Open access peer-reviewed chapter

# 4-Dimensional Canards with Brownian Motion

Written By

Shuya Kanagawa and Kiyoyuki Tchizawa

Reviewed: 20 December 2021 Published: 13 February 2022

DOI: 10.5772/intechopen.102151

From the Edited Volume

Edited by Francisco Bulnes

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## Abstract

Generally speaking, it is impossible to analyze slow-fast system with Brownian motion. If it becomes possible to do using a new approach, we can evaluate the rigidity of the original system. What kind of the behavior of such a system we have? Using non-standard analysis, on a“hyper finite time line” by Anderson, the Brownian motions are described by step functions. Then, the original differential equations are described by the difference equations due to using non-standard analysis. When constructing the difference equations, the corresponding measure is extended topologically. Because the interval of the difference is according to the hyper finite time line, the topological space is well defined. In this paper, we propose a two-region economic model with Brownian motions. This concrete example gives us new results.

### Keywords

• canard solution
• slow-fast system
• nonstandard analysis
• Brownian motion
• stochastic differential equation

## 1. Introduction

Consider a slow-fast system in R4 with a 2-dimensional slow manifold:

εdxdt=hxyεdydt=gxyε,E1

where ε is infinitesimal and

g=g1g2:R4R2,h=h1h2:R4R2,x=xt=x1x2R2,y=yt=y1y2R2.

The above expression form is based on Nelson’s [1]. The slow-fast system (1) is applied to many fields, e.g. electronic circuits, neuron systems, etc. In these applications, effectiveness of random noises always exists. We take up this point of view as being one of the main problems.

Now, let us consider a stochastic differential equation for a slow-fast system with a Brownian motion Bt as the random noises modifying the slow fast system (1): For t0T,T>0

εdx=hxyεdtdy=gxyεdt+σdB,E2

where B=B1B2R2 is a 2-dimensional standard Brownian motion and σ>0 is a positive constant which gives a standard deviation for the Brownian motion Bt.

Since the Brownian motion Bt is almost surely non-differentiable everywhere, it is difficult to analyze slow-fast system (2).

On the other hand, Anderson [2] showed that the Brownian motion is described by step functions using non-standard analysis on a hyper finite time line by the following definition. (See also [3, 4]).

Definition 1. Let Nt=tΔt,0tT and N=NT. Assume that a sequence of i.i.d. random variables ΔBkk=1N has the distribution

PΔBk=Δt=PΔBk=Δt=12E3

for each k=1,,N. An extended Wiener process Btt0 is defined by

Bt=k=1NtΔBk,0tT.E4

Rewriting the system (2) via step functions on the hyper finite time line, the following system (5) is obtained.

εxnxn1=hxn1yn1εΔtynyn1=gxn1yn1εΔt+σΔBn,E5

for n=1,2,,N, where ΔBn=BnΔtBn1Δt, Δt=TN and N is a hyper finite natural number.

Since the system (5) is equivalent to the system (2), taking B(t) in Definition 1, we prove the existence of the solution for the system (2) in Section 3.

## 2. Slow-fast system in R4 with co-dimension 2

We assume that the system (1) sastisfies the following conditions (A1) (A5):

(A1) h is of class C1 and g is of class C2.

(A2) The slow manifold S=xyR4hxy0=0 is a two-dimensional differential manifold and intersects the set V=xyR4dethxxy0=0 transversely.

Then, the pli set PL=xySV is a one-dimensional differentiable manifold.

(A3) Either the value of g1 or that of g2 is nonzero at any point of PL.

Note that the pli set PL devides the slow manifolds S\PL into three parts depending on the signs of the two eigenvalues of hxxy0.

First consider the following reduced system which is obtained from (1) with ε=0:

0=hxy0dydt=gxy0.E6

By differentiating hxy0 with respect to t, we have

hxxy0dxdt+hyxy0gxy0=0.E7

Then (6) becomes the following:

dxdt=hxxy01hyxy0gxy0dydt=gxy0,E8

where xyS\PL. To avoid degeneracy in (8), we consider the following system:

dxdt=dethxxy01hxxy01hyxy0gxy0dydt=dethxxy01gxy0.E9

The phase portrait of the system (9) is the same as that of (8) except the region where dethxxy0=0, but only the orientation of the orbit is different.

