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This paper gives the existence of a duck solution in a slow-fast system in *R*^{2+2} using two ways. One is an indirect way and the other is a direct way. In the indirect way, the original system is once reduced to the slow-fast system in *R*^{2+1}. In the direct one, it has a 4-dimensional duck solution when having an efficient local model. This is already published in [1,2]. Some sufficient conditions are given to get such a good model.

In the

In this section, we shall review some results in Zvonkin and Shubin [6,7]. Let us consider the following system of differential equations

where

Definition 2.1 A solution

We give a necessary condition for the existence of a duck solution close to the extremum point

Proposition 2.2 If there is a duck solution of the system (1) close to the extremum point

We finally obtain the following proposition concerning the existence of duck solutions.

Proposition 2.3 Suppose that

We shall introduce

where

(A1)

intersects the set

(A3)

Let

where

where

As the system (4) is well defined at any point of

(A4) For any point

If we take

(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

Definition 3.1 Let

From now on we use IST [

Definition 3.2 The solution

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

Now, let us consider a slow-fast system (6):

where

First, we assume the following condition

(B1)

Furthermore, we assume that the system (6) satisfies the following generic conditions

(B2) The set

(B3) The value of

(B4) The

for any

Assume

where

On the other hand,

because of

Using (8), the system (9) is described by

Put

where

The system (10) is the time scaled reduced system projected into

(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

As this approach transforms the original system to the time scaled reduced system directly, it is called a direct method.

Definition 4.1 Let

Now, we have to give a description on the definition of the duck solution in

Definition 4.2 Let a point

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

Remark The first part of Definition 4.3 ensures that only one of the eigenvalues of the matrix

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

since the relation

Lemma 5.1 The transversality condition

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

Let

The transversality between

and then put

where

As the gradient vectors satisfy the relation (13),

Let the original system have a generalized pseudo singular saddle point

Note that this system is described on the constrained surface.

Now, let us pull it back to the system in

and using the assumption

The above systems look like having a 1-dimensional slow manifold in

The condition

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.

In this section, we shall give the following two theorems through a local model in

Theorem 6.1 Let

(Proof) As only one of the eigenvalues of the matrix

Let

Theorem 6.2 If the system has a square-linear solution in a local model, for any

(Proof)

In the case

we reduce the system as well in (19) as well in (20).

Multiplying the right hand side of the system (19) by

In fact, doing time scaling

By using the assumptions

Putting ^{3})

where

Note that the conditions

when

In the case

we construct a local model under the conditions:

The corresponding local model is

where

Notice that we assume again that

when

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

It is easy to find that any solutions

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.