INNA SAMUILIK
Department of Engineering Mathematics
Riga Technical University
LATVIA
Abstract: This work introduces a new high-dimensional five-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. The main intent is to use the 2D and 3D projections of the 5D
trajectories on different subspaces, to construct the graphs of solutions for understanding
and managing the system. Visualizations where possible, are provided.
Keywords: chaos, KaplanYorke dimension, Lyapunov exponents, chaotic solution, peri-
odic solution
Received: September 9, 2021. Revised: October 25, 2022. Accepted: November 22, 2022. Published: December 20, 2022.
1 Introduction
Chaos is a phenomenon that is not easily
classified. There is no universally accepted
definition for chaos, [1]. The authors inter-
pret chaos in their way and give their own
definitions for the concept of chaos. But
the authors mention three main features of
chaos: 1) long-term aperiodic (nonperiodic)
behavior; sensitivity to initial conditions;
fractal structure, [1]. One of the features
of irregular regimes is the instability trajec-
tories belonging to a chaotic or strange at-
tractor. The quantitative measure of this in-
stability is the characteristic call exponents,
originally introduced by Lyapunov to study
the behavior of one trajectory. Positive Lya-
punov exponents (LE) and a high KaplanY-
orke dimension DKY guarantee a chaotic be-
havior for long-times, [2]. LE quantifies the
average increment of an infinitely small er-
ror at the initial point. LE > 0 indicates
that the dynamic system is sensitive to the
initial condition; LE = 0 means the sys-
tem is stable; and LE < 0 reflects that the
system tends to stabilize, [3]. The dissipa-
tive dynamical system has at least one nega-
tive Lyapunov exponent, the sum of all Lya-
punov exponents is the negative, [4]. The
same system with different parameters can
have periodic or chaotic solutions, [5]. In
this case, the system has a bifurcation. The
bifurcation is a change in the dynamics sys-
tem, accompanied by the disappearance of
some and the appearance of other regimes.
Firstly, the stable point goes into the peri-
odic regime, then to the chaotic regime, [6].
2 Definitions and propo-
sitions
Definition 2.1. A chaotic system is a
deterministic system that exhibits irregular
and unpredictable behavior, [7].
Definition 2.2. A strange attractor,
(chaotic attractor, fractal attractor) is an
attractor that exhibits sensitivity to initial
conditions, [1].
Definition 2.3. A fractal is an object that
displays self-similarity under magnification
and can be constructed using a simple motif
(an image repeated on ever-reduced scales),
[1].
Lyapunov Exponents and Kaplan-yorke Dimension for Five-
dimensional System
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Such strange objects were identified in
nature. Sunflowers and broccoli (Figure
2), sea shells, fern, snowflakes (Figure 1),
mountain chasms, coastlines, lightning
bolts (Figure 3), tree branches, river beds,
turbulent eddies, human vascular system.
These fractal geometries play a significant
role in the characterization of chaotic
dynamical processes, fractal dimension is
therefore an important attribute of such a
process, [8].
Figure 1: The picture from www.esa.org
Figure 2: Remarkable Romanesco Broccoli.
The picture from www.gardenbetty.com
Proposition 2.1. Dissipative systems ex-
hibit chaos for most initial conditions in a
specified range of parameters. A conser-
vative system exhibits periodic and quasi-
periodic solutions for most values of param-
eters and initial conditions, and can exhibit
chaos for special values only, [9].
Proposition 2.2. Only dissipative dynam-
ical systems have attractors, [10].
Figure 3: The picture from
www.zmescience.com
3Materials and methods
Our consideration is geometrical. The main
intent is to use the 2D and 3D projections of
the 5D trajectories on different subspaces,
to construct the graphs of solutions for
understanding and managing the system.
Visualizations where possible, are provided.
The dynamics of Lyapunov exponents
are shown. Computations are performed
using Wolfram Mathematics, [11]. In the
article for Lyapunov exponents calcula-
tion the program Wolfram Mathematica
“Lynch-DSAM.nb” was used, [1], [5].
