Canard Cycles and Poincaré Index of NonSmooth Vector Fields on the Plane
Abstract.
This paper is concerned with closed orbits of nonsmooth vector fields on the plane. For a subclass of nonsmooth vector fields we provide necessary and sufficient conditions for the existence of canard kind solutions. By means of a regularization we prove that the canard cycles are singular orbits of singular perturbation problems which are limit periodic sets of a sequence of limit cycles. Moreover, we generalize the Poincaré Index for nonsmooth vector fields.
Key words and phrases:
Limit cycles, vector fields, singular perturbation, nonsmooth vector fields, heteroclinic orbits, Poincaré index, canard cycles.1991 Mathematics Subject Classification:
Primary 34C20, 34C26, 34D15, 34H051. Introduction
Piecewisesmooth systems are widespread within application
areas such as engineering, economics, medicine, biology and ecology.
The most common piecewisesmooth
systems involve either a discontinuity in the vector field, or in
the orbit given by the integral solution . In this paper we
consider the former, that is, general systems where the vector field
is independently defined on either side of a smooth codimension one
switching manifold. Three possible regions of the manifold are then
apparent. At a crossing region the component of the vector field
normal to the switching manifold has the same direction on both
sides of the manifold (sometimes called sewing instead of crossing).
At a stable sliding region both normal components of the vector
field point toward the manifold. At an unstable sliding region both
normal components point away from the manifold. Piecewisesmooth
systems with sliding are also known as Filippov systems. Clearly
these three different scenarios lead to vastly different dynamics.
An orbit that meets the switching manifold at a crossing region
passes through it, but is nondifferentiable at the crossing point.
An orbit that impacts at a stable sliding region sticks becomes
constrained (sticks) to the manifold. An orbit in an unstable
sliding region slides along the switching manifold, but will depart
it under any infinitesimal perturbation. Consequently, the only
means by which a stable sliding orbit can escaping the switching
manifold is tangentially, at the boundary of the sliding region.
This leads to the observation that, under parameter variation,
orbits in Filippov systems can undergo a large variety of
bifurcations, commonly called sliding bifurcations.
In this paper we study piecewisesmooth system on open regions on
the plane. Let be an open set and
given by where
is a smooth function having as a regular value (i.e. , for any . Clearly is the separating boundary of the
regions and
. We can assume that
is represented, locally
around a point , by the function
Designate by the space of vector fields on endowed with the topology with or large enough for our purposes. Call the space of vector fields such that
(1) 
where
We write which we will accept to be multivalued in
the points of The trajectories of are solutions of
which
has, in general, discontinuous righthand side. The basic results of
differential equations, in this context, were stated by Filippov in
[5]. Related theories can be found in [6, 9, 12]. In this
paper we consider finite discontinuities, i.e., the vector fields
and are defined in the set . Another kind of
discontinuity of which the vector field tends to infinity when it
approximates to the switching manifold
can be obtained in the equations with impasse (more details in [10]).
In what follows we will use the notation .
We distinguish the following regions on the discontinuity set

is the sewing region if on .

is the escaping region if and on .

is the sliding region if and on .
Consider The sliding vector field
associated to is the vector field tangent to
and defined at by with
being the point where the segment joining and
is tangent to (see Figure 2). It is
clear that if then for and
then we can define the escaping vector field on
associated to by . In what follows
we use the notation for both cases.
Our main interest here is to study a special kind of typical minimal
sets of nonsmooth vector fields which will be called nonsmooth
“canard cycles” (see Figure 2). A canard
cycle is a graphic composed by pieces of orbit of , pieces of
orbit of the sliding vector field and/or pieces of
orbit of . See Section 2 for a more precise definition.
An approximation of the nonsmooth vector field by a parameter family of smooth vector fields is called an regularization of . We give the details about this process in section 4. A transition function is used to average and in order to get a family of smooth vector fields that approximates . The main aim is to deduce certain dynamical properties of the nonsmooth dynamical system from the regularized system. What is familiar may or may not be a matter of taste, at least it depends a lot on the dynamical properties of one’s interest. The regularization process developed by Sotomayor and Teixeira produces a singular problem for which the discontinuous set is a center manifold. Via a blow up we establish a bridge between nonsmooth systems and the geometric singular perturbation theory.
Roughly speaking, the main results of this paper are the following:

In our first result (Theorem 1), for a subclass of nonsmooth vector fields, we provide necessary and sufficient conditions for the existence of canard kind solutions.

