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In a previous paper we have proposed a new method for proving the existence of “canard solutions” for three- and four-dimensional singularly perturbed systems with only one

In the beginning of the eighties, Benoît and Lobry [

In a previous paper entitled “Canards Existence in Memristor’s Circuits” (see Ginoux and Llibre [

The aim of this work is to extend this method to the case of four-dimensional singularly perturbed systems with

The outline of this paper is as follows. In Section

According to Tikhonov [

In the case when

In general we consider that

The dot represents the derivative with respect to the new independent variable

The independent variables

In such case system (

The

Such a normally hyperbolic invariant manifold (

When

According to Fenichel [

A

A

A

According to Wechselberger [

Canards are special class of solutions of singularly perturbed dynamical systems for which normal hyperbolicity is lost. Canards in singularly perturbed systems with two or more slow variables

In order to characterize the “slow dynamics,” that is, the slow trajectory of the

Since, according to Fenichel [

By replacing

This justifies the introduction of the

Now, let

Then, by rescaling the time by setting

Let us notice that Argémi [

By application of the Implicit Function Theorem, let suppose that we can explicitly express from (

As recalled by Guckenheimer and Haiduc [

Thus, for dimensions higher than three, his concept encompasses that of Argémi. Moreover, Wechselberger [

Four-dimensional

The critical manifold equation of system (

By application of the Implicit Function Theorem, let us suppose that we can explicitly express from ((

The

Equations (

By solving the system of (

So, we have the following constrained system:

By rescaling the time by setting

Then, since we have supposed that

Let us notice on one hand that (

The Jacobian matrix of system (

Without loss of generality, it seems reasonable to extend Benoît’s generic hypotheses introduced for the three-dimensional case to the four-dimensional case. So, first, let us suppose that by a “standard translation” the

Then, let us make the following assumptions for the nondegeneracy of the

According to these generic hypotheses ((

Thus, we have the following Cayley-Hamilton eigenpolynomial associated with such Jacobian matrix (

But, according to Wechselberger [

Let

Condition

Following the works of Wechselberger [

We establish in the Appendix for any four-dimensional singularly perturbed systems (

Thus, in his paper Wechselberger [

In the folded saddle case of system (

See Wechselberger [

Since our method does not use the “desingularized vector field” (

If the normalized slow dynamics (

By making some smooth changes of time and smooth changes of coordinates (see the Appendix) we brought system (

Then, we deduce that the condition for the

So, the condition for which the

So, Proposition

The FitzHugh-Nagumo model [

The slow manifold equation of system (

Then, by rescaling the time by setting

From (

According to Tchizawa and Campbell [

The Jacobian matrix of system (

Although the

According to (

Thus, according to Proposition

So, we have conditions

Let us choose arbitrarily

Let us notice that the

The Jacobian matrix

Although the

According to (

Thus, according to Proposition

So, we have conditions

Let us choose arbitrarily

Because of the symmetry of these coupled FitzHugh-Nagumo equations, the Jacobian matrix of system (

The original Hodgkin-Huxley model [

where:

The first equation (

Let us notice that the variables and symbols in ((

According to Suckley and Biktashev [

By using (

Now, in order to apply the

Graph of ^{−1}) against

Figure

So, according to Awiszus et al. [

Then, by posing

Let us notice that the multiplicative parameter

According to the

Then, the

The

The Jacobian matrix of the

So, the determinant of the Jacobian matrix of the

Thus, the condition for the

Therefore

By subtracting (

Plugging this value of

So, the

The

Let us notice that (

Moreover, (

Thus, it appears that (

So, following their works, let us plot the function

Function

Let us notice that this plot (the function

We observe from Figure

This value corresponds to a voltage

For

According to Proposition

According to (

Thus, according to Proposition

So, according to Proposition

In Figures

Phase portrait, canard solution, and

Phase portrait, canard solution, and

Phase portrait, canard solution, and

In a previous paper entitled “Canards Existence in Memristor’s Circuits” (see Ginoux and Llibre [

Changes of coordinates leading to the

Let us consider the four-dimensional

By taking into account extension of Benoît’s generic hypothesis ((

Then, let us make the standard polynomial change of variables:

From (

The time derivative of system (

Then, multiplying the third and fourth equation of (

Since

Then, by replacing in (

Finally, we deduce

This is the result we established in Section

The authors declare that there is no conflict of interests regarding the publication of this paper.

One of the authors (Jaume Llibre) is partially supported by a MINECO Grant MTM2013-40998-P, an AGAUR Grant no. 2014SGR-568, and Grants FP7-PEOPLE-2012-IRSES 318999 and 316338.