3.1 The example of five-
dimensional system
Consider the system
x0
1= tanh(x4−x2)−bx1,
x0
2= tanh(x1+x4)−bx2,
x0
3= tanh(x1+x2−x4)−bx3,
x0
4= tanh(x3−x2)−bx4,
x0
5= tanh(x1−x2−x4−x5)−bx5
(1)
The initial conditions are
x1(0) = 1.4; x2(0) = 0.5;
x3(0) = 1.4; x4(0) = −1; x5(0) = −1.(2)
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Table 1.Lyapunov exponents for the
system (1), 2000 steps
bLE1LE2LE3LE4LE5
0.035 0 0 -0.06 -0.07 -0.08
0.036 0 0 -0.07 -0.07 -0.09
0.037 0 0 -0.06 -0.08 -0.09
0.038 0.02 0 -0.07 -0.09 -0.12
0.039 0.02 0 -0.07 -0.09 -0.11
0.04 0 0 -0.06 -0.09 -0.10
0.041 0.03 0 -0.07 -0.12 -0.15
0.042 0.04 0 -0.09 -0.12 -0.15
0.043 0.03 0 -0.07 -0.13 -0.15
0.044 0.03 0 -0.07 -0.13 -0.16
0.045 0.03 0 -0.07 -0.13 -0.16
0.046 0.01 0 -0.07 -0.13 -0.16
0.047 0.02 0 -0.07 -0.13 -0.16
0.048 0 0 -0.09 -0.11 -0.16
0.049 0 0 -0.06 -0.12 -0.17
0.05 0 0 -0.07 -0.12 -0.17
0.051 0 -0.4 -0.07 -0.09 -0.17
Let calculate the Kaplan-Yorke dimension
using the following formula [12]
DKY =j+1
|Lj+1|
j
X
j=1
Lj(3)
with jrepresenting the index such that
j
X
i=1
Lj>0,
j+1
X
i=1
Lj<0.
The formula (3) is called the Kaplan-Yorke
formula after the names of the researchers
who proposed this formula in 1979. The
initial hypothesis was that this formula al-
lows one to calculate the information dimen-
sion of attractors. For chaotic attractors
of two-dimensional invertible mappings for
which λ1>0 and λ2<0. This proposition
has been proven rigorously. In the general
case, the Kaplan-Yorke proposition cannot
be proved.
Table 2. Kaplan-Yorke dimension for the
system (1)
bDKY
0.035 DKY= 4 + P4
iLEi
|LE5|=2.38
0.036 DKY= 4 + P4
iLEi
|LE5|=2.44
0.037 DKY= 4 + P4
iLEi
|LE5|=2.44
0.038 DKY= 4 + P4
iLEi
|LE5|=2.83
0.039 DKY= 4 + P4
iLEi
|LE5|=2.72
0.04 DKY= 4 + P4
iLEi
|LE5|=2.5
0.041 DKY= 4 + P4
iLEi
|LE5|=2.93
0.042 DKY= 4 + P4
iLEi
|LE5|=2.86
0.043 DKY= 4 + P4
iLEi
|LE5|=2.87
0.044 DKY= 4 + P4
iLEi
|LE5|=2.94
0.045 DKY= 4 + P4
iLEi
|LE5|=2.94
0.046 DKY= 4 + P4
iLEi
|LE5|=2.81
0.047 DKY= 4 + P4
iLEi
|LE5|=2.88
0.048 DKY= 4 + P4
iLEi
|LE5|=2.75
0.049 DKY= 4 + P4
iLEi
|LE5|=2.94
0.05 DKY= 4 + P4
iLEi
|LE5|=2.88
0.051 DKY= 4 + P4
iLEi
|LE5|=0.71
Calculations showed the following:
•if 0.035 ≤b≤0.037, then the system
(1) has a quasi-periodic solution;
•if 0.038 ≤b≤0.039, then the system
(1) has a chaotic solution;
•if b= 0.04, then the system (1) has a
quasi-periodic solution;
•0.041 ≤b≤0.047, then the system (1)
has a chaotic solution;
•if 0.048 ≤b≤0.05, then the system
(1) has a quasi-periodic solution;
•if b= 0.051, then the system (1) has a
periodic solution.
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The projections of 5D trajectories on two-
dimensional subspaces (x1, x5) and (x1, x4)
are in the figures below.
-4
-2
2
4
x1
-5
5
10
x5
Figure 4: The projection of 5D trajectories to
2D subspace (x1, x5), b= 0.042.