In our second result (Theorem 2), following the ideas exposed in [8], we prove that hyperbolic canard cycles are limit sets, according Hausdorf distance, of families of (smooth) hyperbolic limit cycles (this fact is not proved in [8]). The regularization process plus a blow up produce a singular perturbation problem . Our result implies that the canard cycle is the periodic limit set of closed orbits of , with . An open problem is to use the geometric singular perturbation theory proposed by Dumortier and Roussarie (center manifolds obtained via saturation by the flow plus blow up techniques, see [3] for details) to obtain the same result.

In our third result (Theorem 3) we found an analogous for Poincaré Index in the case of nonsmooth vector fields.
2. Preliminaries and statements of the main results
Consider We say that is a regular point if

or

and (that is and it is not a singular point of ).
The points of which are not regular are called
singular. We distinguish two subsets in the set of
singular points: and . We say that is a pseudo equilibrium of if
and we say that is a
contact point if and
( is a contact point of ).
A contact point is a fold
point of if but Moreover,
is a visible (resp. invisible)
fold point of
if and
(resp. ). We say that is a fold point
of
if it is a fold point either of or of .
A pseudo equilibrium is a saddle
provided one of the following condition is satisfied: (i)
and is an attractor for or (ii)
and is a repeller for . A pseudo
equilibrium of is a repeller
(resp. attractor) provided (resp.
) and is a repeller (resp. attractor) for
. A point is a hyperbolic
pseudo equilibrium
of if is a hyperbolic equilibrium point of
Definition 1.
Consider

A curve is a canard cycle if is closed and

contains arcs of at least two of the vector fields , and or is composed by a single arc of ;

the transition between arcs of and arcs of happens in sewing points (and vice versa);

the transition between arcs of (or ) and arcs of happens through fold points or regular points in the escape or sliding arc, respecting the orientation. Moreover if then there exists at least one visible fold point on each connected component of .


Let be a canard cycle of . We say that

is a canard cycle of kind I if meets just in sewing points;

is a canard cycle of kind II if ;

is a canard cycle of kind III if contains at least one visible fold point of .


Let be a canard cycle. We say that is hyperbolic if

is of kind I and where is the first return map defined on a segment with ;

is of kind II;

is of kind III and or or .

In [8] is proved that the regularization of nonsmooth vector fields with hyperbolic canard cycles has hyperbolic limit cycles.
Definition 2.
We remark that for a hyperbolic canard cycle we have that each connected component of has only one fold point (See [8] for more details).
By using the previous notation, our results are:
Theorem 1.
Let be a nonsmooth vector field with presenting only one fold point which is visible. Denote the arc of ( or ) which passes through and call the transversal contact point of with . Then has a canard cycle if and only if the following conditions are satisfied: (i) the component of which passes through is a focal kind arc; (ii) in and (iii) is a linearly independent set in . Moreover, is of kind III.
Theorem 2.
Let be a hyperbolic canard cycle of . Then for any the regularized vector field , has a hyperbolic limit cycle such that when
We remark that the Hausdorff distance between compact sets of is:
Theorem 3.
Let be a hyperbolic canard cycle of the nonsmooth vector field If , , is the set of fixed or pseudo equilibrium points (all hyperbolic) of inside then the index of with respect to is the sum of the index of , for . Moreover, this sum is equal to one.
In section 5 we will define index of nonsmooth vector fields.
The paper is organized as follows. In Sections 3, 4 and 5 we prove Theorems 1, 2 and 3, respectively. In section 6 we apply Theorem 1 to study a class of nonsmooth vector fields with just one focal kind arc and its bifurcation and we use the singular perturbation theory to study hyperbolic canard cycles.
3. Proof of the Theorem 1
In this section we prove the first result of the paper.
Proof. First we prove that (i),(ii) and (iii) imply the existence of the canard cycle. Since in the piece of between and is part of a escaping region or a sliding region. Moreover since is a linearly independent set in the system does not have pseudo equilibrium points in . Without lost of generality, is part of the sliding region like in Figure 7. The curve is a hyperbolic canard cycle of kind III. We remark that this canard cycle takes place in just one side of .
Now we prove that (i),(ii) and (iii) are necessaries conditions for
the existence of this particular kind of canard cycle. Since
is a hyperbolic canard cycle of kind III with just one
fold point, takes place in just one side of
. In fact, if it does not occur, then returns to
at least twice and so there exists at least a second
fold point. Without lost of generality we suppose that
is on the side corresponding to . We denote by
the part of the cycle which is a trajectory of
. Thus we have that is a focal kind arc because
if it is a graphic kind arc then there is another fold
point on (see Figure 7). Since
has no one arc of , the point belongs to an
escaping region or a sliding region and so . Let us assume that . Since meets
in the point , the flow slides via until
the point because there are not another fold point
between and ; therefore in
Moreover, the linear independence of on
follows from the nonexistence of pseudo equilibrium points on
Now, we will define an
auxiliar function which will be useful in the sequel.
Take contained in the escaping or in the sliding region. In consider the point , the vectors and (as illustrated in Figure 8). The straight segment passing through and meets in a point . We define the Capplication
We can choose local coordinates such that is the axis; so and . The direction function on is defined by
We have that is a Capplication and

