Keywords:Slow-Fast System; Singular Perturbation; Duck Solutions; Blowing-Up; Nonstandard Analysis
1. Introduction
In the 
slow-fast system with an invariant manifold, we first assume that this manifold describing limit cycle has a duck solution in a projected 
system in
. We introduce 2-dimensional duck solutions in the Section
, then introduce 4-dimensional duck solutions in the Section 4. The blowing up method which constructs a local model
, is published but revised and extended in this paper, and introduced in the Section
. In general, we do not need the first assumption to get 4-dimensional duck solutions. It is the shortest way to explain these singular solutions. There are some concrete examples in [3-5].
2. Slow-Fast System in R2
In this section, we shall review some results in Zvonkin and Shubin [6,7]. Let us consider the following system of differential equations
(1)
where 
is defined in 
and 
is infinitesimal in the sence of non-standard analysis of Nelson [8]. For the system (1), the graph 
is called the slow curve. We consider the extremum point 
that separates the attracting part and the repelling part.
Definition 2.1 A solution 
of the system (1) is called a duck solution if there exist standard numbers
, 
, 
such that 1)
, where 
denotes the standard part of
2) for 
the segment of the trajectory 
is infinitesimally close to the attracting part of the slow curve3) for
, it is infinitesimally close to the repelling part of the slow curve, and 4) the attracting and repelling parts of the trajectory are not infinitesimal.
We give a necessary condition for the existence of a duck solution close to the extremum point 
of
.
Proposition 2.2 If there is a duck solution of the system (1) close to the extremum point
, then
.
We finally obtain the following proposition concerning the existence of duck solutions.
Proposition 2.3 Suppose that 
has a nondegenerate extremum point
, that is, 
and
. Then there are the corresponding values of the parameter 
satisfying Proposition 2.2 for which there exist duck solutions in the system (1).
3. Slow-Fast System in R3
We shall introduce 
-dimensional duck solutions by E. Benoit to get a concrete image before giving a framework in the 
-dimensional duck solutions. Let us consider the following slow-fast system:
(2)
where
, 
, are variables, and 
is a parameter as the same as in (1). We give the following assumptions in the system (2).
(A1)
, 
are defined on
(A2) The set 
is a 
-dimensional differentiable manifold and the set 
intersects the set 
transversely so that the pli set 
is a 1-dimensional differentiable manifold.
(A3)
, or 
at any point
.
Let 
be a solution of (2). When
, differentiating 
with respect to the time
, the following equation holds:
where
,
. The above system (2) restricted to 
on the neighborhood of 
becomes the following system:
(3)
where
. The system (2) coincides with the system (3) at any point
. In order to avoid the degeneracy of the system (3), let us consider the following system:
(4)
As the system (4) is well defined at any point of
, it is well defined indeed at any point of
. The solutions of the system (4) coincide with those of the system (3) on 
except the velocity when they start from the same initial points.
(A4) For any point
, either of the following holds;
, 
, that is, the surface 
can be expressed as 
or 
in the neighborhood of
. Let
exist, then the projected system (5) is obtained:
(5)
If we take
, it can be analyzed in the same way.
(A5) All the singular points of the system (5) are nondegenerate, that is, the matrix induced from the linearized system of (5) at a singular point has distinct nonzero eigenvalues.
Remark All these points are contained in the set
, which is called the set of pseudo singular points. Note that these points are the singular points in the system (4). The above assumptions 
might be enough to use on the neighborhood of these pseudo singular points, because we aim to analyze only in the neighborhood of the pseudo singular point in the system (2).
Definition 3.1 Let 
and
, 
be two eigenvalues of the matrix associated with the linearized system of (5) at
. The point 
is called pseudo singular saddle if 
and called pseudo singular node if 
or
. When
, 
are complex conjugate, they are called pseudo singular focus.
From now on we use IST [8]. The “transfer principle” is applied for the approximation to the standard analysis. We take the functions
, which are non-standard, that is, they depend on
. The second derivative of
, and the first derivatives of 
have S-continuity. Let the system (2) have a solution 
and let 
be a solution of the system (4) , then a duck solution is defined as follows.
Definition 3.2 The solution 
of the systems (2) is called a duck, if there exist standard 
such that 1)
2) for 
the segment of the trajectory 
is infinitesimally close to the attracting part of the slow curves (the constrained surface)3) for
, it is infinitesimally close to the repelling part of the slow curves, and 4) the attracting and repelling parts of the trajectory are not infinitesimal.
The definitions of attracting and repelling are the same in [9,10].
Theorem 3.3 (Benoit) If the system has a pseudo singular saddle or node point, then it has duck solutions. In the saddle case, the duck solutions are determined uniquely. In the node case, for the distinct eigenvalues they are determined uniquely, if it has no resonance. If the system has a pseudo singular focus point, it has no duck solutions.