-6
-4
-2
2
4
6
x1
-4
-2
2
4
x4
Figure 5: The projection of 5D trajectories to
2D subspace (x1, x4), b= 0.042.
The projections of 5D trajectories on
three-dimensional subspaces (x1, x4, x5),
(x1, x2, x4) and (x1, x3, x4) are in the figures
below.
-5
0
5
x1
-5
0
5
x4
-10
-5
0
5
10
x5
Figure 6: The projection of 5D trajectories to
3D subspace (x1, x4, x5), b= 0.042.
-10
-5
0
5
10
x1
-10
-5
0
5
10
x2
-10
-5
0
5
10
x4
Figure 7: The projection of 5D trajectories to
3D subspace (x1, x2, x4), b= 0.042.
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-5
0
5
x1
-5
0
5
x3
-5
0
5
x4
Figure 8: The projection of 5D trajectories to
3D subspace (x1, x3, x4), b= 0.042.
The projections of 5D trajectories on two-
dimensional subspaces (x1, x5) and (x1, x4)
are shown in Figure 9 and Figure 10.
-3
-2
-1
1
2
x1
-2
2
4
6
8
x5
Figure 9: The projection of 5D trajectories to
2D subspace (x1, x5), b= 0.051.
-2
2
4
x1
-3
-2
-1
1
2
x4
Figure 10: The projection of 5D trajectories
to 2D subspace (x1, x4), b= 0.051.
The projections of 5D trajectories on
three-dimensional subspaces (x1, x4, x5),
(x1, x2, x4) and (x1, x3, x4) are shown in
Figure 11, Figure 12 and Figure 13.
-5
0
5
x1
-5
0
5
x4
-5
0
5
x5
Figure 11: The projection of 5D trajectories
to 3D subspace (x1, x4, x5), b= 0.051.
Solutions (x1(t), x2(t), x3(t), x4(t), x5(t))
of the system (1), b= 0.042 are
shown in Figure 14. Solutions
(x1(t), x2(t), x3(t), x4(t), x5(t)) of the
system (1), b= 0.042 are shown in Figure
15.
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-5
0
5
x1
-5
0
5
x2
-5
0
5
x4
Figure 12: The projection of 5D trajectories
to 3D subspace (x1, x2, x4), b= 0.051.
-5
0
5
x1
-5
0
5
x3
-5
0
5
x4
Figure 13: The projection of 5D trajectories
to 3D subspace (x1, x3, x4), b= 0.051.
100
200
300
400
500
600
700
t
-10
-5
5
10
8x1, x2, x3, x4, x5<
Figure 14: Solutions
(x1(t), x2(t), x3(t), x4(t), x5(t)) of the system
(1), b= 0.042.
The dynamics of Lyapunov exponents are
shown in Figure 16 and Figure 17.
100
200
300
400
500
600
700
t
-6
-4
-2
2
4
6
8
8x1, x2, x3, x4, x5<
Figure 15: Solutions
(x1(t), x2(t), x3(t), x4(t), x5(t)) of the system
(1), b= 0.051.
500
1000
1500
2000
-0.3
-0.2
-0.1
0.1
0.2
Figure 16: b= 0.042.
500
1000
1500
2000
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
Figure 17: b= 0.051.
4 Conclusions
Mathematical systems with chaotic behav-
ior are deterministic, that is, they obey
some strict law and, in a sense, are ordered.
This use of the word “chaos” differs from
its usual meaning. Chaos theory says that
complex systems are extremely dependent
on initial conditions and small changes
in the environment lead to unpredictable
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consequences. To study chaos, general
mathematical principles and computer
modeling are used.
This article showed that changing one
parameter of the system (1) changes the
solution of the system (a quasi-periodic
solution, a chaotic solution, a periodic
solution).
Lyapunov exponents are calculated using
Wolfram Mathematica. Lyapunov expo-
nents are one of the most useful diagnostic
tools available for analyzing dynamical
systems.
Visualizations of solutions of the sys-
tem (1) are shown, projections onto
two-dimensional and three-dimensional
subspaces are provided.
Kaplan-Yorke dimension is calculated for
the system (1) with different parameters.
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