Definition 2. A singular point of (9), which is on PL, is called a pseudo singular point of (1).

(A4) rankhxxy0=2 for any xyS\PL.

From (A4), the implicit function theorem guarantees the existence of a unique function y=ξx such that hxξx0=0. By using y=ξx, we obtain the following system:

dxdt=dethxxξx01hxxξx01hyxξx0gxξx0.E10

(A5) All singular points of (10) are non-degenerate, that is, the linearization of (10) at a singular point has two nonzero eigenvalues.

Definition 3. Let λ1,λ2 be two eigenvalues of the linearization of (10) at a pseudo singular point. The pseudo singular point with real eigenvalues is called a pseudo singular saddle point if λ1<0<λ2 and a pseudo singular node point if λ1<λ2<0 or λ1>λ2>0.

The following theorem is established (see, e.g. [5]).

Theorem 1. Let x0y0 be a pseudo singular point. If tracehxx0y00<0, then there exists a solution which first follows the attractive part and the repulsive part after crossing PL near the pseudo singular point.

Remark 1. The condition tracehxx0y00<0 implies that one of eigenvalues of hxx0y00 is equal to zero and the other one is negative. Notice that the system has two kinds of vector fields: one is 2-dimensional slow and the other is 2-dimensional fast one. The condition provides the state of the fast vector field.

Remark 2. The singular solution in Theorem 1 is called a canard in R4 with 2-dimensional slow manifold. As a result, it causes a delayed jumping. The study of canards requires still more precise topological analysis on the slow vector field.

In the next section, we show that a canard exists for the system (2) in which the orbit of the canard of the system (1) is moved to another one by a Brownian motion Bt.

## 3. Canards with Brownian motion

Let us prove the following theorem.

Theorem 2. In the system (3), if there exists kn such that

xnxn1εkn,n=1,2,,NE11

and

sup1nNknKE12

for some K hyper finite, then there exists a solution of (5) which is called canard in the sense of Remark 2.

Proof. From the condition (11), we have

xnxn1=1εhxn1yn1εΔtεkn.E13

ε is an arbitrary constant, therefore putting ε=1N we have from (13)

hxn1yn1εε2knΔtknN,E14

for each n1.

From Definition 2, the following is satisfied for the pseudo-singular point x0y0 of (1);

dethxx0y001hxx0y001hyx0y00gx0y00=0hx0y00=0.E15

Assume that σ2 in the Brownian motion Bt is sufficiently small. Let xξyξ be a pseudo-singular point of (2) or (5). Note that (2) is equivalent to (5) in the sense of nonstandard analysis. Then there exists a positive number ξ such that

dethxxξ,yξ01hxxξ,yξ01hyxξ,yξ0gxξ,yξ0Δt+σΔBξ0,E16

where Δt=TN .

In this situation, as σ0

xξyξx0y0.E17

Therefore, the eigenvalues of the linearized system (2) at the point xξyξ keeps the almost same as the eigenvalues of the system (1) at the point x0y0.

On the other hand, there exists a canard of (1) from Theorem 1. Since knN is small enough, the solution of (5) also first follows the attractive part and the repulsive part follows after crossing PL near the pseudo singular point like as the canard of (1).

## 4. Concrete models

### 4.1 Two-region business cycle model

As a concrete model, we consider a two-region business cycle model between two nations A and B including a Brownian motion Bt=B1tB2t as followings. See [6] for more details of the two-region business cycle model.

for 0tT, where x1t and x2t are exports of A and B, m1t and m2t are imports of A and B, y1tq1α and y2tq1α are national income identities of A and B for some constants q and α, respectively. (See [6] for more details.)

Now, let us introduce a difference equations for the system (18). Then, the relations Δt=TN and tk=kTN,k=0,1,,N are satisfied, where N is a hyper finite. Put

εΔtx1tkx1tk1=1α+m1θy1tk1+m2θy2tk1εθ+1αx1tk1a+1n1θφ1x1tk1a+n2θφ2x2tk1aεΔtx2tkx2tk1=m1θy1tk11α+m2θy2tk1+1n1θφ1x1tk1aεθ+1αx2tk1a+1n2θφ2x2tk1ay1tky1tk1=x1tk1aΔt+σB1tkB1tk1y2tky2tk1=x2tk1aΔt+σB2tkB2tk1.E19

Furthermore put

φ1xα=φ2xα=1αθx+x2x33.E20

### 4.2 Simulation results

In this section, let us provide computer simulations for the two-region business cycle model using the above Eqs. (19) and (20. In (19), we assume that two Brownian motions B1t and B2t are mutually independent and note that

B1tkB1tk1N0Δtσ12,B2tkB2tk1N0Δtσ22,E21

for each 1kTΔt.

