On a six-dimensional Artificial Neural Network Model

INNA SAMUILIK

Department of Engineering Mathematics

Riga Technical University

LATVIA

Abstract: This work introduces a new six-dimensional system with chaotic and periodic

solutions. For special values of parameters, we calculate the Kaplan-Yorke dimension and

we show the dynamics of Lyapunov exponents. Some deﬁnitions and propositions are

given. Visualizations where possible, are provided.

Keywords: artiﬁcial neural network, Kaplan-Yorke dimension, Lyapunov exponents,

chaotic solution, periodic solution

Received: October 13, 2022. Revised: January 15, 2023. Accepted: February 18, 2023. Published: March 24, 2023.

1Introduction

Mathematical modeling of nonlinear dy-

namic systems is an interdisciplinary tool

for studying various processes in nature and

society. Artiﬁcial neural network (ANN in

short) models are a simpliﬁcation of human

neural systems. ANN consists of computing

units. These blocks are called artiﬁcial

neurons. Artiﬁcial neurons are similar to

neurons in the biological nervous system,

[1]. The ANNs model is the most common

emerging tool for modeling environmental

concerns, particularly, in water quality

modeling, [2]. Also, the ANN model for

the crude oil distillation column was con-

structed based on the results of simulations,

[3]. ANN are used in agriculture, medicine,

marketing and other industries. Their

mathematical models can be formulated in

terms of systems of diﬀerential equations of

the form (1)











1= tanh(w11x1+. . . +w1nxn)−b1x1,

2= tanh(w21x1+. . . +w2nxn)−b2x2,

. . .

n= tanh(wn1x1+. . . +wnnxn)−bnxn.

(1)

Each dependent variable is associated with

a neuron. It accepts signals from other neu-

rons and elaborates its signal which is sent

to a network, [4]. In such systems, peri-

odic solutions, quasi-periodic solutions, and

also chaotic solutions are possible. This

article uses the Lyapunov exponents and

the Kaplan-Yorke formula to determine the

chaotic solutions of the system (1).

2 Lyapunov exponents

and Kaplan-Yorke di-

mension

Lyapunov exponents are a useful tool to

distinguish between regular and chaotic dy-

namics. The generally accepted convention

is to write the Lyapunov exponents in de-

scending order

λ1≥λ2≥. . . λn.

Lyapunov exponent measures how quickly

an inﬁnitesimally small distance between

two initially close states grows over time

Ft(x0+²)−Ft(x0)≈²eλt.

The left-hand side is the distance between

two initially close states after tsteps, and

the right-hand side is the assumption that

the distance grows exponentially over time.

The exponent λmeasured for a long period

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of time is the Lyapunov exponent. Each

indicator can be interpreted as indicator of

the rate of stretching (if λ > 0), small dis-

tances grow indeﬁnitely over time. If λ < 0,

small distances don’t grow indeﬁnitely, i.e.,

the system settles down into a periodic tra-

jectory, [5].

Some facts about Lyapunov exponents:

•The number of Lyapunov exponents is

equal to the number of phase space di-

mensions, or the order of the system of

diﬀerential equations, [6], [7].

•The largest Lyapunov exponent of a

stable system does not exceed zero, [8].

•A hyperchaotic system is deﬁned as a

chaotic system with at least two pos-

itive Lyapunov exponents. Combined

with one null exponent and one nega-

tive exponent, the minimal dimension

for a hyperchaotic system is four, [9].

•A strictly positive maximal Lyapunov

exponent is often considered as a deﬁ-

nition of deterministic chaos.

Knowing the Lyapunov exponents allows us

to conclude how the system develops over

time. Often enough to know the sign the

highest exponent, and the sum of the Lya-

punov exponents.

In 1979, the Kaplan-Yorke formula was pro-

posed to estimate the fractal size - in terms

of Lyapunov exponents, [10], [11],[12].

DKY =j+1

|λj+1|

j=1

λj(2)

with jrepresenting the index such that

i=1

λj>0,

j+1

i=1

λj<0.

The result obtained by this formula is

called the Kaplan-Yorke dimension or the

Lyapunov dimension. A valuable advan-

tage of this dimension is that in order to

calculate it, one needs to know only the

spectrum of Lyapunov exponents of the

given attractor, [13].

Deﬁnition 2.1. A chaotic system is a

deterministic system that exhibits irregular

and unpredictable behavior, [8].

Deﬁnition 2.2. An attractor is the limit-

ing trajectory of the representing point in

the phase space, to which all initial modes

tend [14].

Each attractor has a basin of attraction

that contains all the initial conditions which

will generate trajectories joining asymptot-

ically this attractor [15].

Deﬁnition 2.3. A chaotic attractor is an

attractor that exhibits sensitivity to initial

conditions, [16].

Proposition 2.1. In dynamical systems

that include three or more equations, there

may be even more unusual attractors, which

are commonly called strange or chaotic

attractors.

Floris Takens (1940-2010) a Dutch math-

ematician known for contributions to the

theory of diﬀerential equations, the theory

of dynamical systems, chaos theory and

ﬂuid mechanics. Introduced the concept

of a “strange attractor”. He was the ﬁrst

to show how chaotic attractors could be

learned by neural networks [17].

Proposition 2.2. It is possible to ﬁnd a

chaotic attractor in diﬀerential systems pre-

senting chaotic behavior [18], [19].