Remark Note that there are some important conditions on the standardness of the functions. At around the pseudo singular point, we blow up the variables in order to get a local model which is described in the Section
. In this case, it is determined uniquely. Through the local model, we can get an exact solution as is approximation in the original system. Using transfer principle, we can confirm the existence of 3-dimensional duck solutions. See [9].
4. Slow-Fast System in R4
Now, let us consider a slow-fast system (6):
(6)
where 
and 
are standard defined on 
and 
is infinitesimal.
First, we assume the following condition 
to get an explicit solution.
(B1) 
is of class 
and 
is of class
.
Furthermore, we assume that the system (6) satisfies the following generic conditions
:
(B2) The set 
is a 2-dimensional differentiable manifold and the set 
intersects the set
, which is a 
-dimensional differentiable manifold, transversely so that the generalized pli set 
is a 
-dimensional differentiable manifold.
(B3) The value of 
is nonzero at any point
.
(B4) The 
for any
, and the 
for any
. Then, the surface 
can be expressed as 
in the neighborhood of
. On the set
, 
or
, then 
and
, where we use the notations
, and
.
Assume
. On the set
, differentiating both sides of 
with respect to
,
(7)
where 
is a derivative with respect to
, thus the following is established:
(8)
On the other hand,
because of
. We can reduce the slow system to the following:
(9)
Using (8), the system (9) is described by
Put 
simply, then
(10)
where 
is a cofactor matrix of
, that is,
. 
is a cofactor of
.
The system (10) is the time scaled reduced system projected into
. Again, we assume the set 
.
(B5) All the singular points of the system (10) are nondegenerate, that is, the matrix induced from the corresponding linearized system at the singular point has distinct nonzero eigenvalues.
Remark All these points are contained in the set
, which is called the set of generalized pseudo singular points.
As this approach transforms the original system to the time scaled reduced system directly, it is called a direct method.
Definition 4.1 Let 
and
, 
be two eigenvalues of the matrix associated with the linearized system of 
at
. The point 
is called generalized pseudo singular saddle if 
and called generalized pseudo singular node if 
or
. It is called generalized pseudo singular focus if they are compex conjugate.
Now, we have to give a description on the definition of the duck solution in 
along the direct method. The method induces a 
-dimensional projected space directly. Note that we can also once project the original system into a 
-dimensinal space. It is called the indirect method.
Definition 4.2 Let a point 
be in GPS. If a trajectory follows first the attractive surface before this point and the saddle point, and then it goes along the slow manifold, which is not infinitesimal, it is called a duck solution in
.
Furthermore, we assume that the following.
(B6) We assume that there exists the set co-GPL, which may contain GPS and then the transversality condition is also established on co-GPL. In the situation, we assume that the invariant manifold through GPS intersects GPL and co-GPL transversely.
Definition 4.3 If the trajectory near the point of GPS passes through along the slow manifold with not infinitesimal and after that it jumps away, it is called a single duck solution. If there exists a co-GPL in 
within the interval, it is called a double duck solution.
Remark The first part of Definition 4.3 ensures that only one of the eigenvalues of the matrix 
on the slow manifold takes zero on GPS, because the fast vector field has saddle after GPS. On another GPL, however, the other eigenvalue takes zero. Note that these two eigenvalues of 
are negative when the fast vector field is attractive, and are positive when it is repulsive. It occurs such a state satisfying the assumption
. When they have different sign, it is saddle.
5. Lemmas
In this section, we give two Lemmas to make it clear the structure of the 4-dimensional system and the 3-dimensional projected system.
Let the latter of 
be satisfied, then the following two projected systems (11), (12) in 
are induced. We assume that 
are limited, that is, 
tends to zero as 
tends to zero.
(11)
since the relation 
is established from the above assumption. First, we can analyze the vector field of the system (11) on the constrained surface. Then, we use 
instead of 
as an approximation. Because we have to avoid redundancy for the system as is using
. Actually, we need the above condition: 
are limited, in such a case. Therefore, this approach is called an indirect method. Using the other relation
, we can get the following:
(12)
Lemma 5.1 The transversality condition 
is established if and only if the transversality condition 
in Section 
is satisfied in the systems (12) and (11) at the common pseudo singular point.
Lemma 5.2 The system (11) or (12) have a pseudo singular saddle (or pseudo singular node) point, if the system (6) has a generalized pseudo singular saddle or node point and if the trajectory follows first the attractive surface before this point and saddle or repulsive one after the point having
, or 
on
.
