Bifurcation on 4-dimensional Canards with Hyper Catastrophe

Abstract: In 4-dimensional slow-fast system, under the condition of ”symmetry”, there exists ”structural stabil-

ity”. It is, however, of one parameter for the slow vector. On other parameters for all slow/fast vectors, it is not yet

discussed still now as it is very complicated geometrical structure. In the ”Hyper catastrophe on 4-dimensional

canards”, it is confirmed due to the existence of ”bifurcation”, because ”catastrophe” is a bifurcation problem

itself. In the beginning of catastrophe theory, the word ”structural stability” is used for the original differential

equations and not be used for the multi variable functions. In the slow-fast system having canards, what kinds of

structure are there? It is used for the parameter, which depends on the existence of canards. Through this paper,

it will become clear.

1 Introduction

Since 4-dimensional slow-fast system is analyzed on

”hyper finite time line” in [1] and [2] or done by using

”non-standard analysis”, it is called ”Hyper Catastro-

phe”. In the slow-fast system which includes a very

small parameter ϵ, it is difficult to do precise analysis.

Thus, it is useful to get the orbits as a singular limit.

When trying to do simulations, it is also faced with

difficulty due to singularity. Using very small time

intervals corresponding small ϵ, we shall overcome

the difficulty, because the ”difference equation” on

the small time interval adopts the standard ”differen-

tial equation”. These small intervals are defined on

hyper finite number N, which is nonstandard. As ϵ

and the intervals are linked to use 1/N, the simula-

tion should be done exactly. In our previous paper

”Hyper catastrophe on 4-dimensional canards” ([3]),

a neuron system induced from the FitzHugh-Nagumo

equation is taken up as a concrete system, but there

is no simulations. In this paper, bifurcation struc-

ture having catastrophe developed by [4], [5], [6] will

be described through the neuron system with simula-

tions. As a result, a new structure on the bifurcation

parameters b3,b4will be provided. For more details

of the catastrophe, see e.g. [7], [8], [9], [10], [11] and

[12].

2 Bifurcation on Slow/Fast Vectors

Let us consider the following system extended having

parameters for slow/fast vectors:











εdx

dt =h(αx, βy, ϵ)

dt =g(αx, βy, ϵ)

,(1)

where α= (b1, b2),β= (b3, b4).

The following is established, see [3],

det ∂h

∂x=b1b2∂h1

∂x1

∂h2

∂x2−∂h1

∂x2

∂h2

∂x1.(2)

In fact,

∂h1(b1x1, b2x2, b3y1, b4y2,0)

∂x1

=(3)

∂h1(x1, x2, b3y1, b4y2,0)

∂x1

∂h1(b1x1, b2x2, b3y1, b4y2,0)

∂x2

KIYOYUKI TCHIZAWA1, SHUYA KANAGAWA2

1Institute of Administration Engineering, Ltd.

Sotokanda 2-2-2, Chiyoda-ku, Tokyo 101-0021

JAPAN

2Department of Mathematics

Tokyo City University

1-28-1 Tamazutsumi, Setagaya-ku,Tokyo 158-8557

JAPAN

.H\:RUGVFDQDUGVIO\LQJGLPHQVLRQDOVORZIDVWV\VWHPK\SHUFDWDVWURSKH

Received: April 25, 2024. Revised: September 17, 2024. Accepted: October 9, 2024. Published: November 6, 2024.

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∂h1(x1, x2, b3y1, b4y2,0)

∂x2

∂h2(b1x1, b2x2, b3y1, b4y2,0)

∂x2

∂h2(x1, x2, b3y1, b4y2,0)

∂x2

∂h2(b1x1, b2x2, b3y1, b4y2,0)

∂x2

∂h2(x1, x2, b3y1, b4y2,0)

∂x2

Lemma 1.

det ∂h

∂x=∂h1

∂x1

∂h2

∂x2−∂h1

∂x2

∂h2

∂x1

= 0.(4)

It does not depend on bifurcation parameter b1, b2.

Theorem 1. If the system having parameters for

slow/fast vectors is ”symmetric”, then it has a poten-

tial classified by [4].

Proof. Under the rank condition (A4) in [3], proceed-

ing a projection (changing the co-ordinates):

X

Y=P









, P =1 1 0 0

0 0 1 1(5)

like as y1+y2=Y, x1+x2=X, then, the potentials

are obtained as ”elementary catastrophe” under the

following conditions.

At around (x0, y0)∈P S,

(i) ∂3h1

∂x3

1= 0,∂h1

∂x1= 0.

(ii) ∂4h1

∂x4

1= 0,∂2h1

∂x2

1= 0,∂h1

∂x1= 0.

