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4-Dimensional Canards with Brownian Motion

Written By

Shuya Kanagawa and Kiyoyuki Tchizawa

Reviewed: December 20th, 2021Published: February 13th, 2022

DOI: 10.5772/intechopen.102151

Advanced Topics of TopologyEdited by Francisco Bulnes

From the Edited Volume

Advanced Topics of Topology [Working Title]

Dr. Francisco Bulnes

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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.


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

1. Introduction

Consider a slow-fast system in R4with a 2-dimensional slow manifold:


where εis infinitesimal and


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 Btas the random noises modifying the slow fast system (1): For t0T,T>0


where B=B1B2R2is a 2-dimensional standard Brownian motion and σ>0is a positive constant which gives a standard deviation for the Brownian motion Bt.

Since the Brownian motion Btis 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.LetNt=tΔt,0tTandN=NT. Assume that a sequence of i.i.d. random variablesΔBkk=1Nhas the distribution


for each k=1,,N. An extended Wiener process Btt0is defined by


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


for n=1,2,,N, where ΔBn=BnΔtBn1Δt, Δt=TNand Nis 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 R4with co-dimension 2

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

(A1) his of class C1and gis of class C2.

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

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

(A3) Either the value of g1or that of g2is nonzero at any point of PL.

Note that the pli set PLdevides the slow manifolds S\PLinto 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:


By differentiating hxy0with respect to t, we have


Then (6) becomes the following:


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


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=2for any xyS\PL.

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


(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,λ2be 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<λ2and a pseudo singular node point if λ1<λ2<0or λ1>λ2>0.

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

Theorem 1.Let x0y0be 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<0implies that one of eigenvalues of hxx0y00is 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 R4with 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 existsknsuch that




for some Khyper 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


εis an arbitrary constant, therefore putting ε=1Nwe have from (13)


for each n1.

From Definition 2, the following is satisfied for the pseudo-singular point x0y0of (1);


Assume that σ2in the Brownian motion Btis sufficiently small. Let xξyξbe 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 ξsuch that


where Δt=TN.

In this situation, as σ0


Therefore, the eigenvalues of the linearized system (2) at the point xξyξkeeps 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 knNis small enough, the solution of (5) also first follows the attractive part and the repulsive part follows after crossing PLnear 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=B1tB2tas followings. See [6] for more details of the two-region business cycle model.


for 0tT, where x1tand x2tare exports of A and B, m1tand m2tare imports of A and B, y1tq1αand y2tq1αare national income identities of A and B for some constants qand α, respectively. (See [6] for more details.)

Now, let us introduce a difference equations for the system (18). Then, the relations Δt=TNand tk=kTN,k=0,1,,Nare satisfied, where Nis a hyper finite. Put


Furthermore put


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 B1tand B2tare mutually independent and note that


for each 1kTΔt.

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

Figure 1.

Pli set PL.

Putting some parameters in (19), we have the following results for some orbits of x1tx2t0tTsatisfying the Eq. (1) or (2).

Figure 2 shows an orbit of x1tx2t0tT=0.8satisfying the Eq. (5) with σ1=σ2=0and starting from 0.80.8near the pseudo singular point 11. In Figures 24, ε=0.01. From Figure 2 the speed of the orbit x1tx2tfor 0t0.2is not only very fast, but also the orbit jumps near the pseudo singular point 11. The orbit turns at the point 22and returns on the line. ∆t = 0.001 in Figures 26.

Figure 2.


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

Figure 3.


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

Figure 4.


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

Figure 5.


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

Figure 6.



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.



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.


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Written By

Shuya Kanagawa and Kiyoyuki Tchizawa

Reviewed: December 20th, 2021Published: February 13th, 2022