2.1 The six-dimensional sys-

tems

Consider the system

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









1= tanh(w11x1+w12x2+w13x3+w14x4+w15x5+w16x6)−b1x1,

2= tanh(w21x1+w22x2+w23x3+w24x4+w25x5+w26x6)−b2x2,

3= tanh(w31x1+w32x2+w33x3+w34x4+w35x5+w36x6)−b3x3,

4=tanh(w41x1+w42x2+w43x3+w44x4+w45x5+w46x6)−b4x4,

5= tanh(w51x1+w52x2+w53x3+w54x4+w55x5+w56x6)−b5x5,

6= tanh(w61x1+w62x2+w63x3+w64x4+w65x5+w66x6)−b6x6

(3)

and the regulatory matrix







0−10100

100100

1 1 0 −1 0 0

0−11000

1−1 0 −1−1 0

1 0 −1 0 0 −1







(4)

The initial conditions are

x1(0) = 1.1; x2(0) = 0.5; x3(0) = 1.1;

x4(0) = −1; x5(0) = −1; x6(0) = −1.(5)

Parameters are b1=b2=b3=b4=b5=

b6= 0.042.

The graph of the system (3) with the

regulatory matrix (4) is considered in

Figure 1.

Figure 1: The graph of the system (3) with

the regulatory matrix (4).

The projections of 6D trajecto-

ries on three-dimensional subspace

(x1, x4, x5) are in Figure 2. Solutions

(x1(t), x2(t), x3(t), x4(t), x5(t), x6(t)) of the

system (3) with the regulatory matrix (4)

are shown in Figure 3.

The dynamics of Lyapunov exponents are

shown in Figure 4.

-5

-10

-5

Figure 2: The projection of 6D trajectories to

3D subspace (x1, x4, x5), b= 0.042.

100

200

300

400

-10

-5

8x1, x2, x3, x4, x5,x6<

Figure 3: Solutions

(x1(t), x2(t),x3(t), x4(t), x5(t), x6(t)) of

the system (3) with the regulatory matrix

(4), b= 0.042.

Lyapunov exponents are λ1= 0.03; λ2=

0.00; λ3=−0.07; λ4=−0.13; λ5=−0.15

and λ6=−0.24.

λ1+λ2>0;

λ1+λ2+λ3<0.

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200

400

600

800

1000

1200

1400

-0.2

0.2

0.4

Figure 4: The dynamics of Lyapunov expo-

nents.

The Kaplan-Yorke dimension is

DKY = 2 + λ1+λ2

|λ3|=2.29.

The presence of a positive λ1indicates the

chaotic nature of the dynamics. Lyapunov

exponents λ3, λ4, λ5and λ6are negative

and are responsible for compression phase

volume and the approximation of phase

trajectories to the attractor. An estimate of

the Kaplan-Yorke dimension from spectrum

of Lyapunov exponents gives for a given

attractor DKY = 2.29.

Let change the value of the element w11

from 0 to 1 of the regulatory matrix (4).

The graph of the system (3) with the regu-

latory matrix (4), w11 = 1, is considered in

Figure 5.

Figure 5: The graph of the system (3) with

the regulatory matrix (4), w11 = 1.

The projections of 6D trajecto-

ries on two-dimensional subspace

(x5, x6) are in Figure 6. Solutions

(x1(t), x2(t), x3(t), x4(t), x5(t), x6(t)) of

the system (3) with the regulatory matrix

(4), w11 = 1, are shown in Figure 7.

-15

-10

-5

-15

-10

-5

Figure 6: The projection of 6D trajectories to

2D subspace (x5, x6), b= 0.042.

100

200

300

400

500

-20

-10

8x1, x2, x3, x4, x5,x6<

Figure 7: Solutions

(x1(t), x2(t),x3(t), x4(t), x5(t), x6(t)) of

the system (3) with the regulatory matrix

(4),w11 = 1, b= 0.042.

The dynamics of Lyapunov exponents are

shown in Figure 8.

Lyapunov exponents are λ1= 0.00; λ2=

−0.03; λ3=−0.04; λ4=−0.05; λ5=−0.54

and λ6=−0.64.

λ1= 0; λ2<0 the system (3) with the

regulatory matrix (4), w11 = 1, has periodic

solutions.

In dissipative dynamical system, the values

of all Lyapunov exponents should sum

to a negative number, [6],[7] . The sys-

tem (3) with the regulatory matrix (4),

w11 = 1, is a dissipative dynamical system

λ1+. . . +λ6<0.

Let change the value of the element w11

from 0 to −1 of the regulatory matrix (4).

The graph of the system (3) with the regu-

latory matrix (4), w11 =−1, is considered

in Figure 9.

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200 400 600 800 1000 1200 1400

-1.0

-0.5

Figure 8: The dynamics of Lyapunov expo-

nents.

Figure 9: The graph of the system (3) with

the regulatory matrix (4), w11 =−1.

The dynamics of Lyapunov exponents are

shown in Figure 10.

200 400 600 800 1000 1200 1400

-1.5

-1.0

-0.5

Figure 10: The dynamics of Lyapunov expo-

nents.

Lyapunov exponents are λ1=−0.04; λ2=

−0.36; λ3=−0.36; λ4=−0.40; λ5=−1.04

and λ6=−1.04.

λ1<0 the system (3) with the regulatory

matrix (4), w11 =−1 has stable ﬁxed point.

The system (3) with the regulatory matrix

(4), w11 =−1, is a dissipative dynamical

system λ1+. . . +λ6<0.

3 Conclusions

The six-dimensional system with diﬀerent

regulatory matrices is considered. The three

examples are provided. Changing the value

of an element w11 changes the system (3) so-

lutions. In this paper, the periodic and the

chaotic attractor is considered. Some def-

initions and propositions about Lyapunov

exponents and Kaplan-Yorke dimension is

given. The dynamics of Lyapunov expo-

nents are shown. Visualizations of peri-

odic, chaotic solutions of the system (3) are

shown. Projections of 6Dtrajectories to 2D

or 3Dsubspaces are shown.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to

declare that is relevant to the content of this

article.

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