(iii) ∂5h1

∂x5

1= 0,∂2h1

∂x2

1= 0,∂h1

∂x1= 0,∂4h1

∂x4

= 0 .

(iv) ∂6h1

∂x6

1= 0,∂4h1

∂x4

1= 0,∂3h1

∂x3

1= 0,

∂2h1

∂x2

1= 0,∂h1

∂x1= 0.

(v) ∂3h1

∂x2

1∂x2= 0,∂3h2

∂x3

2= 0,∂2h1

∂x2

1= 0,

∂h2

∂x2= 0,∂3h1

∂x2

2∂x1

= 0,∂2h1

∂x1∂x2

= 0.

(vi) ∂3h1

∂x2

1∂x2= 0,∂3h2

∂x3

2= 0,∂2h1

∂x2

1= 0,

∂h1

∂x2= 0,∂h1

∂x1= 0.

(vii) ∂3h1

∂x2

1∂x2= 0,∂4h2

∂x4

2= 0,∂2h2

∂x2

2= 0,

∂2h1

∂x1

2= 0,∂h2

∂x2= 0,∂h1

∂x1= 0.

Remark 1. As the system is ”symmetric”, the condi-

tions are described exclusively, for example, the con-

dition (1) is as the following,

∂3h2

∂x3

2= 0,∂h2

∂x2= 0.

Remark 2. It is called ”Hyper Catastrophe”, which

is composed of the potential reduced from the slow

manifold ( ϵ= 0). Although it is using non-standard

analysis, for example ϵis infinitesimal, ”Transfer

Principle” ensures that it is established in standard

analysis. Then, the slow manifold is obtained as the

singular limit (ϵtends to zero). They are ”dynamical

catastrophe” but not ”statical one”.

Theorem 2. The capital Y=y1+y2in the equation

(5) includes bifurcation parameters αand βimplic-

itly. Because the implicit function theorem ensures

that there exists a function y=ϕ(x, α, β)where

y= (yi=ϕi(x, b1, b2, b3, b4))(i= 1,2) by tak-

ing ϵ= 0. Then, the corresponding capital Xaxis,

which satisfies Y= 0, sometimes changes the sign,

i.e., switching the direction. It causes another bifur-

cation.

3 Concrete Example

Consider the equation, for 0⩽t⩽T











εdx1

dt =b2x2+b3y1−b3

1x3

εdx2

dt =b1x1+b4y2−b3

2x3

dy1

dt =−1

c(b1x1+b3y1)

dy2

dt =−1

c(b2x2+b4y2)

,(6)

where bi(i= 1, ..., 4) are bifurcation parameters and

cis a positive constant. Changing the coordinates by

X

Y=Px

y, P =1 1 0 0

0 0 3b3

3b4

2,(7)

like as











x1+x2=X

3b3

y1+3b4

y2=Y

,(8)

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then on the axis X, getting the function

Y=x3

1+x3

2−3b2

x2−3b1

x1.(9)

This potential function is ”hyperbolic umbilic” clas-

sified by [4].

Using variables X, Y , when satisfying b1=b2,

Y=X3−kX +O(X3)and k=3

.(10)

Remark 3. One of the pseudo singular point

(x0, y0) = (1,−1) is structurally stable, and the

other one is (1,1) is unstable. At around the pseudo

singular point, the above function keeps approxima-

tion. In [3], p201, the equations (17), (18), (19) are

mistyped, and the above form is correct one.

Theorem 3. The system induced from FitzHugh-

Nagumo equation has ”Hyper catastrophe” at around

the pseudo singular point. The multi variable function

does not depend on the parameter b4but depends on

b1, and b2. If b4<−3/4 the pseudo singular point is

saddle, and it is structurally stable. If −3/4 < b4<0

it is node, which is unstable. In case b4>0, they are

unstable.

Theorem 4. The changing coordinates in (8) includes

another bifurcation structure with respect to the pa-

rameters b3and b4.

Proof. On the capital Xaxis, that is Y= 0, there

exist two cases 3b3

3b4

>0 (or < 0).

Remark 4. Take a notice that y1=b3

3b3

1−b2

x2,

and y2=b3

3b4

2−b1

x1are satisfied.

Remark 5. The potential function depends on the pa-

rameters b1, and b2. It implies that the value of b4may

take some constant on the function. The condition on

b4is ensured in [3].

