
1)2−εt1λ±1+ε2d1
λ±1−εa22
and
c±ε =
ε
b2−ελ±b1a21
1−εa22 +ε2λ± a21d3
1(λ±1−εa22)
ε
−λ±b1
1−εa22 +ελ± d3 1(λ±1−εa22)
.
We remark that, the meaning of±1 in the matrixBε± is that it stands for 1 in the matrixBε+ and for−1 in the matrixBε−.
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 timetin system (20) by the slow timeτ=εt.We note that from the expressions ofλ+1 andλ−1 appearing in Lemma 3.1it follows that
1
ε 1− λ+12
−εt1λ+1 +ε2d1
λ+1 −εa22
!
=a11−a13+O(ε), and
1
ε −1− λ−12
−εt1λ−1 +ε2d1
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 εtends 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–
planeSε+.Letpbe a point inR3∩ {v >0}and letv+1 be the eigenvector associated to the eigenvalueλ+1.Sincev+1 is not parallel to Sε+,the pointpcan be expressed as a sum of a point q in Sε+ and a point q1 in the straight line {rv+1 : r ∈ R},
i.e. p=q+q1, with q1=rv+1. As far asϕ(t;p) remains in{v ≥0} andϕ(t;q) remains inSε+ it can be expressed as
ϕ(t;p) =eA+εtq+ Z t
0
eA+ε(t−s)bεds+reA+εtv+1 =ϕ(t;q) +reλ+1tv+1. Hence, we conclude that
d(ϕ(t;p),Sε+) =d(p,Sε+)eλ+1t, 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+rv−1, where q∈ Sε−, r∈Randv−1 is the eigenvector associated to the eigenvalueλ−1.
We end this section with the proofs of Theorems1.1and1.2.
Proof of Theorem 1.1. The statement (a) of the theorem is a straightforward con- sequence 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
(λ+1 −εa22)u1+εa12u2=−b1ε+ d3
λ+1ε2 (λ−1 −εa22)u1+εa12u2=−b1ε+ d3
λ−1 ε2
(21)
whose determinant is λ+1 −λ−1 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 onpc.Since it depends on the sign of the first component ofpc,see (13), we conclude that ifd3>0,then the orbitγpc
throughpcgoes fromSε− toSε+.SinceSε− is the stable branch of the slow manifold Sε andSε+ is the unstable branch ofSε,the orbitγpc 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 throughpc from one branch to the other ofSε. Therefore no primary canards exist.
Ifa12= 0 then the system (21), forεsmall enough, has no solution unlessb1= 0.
If it is the case, thend3 = 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 a226= 0, then the straight line is made up by the critical point (0,−b2/a22,0) plus the solutions (0,−b2/a22±eεa22t,0). Otherwise, when b2 6= 0 the straight line contains the orbit{(0, εb2t,0) : t∈R}; and whenb2= 0 each point of the straight line is a critical point.
4. Example of canard cycle. In this section we apply Theorem1.1and Theorem 1.2to a particular family of singularly perturbed piecewise linear differential system, (5), given bya11=a13=b1= 0, a12<0, a21=a22=a23= 0 and b2= 1,i.e.
u′1=εa12u2, u′2=ε, v′=u1+|v|.
(22) We note that system (22) is a piecewise linear version of the differential system x′=−2ε y, y′=ε, z′ =x+z2, which is considered in [26].
The matrices of the linear problems associated to (22) are A+ε =
0 εa12 0
0 0 0
1 0 1
, A−ε =
0 εa12 0
0 0 0
1 0 −1
,
and their eigenvalues are λ+1 = 1, λ+2 = λ+3 = 0 and λ−1 = −1, λ−2 = λ−3 = 0, respectively.
According to Theorem 1.1, the slow manifold Sε of (22) is the union of the unstable half–plane
Sε+=n
(u1, u2, v)∈R3 :v≥0, u1+εa12u2+v=−ε2a12
o
and the stable half–plane Sε− =n
(u1, u2, v)∈R3 :v≤0, −u1+εa12u2+v=ε2a12
o.
These half–planes intersect at a unique pointpc = (−ε2a12,0,0).Sinced3=−a12>
0,from Theorem1.2, the orbitγpc 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 ofϕ(t;p) can be written as
ϕ1(t;p) =p1+εta12p2+ε2t2 2 a12, 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 whenp1 >0, and downward when p1 <0 (see the third equation in (22)). Therefore, settingR+={p∈R3:p3>0, orp3= 0 andp1>0} andR− ={p∈R3 :p3<0, orp3= 0 andp1<0},it follows that:
ifp∈ R+,then ϕ3(t;p) = et−1
p1+εa12 et−1−t
p2+etp3+ε2a12
et−1−t−t2 2
, fort∈(t−p, tp); and ifp∈ R−,then
ϕ3(t;p) = 1−e−t
p1+εa12 e−t−1 +t
p2+e−tp3−ε2a12
e−t−1 +t−t2 2
,
fort∈(t−p, tp).The endpoints of the intervals of definitiont−p≤0≤tpcorrespond 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.
Sincepc ∈ R+,the next proposition is a direct consequence of the expression of the flow shown above.
Proposition 1. The canard orbitγpc is given by ϕ(t;pc) =
−ε2a12
1−t2
2
, εt, −ε2a12t
1 + |t|
2
for t∈R.
From Proposition1 and the expression of the slow manifoldSε = Sε+∪ Sε−, it is easy to check that the canard orbit γpc remains in Sε+ fort > 0 and in Sε− for t <0.In Figure3 we represent the canard orbitγpc for the parametersa12=−1.3 andε= 1e−1.