5.1. Proof of Lemma 5.1
Let 
denote a gradient vector of
. The transversality between 
and 
at the generalized pseudo singular point 
is checked as follows:
(13)
The transversality between 
and 
in the system (11) and (12) are checked as follows. Put
and then put
(14)
where
,
(15)
As the gradient vectors satisfy the relation (13), 
holds. In fact, the gradient vectors in (14) and (15) are independent, since the assumption 
ensures that only the coordinates are changed. Conversely, pulling back the equations (14), (15) to
, that is, embedding the corresponding 2-dimensional manifold into the original
, we can confirm that the relation (13) holds. In fact, the second equation in (14), (15) is equivalent to the third one in (13). The proof is complete.
5.2. Proof of Lemma 5.2
Let the original system have a generalized pseudo singular saddle point
, that is, the point 
is a singular point of the system (10) satisfying
Note that this system is described on the constrained surface.
Now, let us pull it back to the system in
. In the case of
, or
, using the assumption
, the following slow-fast system describes the current state.
(16)
and using the assumption
,
(17)
The above systems look like having a 1-dimensional slow manifold in
, however, they are tangent each other, because they have a still 
-dimensional differentiable manifold in
. Therefore, the orbits of the linearized systems (16), (17) are equivalent to the eigenvectors of the time scaled reduced system in the system (10).
The condition 
on the set 
ensures that the sign of 
changes 
to 
when one of the eigenvalues in 
on the slow manifold changes as the same sign. If 
on the set 
changes the sign, then the value of 
changes in the same way. Therefore, the system (11) has a pseudo singular saddle.
In fact, the system (16) is equivalent to the system (11) and the system (17) is also equal to the system (12). In the case of the node point, the proof is similar. The proof is complete.
6. Local Models
In this section, we shall give the following two theorems through a local model in
. See [1].
Theorem 6.1 Let 
be saddle or node. If the matrix 
has one zero eigenvalue and the other one has negative with a local model satisfying the conditions:
,
, there exists a duck solution in
.
(Proof) As only one of the eigenvalues of the matrix 
on the slow manifold takes zero on GPS, the assumptions
, 
ensure that two eigenvalues of 
are negative in the fast vector field before GPS. They are maybe negative, respectively positive after GPS. When each coefficient on GPS is limitted, a local model shows a precise structure as an approximation of the original system. Then, the property on GPS reflects directly the whole system. It can be shown that the time scaled reduced system 
is an apprximated one with a singular solution of the whole system
, because the corresponding solutions are very close to each other under the only two conditions. Therefore, we can conclude that there exists a duck solution.
Let 
be saddle or node. When changing the variables correspond to microscopes
:
, 
, the original system is reduced to the system with variables
, 
, 
,
. Then there exist local models which describe the 
-dimensional duck solutions.
Theorem 6.2 If the system has a square-linear solution in a local model, for any
, there exist essentially two local models describing the explicit duck solutions.
(Proof)
In the case
, 
, 
, 
, changing variables:
(18)
we reduce the system as well in (19) as well in (20).
(19)
Multiplying the right hand side of the system (19) by
,
(20)
In fact, doing time scaling
, then
. It is easy to show that Formula (20) is equivalent to (19).
By using the assumptions 
and
, we construct a local model under the most simple conditions:
(21)
Putting 
infinitesimal to δ simply, that is δ = L(ε3)
(22)
where
,
,
.
Note that the conditions 
imply that 
is saddle. See Definition 4.3. The corresponding solutions in the local model are as follows: when
,
(23)
when
,
(24)
In the case
, 
, 
, 
, changing variables:
(25)
we construct a local model under the conditions:
(26)
The corresponding local model is
(27)
where
,
,
,
,
.
Notice that we assume again that
, because the fast vector field has one zero eigenvalue and the other one is negative. The corresponding solutions in the local model are as follows: when
,
(28)
when
,
(29)
In another case, it is impossible to get an explicit solution with a square-linear one but a cubic-linear (or much higher order) one.
In this approach, an invertible affine transformation must be needed for a general point
, because the conditions (21), (26) are assumed at only
. These conditions may not be satisfied at the general pseudo singular point. We have to change the coordinates from the point 
to
. Notice that we do not know if the corresponding affine transformation keeps the conditions (21). In many cases, however, it is feasible.
7. Remark
It is easy to find that any solutions 
at the same time 
in (23) and (24) are very near. This fact implies that the time scaled reduced system is an approximated one. As blowing up the coordinates, the microscopes give a freedom on the solutions with respect to the initial values. The corresponding local model has higher possible polynomial solutions (not to be unique generally). We choose the smallest polynomial order.
Acknowledgements
We would thank I. V. D. Berg who read through our preprint carefully and gave many suggestions to make it better. H. Nishino and Dr student H. Miki gave us valuable comments especially in the Section 6.