4 Simulation results induced from

FitzHugh-Nagumo equations

In this section, let us provide computer simulations for

the equation (6) using the following difference equa-

tion. For tk=k∆t, k = 1,2,··· , NT,











ε{x1(tk)−x1(tk−1)}

={b2x2(tk−1) + b3y1(tk−1)

−b1x1(tk−1)3

3∆t

ε{x2(tk)−x2(tk−1)}

={b1x1(tk−1) + b4y2(tk−1)

−b2x2(tk−1)3

3∆t

y1(tk)−y1(tk−1)

=−1

c{b1x1(tk−1) + b3y1(tk−1)}∆t

y2(tk)−y2(tk−1)

=−1

c{b2x2(tk−1) + b4y2(tk−1)}∆t

,(11)

where ∆t= 1/Nand N is a hyper number in the

sense of nonstandard. When doing simulations N

takes standard number in the equation (11).

In Figure 1, Figure 2, Figure 3, Figure 4 and

Figure 5 in the Appendix, the line x1=x2is an

invariant manifold and two red points are pseudo

singular points. Furthermore, ε= 0.01,c= 1 and

∆t= 0.0001 in (11). The curves, which satisfy

x1x2= 1 and x1x2=−1, respectively, are Pli set.

Figure 1. Figure 1 in Appendix shows an orbit

of {(x1(t), x2(t)) ,0≤t≤T= 5}satisfy-

ing the equation (11) with b1=b2=b3=

b4= 1 and starting from (x1(t0), x2(t0)) =

(0.5,1.5) near the pseudo singular point

1

23−√5,1

23 + √5. The orbit con-

verges to the invariant manifold x1=x2. Then,

corresponding potential is Y=x3

1+x3

2−3x1−3x2.

Figure 2. Figure 2 in Appendix shows an orbit

of {(x1(t), x2(t)) ,0≤t≤T= 5}satisfying the

equation (11) with b1= 1.4, b2= 0.6, b3= 0.6, b4=

1.4and starting from (0.5,1.5) near the pseudo

singular point 1

23−√5,1

23 + √5.

From Figure 2 (Appendix) we observe that the orbit

converges to the invariant manifold x1=x2.

Figure 3. Figure 3 in Appendix shows an orbit

of {(x1(t), x2(t)) ,0≤t≤T= 5}satisfying

the equation (11) with b1= 0.6, b2= 0.8, b3=

−0.6, b4=−0.8and starting from (−1.2,1.2) near

the pseudo singular point.

Figure 4. Figure 4 in Appendix shows an orbit

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of {(x1(t), x2(t)) ,0≤t≤T= 5}satisfying the

equation (11) with b1= 1.4, b2= 0.8, b3= 0.1, b4=

0.8and starting from (0.8,1.5) near the pseudo singu-

lar point. From Figure 4 (Appendix) we observe that

the orbit converges to the invariant manifold x1=x2.

Figure 5. Figure 5 in Appendix shows an orbit

of {(x1(t), x2(t)) ,0≤t≤T= 5}satisfying

the equation (11) with b1= 0.6, b2= 0.8, b3=

−0.1, b4=−0.8and starting from (1.2,−1.2) near

the pseudo singular point.

5 Conclusion

In the system induced from the FitzHugh-Nagumo

equation, when b1=b2=b3= 1 it is composed

of only one parameter b4. Then, this state is quite

the same as the system in [3]. Bifurcation problem

on 4-dimensional canards makes its appearance

through constructing ”Hyper catastrophe”, which

is a dynamical model, not a statical one. Notice

that there is no parameter b4in the multi variable

function but it is fixed. Notice that Figure 3 and

Figure 5 (Appendix), which satisfy b3<0,b4<0,

provide a new jumping direction along x2=−x1.

The parameters b3,b4give a new bifurcation along

the orthogonal complement of the invariant set

different from our previous paper. When satisfy-

ing b3= 1,b4changes the positive sign to negative

one, corresponding canards are flying on the function.

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[3] K. Tchizawa and S. Kanagawa, (2024) Hyper

Catastrophe on 4-Dimensional Canards, Advances

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[11] Th. Br¨ocker and L. Lander, (1975) Dif-

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5HIHUHQFHV

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-2-1 1 2

-2

-1

Figure 1: (x1(0), x2(0)) = (0.5,1.5),b1=b2=

b3=b4= 1

-1.5 -1.0 -0.5 0.5 1.0

-3

-2

-1

Figure 2: (x1(0), x2(0)) = (0.5,1.5),b1= 1.4, b2=

0.6, b3= 0.6, b4= 1.4

-3-2-1 1 2 3

-3

-2

-1

Figure 3: (x1(0), x2(0)) = (1.2,−1.2),b1=

0.6, b2= 0.8, b3=−0.6, b4=−0.8

-2-1 1

-2

-1

Figure 4: (x1(0), x2(0)) = (0.5,1.5),b1= 1.4, b2=

0.8, b3= 0.1, b4= 0.8

-1 1 2

-2

-1

Figure 5: (x1(0), x2(0)) = (1.2,−1.2),b1=

0.6, b2= 0.8, b3=−0.1, b4=−0.8

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