–1.5 –1 –0.5 0.5 1 1.5
–1.5 –1
0.5 1 1.5
–1.2–1.4 –0.8–1 –0.6
γpc
Sε− Sε+
Figure 3. Canard orbit γpc crossing from the stable half–plane Sε− through the unstable half–planeSε+of the piecewise linear dif- ferential system (22).
Hence, forγpc to be a periodic orbit, it is necessary thatγpc leavesSε− andSε+
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 numberη, we consider the four–pieces linear differential system
x′ =
Fu(x) ifv≥η, Fo(x) if|v| ≤η, Fl(x) ifv≤ −η,
(23) wherex= (u1, u2, v)T,
Fu(x) =
εa12u2+a1(v−η) ε−a22(v−η) u1+η+a23(v−η)
, Fl(x) =
εa12u2+a1(v+η) ε+a22(v+η) u1+η−a23(v+η)
, a1, a2, a3∈R,andFo(x) is the piecewise linear vector field defined by the differen- tial system (22).
Let ˜ϕ(t;p) be the flow defined by the piecewise linear differential system (23).
It is easy to conclude that ˜ϕ coincides with ϕ when we restrict ˜ϕ to the central region{(u1, u2, v) :|v| ≤η}.Therefore, the slow manifold ˜Sε of the system (23) is the union of the unstable branch
S˜ε+=n
(u1, u2, v)∈R3 : 0≤v≤η, u1+εa12u2+v=−ε2a12
o
and the stable branch S˜ε− =n
(u1, u2, v)∈R3 :−η≤v≤0, −u1+εa12u2+v=ε2a12
o, which are locally invariants under the flow ˜ϕ.In fact, the slow manifold has bound- aries at{v= 0}and at{v=±η}(border planes) through which the flow leaves the manifold, see Figure4. Moreover, we emphasize that the orbitγpc is also a canard in this new setting, and its expression, given in Proposition1, is correct while the orbit remains in the central region{|v| ≤η}.That is, for
|t| ≤t∗= 1 ε
r
ε2− 2η a12
−1. (24)
pc
γpc
p+c C(p2)
qc
v
u1
u2
{v=η}
{v= 0}
{v=−η}
S˜ε+
Figure 4. Representation of the canard cycleγpc,slow manifolds S˜ε∪S˜εand 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 ofγpc 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 involutionR(u1, u2, v) = (u1,−u2,
−v).This means that the flow ˜ϕ(t;p) is reversible in the sense that R( ˜ϕ(t;p)) = ˜ϕ(−t;R(p)).
We note that the set of fixed points of involution R corresponds to the u1–axis.
Therefore, sincepc is on the u1–axis, a sufficient condition for the orbitγpc to be periodic is that it intersects theu1–axis at a new point, which we denote byqc,see Figure4.
Let us seek now the initial conditionsp= (p1, p2, η) on the plane{v=η} such that orbits through them, reach theu1–axis. That is,ϕ2(t;p) = 0 andϕ3(t;p) = 0,
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+etη+ε2a12
et−1−t−t22 . From this, we get the relationp1=C(p2),where
C(p2) =
1−e−p2ε −1
e−p2ε (η+εa12(p2+ε)) +1
2a12p22−ε2a12
. (25)
Let p+c = ˜ϕ(t∗,pc), where t∗ is the value of the time defined in (24). Hence, p+c is the point of intersection of the canard orbitγpc with the plane{v=η}, see Figure4. Then
p+c =
−η−εa12
r
ε2− 2η a12
, −ε+ r
ε2− 2η a12
, η
.
Therefore, a sufficient condition forγpc to be a periodic orbit is the existence of a value of the timet+c >0 such that the orbit throughp+c intersect the plane{v=η}
just at the graph of the functionC(p2),see Figure4, that is
˜
ϕ(t+c,p+c)∈
(C(p2), p2, η) :p2∈R− , (26) and ˜ϕ3(t,p+c)> η fort∈(0, t+c).
Condition (26) leads us to a system of two equations
˜
ϕ1(t+c;p+c) =C( ˜ϕ2(t+c;p+c)),
˜
ϕ3(t+c;p+c) =η, (27)
with seven unknowns ε, η, a12, a1, a2, a3 andt+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 sys- tems 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 func- tions. 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 anda2 = 3.4, and solving (27) for a3 andt+c,we obtain the valuesa3≈0.583695486652 andt+c ≈2.3372454.
Figure 5 contains different views of the canard cycleγpc obtained by using the software package Dynamic Solver [1]. Pictures (a), (b) and (c), represent the pro- jection 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 variablev versus the timet, respectively.
We recall that fast dynamics takes place on perturbed vertical straight lines, while the slow dynamics follows the slow manifoldSε=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 variablev versus time. As it can be observed, along a period, the time interval in which the variablev increases (the slow phase)
(a) (b) (c)
(d) (e)
✒
◆◆ ✒
◆◆
❄❄
✯
❨
✻
✒
❲❲
❲❲
Figure 5. Different views of the canard cycle γpc exhibited by system (23) withε= 0.1, a12=−1.3, η= 1, a1=−1, a2= 3.4 and a3≈0.583695486652.(a) Projection ofγpcon the plane (u2, v). (b) Projection ofγpc on the plane (u1, v). (c) Projection ofγpc on the plane (u1, u2). (d) Three dimensional view of the canardγpc. (e) Representation of the variablevversus the time.
is, approximately, one tenth of the time interval in which the variable v decreases (the fast phase). This is becauseε= 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.
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[3] V. Carmona, F. Fern´andez-Sanchez and A. E. Teruel,
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