if then the orientation of in a small neighborhood of is from to ;

if then ;

if then the orientation of in a small neighborhood of is from to .
Simple calculations shows that .
Assuming all the hypothesis of Theorem 1 we have the following corollary.
Corollary 1.
The nonsmooth vector field has a canard cycle if and only if the direction function is a well defined function and it has no zeros. Moreover, is of kind III.
4. Proof of Theorem 2
First of all we present the concept of regularization of nonsmooth vector fields. It was introduced by Sotomayor and Teixeira in [9]. The regularization gives the mathematical tool to study the stability of these systems, according to the program introduced by Peixoto. The method consists in the analysis of the regularized vector field which is a smooth approximation of the nonsmooth vector field. Using this process we get a parameter family of vector fields such that for each fixed we have

is equal to in all points of whose distance to is bigger than

is equal to in all points of whose distance to is bigger than .
Definition 3.
A function is a transition function if for , for and if The regularization of is the 1parameter family given by
(2) 
with for
In order to prove Theorem 2 we need to construct a special
neighborhood of arbitrary diameter for hyperbolic canard cycles.
Construction of a neighborhood of diameter
around a hyperbolic canard cycle. Here we describe a method to
construct a tubular neighborhood of diameter around a
hyperbolic canard cycle. This presentation is done for canard cycles
of kind III, but the ideas can also be extended for kinds I or II.
We will be particularly interested in two of them: the ones that
take place on just one side of and with just one visible
fold point and the ones that take place on the two sides of
with two visible fold points (one for and
another one for ).
Case 1 One fold point. Denote by the hyperbolic canard cycle of kind III with just one fold point and with orientation showed in Figure 9 (the reverse orientation is treated in a similar way). Consider the strip of diameter around . Let and be points in . Take an arc of the vector field passing to the point in such a way that stays on the left of and such that returns to the line in a point which is in a neighborhood of . Take this trajectory satisfying (this is possible by the continuity of ). Let be the point where the arc of through by first meets the straight line for negative time. Analogously take an arc of the field passing by the point in such a way that stays on the left of and such that has second return to in a point which is in a neighborhood of . Take this trajectory satisfying . On , on the left of the fold point , the flow of is oriented to up, so it is possible to construct a transversal section joining to the straight line in such a way that the same trajectories of cross transversally and the segment . Take the point where meets the straight line satisfying , as before. Moreover, on , on the right of the point , the flow of is oriented for down, so it is possible to construct a transversal section joining to the straight line in such a way that the trajectories of that cross transversally do not cross the segment . Let be the point where meets the straight line (here we also need to take care for ). Since is a sliding region, the flow of is transversal to . In the straight line , consider the points and ( is on the left of ) satisfying that the trajectories of crossing the transversal sections and meet transversally at the segment (again, we need to take care for and ).
In this way, the strip defined by the closed curve and by the closed curve is a tubular neighborhood of
of diameter . Note that the flow of is
arriving in this neighborhood and never it departs from it.
Case 2 Two fold points. Now we study the hyperbolic canard cycles of kind III with two visible fold points, being one for and the other one for , like showed in Figure 10. We work with canard cycles that have only escaping regions on (the case with sliding regions is treated similarly). Consider the strip of diameter around . Let and be points in . Take an arc of the vector field through satisfying that stays on the left of and such that returns to the line in a point which is in a neighborhood of . Take this trajectory satisfying that . Take an arc of the vector field through satisfying that stays on the right of and such that has second return on the straight line in a point , even take this trajectory with the particularity that . We repeat the same argument for the vector field and we found the points , , , , and respectively, and the curves and . Let be the point on , be the point on , be the point on and be the point on as indicated in Figure 10. On , take the points on the left of , between and , between and and on the right of ; satisfying that the arcs (joining to ), (joining to ), (joining to ) and (joining to ) are transversal sections for and the arcs (joining to ), (joining to ), (joining to ) and (joining and ) are transversal sections for with the distance from to any one of this arcs less than .
In this way, the strip defined by the closed curve and by the closed curve is a tubular neighborhood for of diameter . Note that the flow of is departing from the tubular neighborhood and never it arrives in it.

