## 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

where

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

where

Since the Brownian motion

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]).

for each

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

for

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 R 4 with co-dimension 2

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

(A1)

(A2) The slow manifold

Then, the pli set

(A3) Either the value of

Note that the pli set

First consider the following reduced system which is obtained from (1) with

By differentiating

Then (6) becomes the following:

where

The phase portrait of the system (9) is the same as that of (8) except the region where

** Definition 2.** A singular point of (9), which is on

(A4)

From (A4), the implicit function theorem guarantees the existence of a unique function

(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

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

** Theorem 1.** Let

** Remark 1.** The condition

** Remark 2.** The singular solution in Theorem 1 is called a canard in

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

## 3. Canards with Brownian motion

Let us prove the following theorem.

* In the system* (3)

, if there exists

such that

and

for some

* Proof.* From the condition (11), we have

for each

From Definition 2, the following is satisfied for the pseudo-singular point

Assume that

where

In this situation, as

Therefore, the eigenvalues of the linearized system (2) at the point

On the other hand, there exists a canard of (1) from Theorem 1. Since * 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

for

Now, let us introduce a difference equations for the system (18). Then, the relations

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

for each

Figure 1, except for the axes, shows the pli set

Putting some parameters in (19), we have the following results for some orbits of

Figure 2 shows an orbit of

Figure 3 shows an orbit of

Figure 4 shows an orbit of

Figure 5 shows an orbit of

Figure 6 shows an orbit of

## 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 1–6 obtained by the simulations, we observe the effects of Brownian motions which change the orbit of

## 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.

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