Figure 1, except for the axes, shows the pli set PL=xySV with the pseudo singular point 11 of (1) defined by (9).

Putting some parameters in (19), we have the following results for some orbits of x1tx2t0tT satisfying the Eq. (1) or (2).

Figure 2 shows an orbit of x1tx2t0tT=0.8 satisfying the Eq. (5) with σ1=σ2=0 and starting from 0.80.8 near the pseudo singular point 11. In Figures 24, ε=0.01. From Figure 2 the speed of the orbit x1tx2t for 0t0.2 is not only very fast, but also the orbit jumps near the pseudo singular point 11. The orbit turns at the point 22 and returns on the line. ∆t = 0.001 in Figures 26.

Figure 3 shows an orbit of x1tx2t0tT=0.8 satisfying the Eq. (5) with σ1=σ2=0.4 and starting from 0.80.8 near the the pseudo singular point 11. From Figure 3 we observe that the orbit moves on the line from 0.80.8 and separates from the line at t=0.2 by the Brownian motion Bt.

Figure 4 shows an orbit of x1tx2t0tT=4.64 satisfying the Eq. (5) with σ1=σ2=0 and starting from 0.80.8 near the pseudo singular point 11. The orbit separates from the line at t=4.61.

Figure 5 shows an orbit of x1tx2t0tT=2.75 satisfying the Eq. (5) with σ1=σ2=0 and starting from 0.80.8 near the pseudo singular point 11, where ε=0.004. The orbit with ε=0.004 separates from the line at t=2.61. On the other hand the orbit with ε=0.01 separates from the line at t=4.61 in Figure 4. Therefore we see that the orbit changes according to ε.

Figure 6 shows an orbit of x1tx2t0tT=0.8 satisfying the Eq. (5) with σ1=σ2=0.4 and starting from 0.80.8 near the pseudo singular point 11, where ε=0.004. The orbit with ε=0.004 separates from the line at t=0.1. On the other hand, in Figure 3, the orbit with ε=0.01 separates from the line at t=0.2. Then, the orbit changes according to ε also in the non-random case.

## 5. Conclusion

Brownian motions are described by non-differentiable functions almost surely. In order to overcome the difficulty in the system (2) we consider the system (5) using nonstandard analysis. The system (5) makes us possible to analyze the canard with Brownian motions. As the difference equations is determined by according to the hyper finite time line, the measure is extended effectively to do this analysis. In Figures 16 obtained by the simulations, we observe the effects of Brownian motions which change the orbit of x1tx2t.

## Acknowledgments

The authors would like to express the reviewer’s comments which are useful to explain the structure of canards. The first author is supported in part by Grant-in-Aid Scientific Research (C), No.18 K03431, Ministry of Education, Science and Culture, Japan.

## The 2020 AMS classification

ordinary differential Eqs., dynamical systems and ergodic theory, difference and functional equations.

## References

1. 1. Nelson E. Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society. 1977;83:1165-1198
2. 2. Anderson RM. A non-standard representation for brownian motion and ito integration. Israel Journal of Mathematics. 1976;25(2):1546
3. 3. Kanagawa S, Tchizawa K. Proof of Ito’s Formula for Ito’s Process in Nonstandard Analysis. Applied Mathematics. 2019;10(2):561-567
4. 4. Kanagawa S, Tchizawa K. Extended wiener process in nonstandard analysis. Applied Mathematics. 2020;11(2):247254
5. 5. Tchizawa K. Four-dimensional canards and their center manifold. Extended Abstracts Spring 2018. Trends in Mathematics. 2019;11:193-199. Springer Nature Switzerland AG
6. 6. Miki H, Nishino H, Tchizawa K. On the possible occurrence of duck solutions in domestic and two-region business cycle models. Nonlinear Studies. 2012;18:39-55

Written By

Shuya Kanagawa and Kiyoyuki Tchizawa

Reviewed: 20 December 2021 Published: 13 February 2022