Since this neighborhood bounds a region where the nonsmooth vector field is arriving in or it is departing from them, it makes sense to say attractor canard cycle or repeller canard cycle.

In the neighborhoods constructed before we allow that trajectories can make part of them, however it is possible to do it with the flow of being transversal to the boundaries of the tubular neighborhoods. In fact, it is enough to replace the trajectories by transversal curves. It is important for the construction of the tubular neighborhood of the canard cycles of kind I. Thus we make a construction like we made before but now we can use for this, the first return application and thus if we have an attractor canard cycle and if we have a repeller canard cycle.

For canard cycles of kind II is enough to take the strip of diameter in the beginning of the construction as the tubular neighborhood.

Any other hyperbolic canard cycle is an arrangement of pieces of the canard cycles described above and so we can construct a tubular neighborhood for it arranging the previous tubular neighborhoods.
Proof of Theorem 2. Let be a
canard cycle of and let be a tubular neighborhood of
diameter around . Since is transversal to the
boundary of , by continuity, the regularized vector field
also is transversal to the boundary of . Assume that
is an attractor canard cycle, so the flow of is
arriving in the neighborhood and consequently the flow of
also is arriving in the neighborhood . As there are not
fixed points in , applying the PoincaréBendixson Theorem
we conclude that there exists an attractor limit cycle
inside Moreover with a more detailed analysis we can prove
that it is hyperbolic (see [8] for instance). Since every
paths that compose depends continuously of we have that
the diameter of the tubular neighborhood is a continuous function of
the variable . Therefore making we
conclude that (see Figure
11).
We remark that if is an attractor (resp. repeller) hyperbolic canard cycle of , then the same occurs for and
5. Proof of Theorem 3
Now we start our discussion about the third result. Let be an interval and be an oriented closed continuous path. Suppose that there are no critical points of on . Let us move a point along the curve in the counterclockwise direction. The vector will rotate during the motion. When returns to its starting place after one revolution along the curve , also returns to its original position. During the journey will make some whole number of revolutions. Counting these revolutions positively if they are counterclockwise, negatively if they are clockwise, the resulting algebraic sum of the number of revolutions is called the index of with respect to , and is denoted by .
To calculate is convenient normalize as an unit vector at the origin. In this way, we can define a function such that
for every . The function is called angle function.
We observe that in the case of smooth vector fields, the angle function is always continuous, but in the case of nonsmooth vector field it admits a “jump” when the path pass to a point , . Therefore, we establish a rule for this jump; at the angle function oscillates from to . If and , where , we add to the the number ; if and we add to the number . We always consider that the jump of the vector to the vector occurs by the smallest angle between this vectors.
The difference is a multiple of , and
is an integer independent of the chosen parametrization.
This number also is called Poincaré Index of the curve
with relation to the nonsmooth vector field .
Our interest here is to calculate the index of canard cycles
surrounding fixed or pseudo equilibrium points that are the critical
points of . We will see that, different from the smooth case,
given two canard cycles and surrounding the
same critical points of
we have in general.
Example. Consider the configuration described in
Figure 13. Let be
an arc of joining the fold points and ;
the fold point of ; and points in the escaping
region ; , and points in the sliding
region where is a invisible fold
point of ; the arc of joining and ,
the arc of joining and ; and
the arc of joining and . Consider the
fixed point of named by in Figure
13. We can choose closed curves
such that the Poincaré index to nonsmooth vector fields is any
natural number. For example, take the path and the index is
; in an analogous way if we take the path the
index is . Repeating this argument we can found closed curves
such that the Poincaré index
is any integer number that we wish.
The previous situation just occurs because the canard cycle given, for example, by , is nonhyperbolic because in its composition we can found pieces of escaping region and pieces of sliding region. If we eliminate this possibility we have the next theorem, which has an analogous in the case of smooth vector fields.
Remark 1.
We recall that if has a hyperbolic saddle (or a hyperbolic focus) then the regularized vector field has a hyperbolic saddle (hyperbolic focus) where when (for details see [8]). Moreover, we can verify that if is a saddle or a saddle then we can take a sufficiently small closed path around and prove that (and if is a focus or a attractor or a repeller then we can take a sufficiently small closed path around and prove that ). When the path is sufficiently small to have just one critical point of , named , in its interior we use the notation to denote its Poincaré Index.
5.1. Proof of Theorem 3
We want to prove that if is a hyperbolic canard cycle of the nonsmooth vector field and if , , are the only ones critical points (all hyperbolic) of inside then is well defined and
First of all we assume that the index is well defined. Let be a nonsmooth vector field with a hyperbolic canard cycle and hyperbolic critical points of inside . Thus, the regularized vector field has a hyperbolic limit cycle and hyperbolic fixed points inside . So, by the Poincaré Index Theorem for smooth vector fields the index calculated in in relation to is the sum of the index of the fixed points of inside and this sum is equal to . Since we conclude that the index calculated in in relation to is the sum of the index of the critical points of inside and this sum is equal to 1 (see remark 1). In order to finish the proof we verify that the index is well defined in the case that the closed curve is a hyperbolic canard cycle. In fact, let be a hyperbolic canard cycle of . Let us assume that is of one kind described in Figures 15 or 15 below.
We will prove that there is not danger of ambiguity in the choose of the closed paths, differently that what happens in the previous example. In Figure 15, we consider the hyperbolic canard cycle given by . Any canard cycle of having pieces of sliding region must to pass by the fold point , now walk in (which is the only one possibility that we have!) and meet the point on . The unique choice we have is return to , closing the path, without ambiguity. For Figure 15 the analysis is more interesting. Obviously we can use the trick of take the vector field and obtain an analogous result that the previous, however we prefer give here a complete idea to the case described in Figure 15. Since in the semi straight line we have an escaping region, the canard cycle must have only escaping region in its composition. Note that, if we choose to depart from by a point in the segment then this path does not return to (it will move spirally around the focus ), if we choose going out from by a point after in then this path also does not return to (to this path return to it must return in a sliding region, what is not allowed because only escaping regions compose this hyperbolic canard cycle). So we must leave by the point and to close the curve. Therefore in any case there is not danger of ambiguity in the choose of the closed curve and so the Poincaré index for nonsmooth vector fields is well defined. To hyperbolic canard cycles of kind III with another particularities is enough repeat the ideas exposed here. To hyperbolic canard cycles of kinds I and II clearly there is not danger of ambiguity in the choose of the closed curves once there are not escaping or sliding region in its composition.
Corollary 2.
Under the hypothesis of the previous theorem and assuming that all canard cycles of are hyperbolic, we have that:

If is a canard cycle then inside there exist critical points of , being saddles or saddles and focus, repeller or attractor.

If all critical points of are saddle or saddle then does not have canard cycles.
Proof. Since the index of each
saddle and each saddle point is equal to and the index
of each other critical point of is equal to the result is
an immediate consequence of
Theorem 3.
6. Applications and Examples
6.1. Heteroclinic Orbits
Consider the notation of the Theorem 1. We give now an
example of a curve that satisfies all the hypothesis in this theorem
except that there exists a point such that the
vectors and are not linearly independent;
instead of obtained in the theorem we have here a
“loop”, that is a
saddleattractor with connection
between separatrices.
Example. Consider the nonsmooth vector field
with ,
and
discontinuity set given by the axis, i.e., . On ,
we have and and so,
(3) 
In this way, we can conclude that is a fold point of which determines a focal kind arc. For we have that is a sliding region and for it is a sewing region (see Figure 17). We show now that there exists a point in the semi straight line for which . In fact, if then . The graphic of is given in Figure 17. We observe that is equal to , where is the direction function defined previously. So, we have the situation described in the Theorem 1, except that and are not linearly independent in where has an equilibrium point. The orientation of is in direction to the fold point because for we have , and so, the direction function is negative (