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International Journal of Advanced Technology & Engineering Research (IJATER)

2nd

International e-Conference on Emerging Trends in Technology

A NEW LORENZ UNLIKE CHAOTIC ATTRACTOR

Angelo A. Beltran Jr. School of Graduate Studies

School of Electrical Electronics Computer Engineering Mapua Institute of Technology

Manila, Philippines abeltranjr@hotmail.com

Abstract

This paper presents the new Lorenz unlike chaotic attractor

which is constructed by a three non linear first order

differential equations. These equations are arranged in a three

dimensional autonomous systems. The dynamic behavior of the

new chaotic system is shown such as time series, strange

attractors, and bifurcations. Numerical experience also shows

that when the parameter d is varied, the global non linear

amplitude is also varying. The paper ends with some possible

research and development recommendations.

Graphical Abstract

I. Introduction

Chaos exhibits non linear operation with a property of

highly sensitivity to initial conditions and very complex

dynamic behaviour. A little difference in initial conditions

produces chaos. It is characterized by non-linear systems that

have non-stable periodic motion, deterministic, unpredictable,

parameter variations, and unstable dynamic behaviour [1 – 3,

30]. The first chaotic system generated is described by three

dimensional autonomous equations, which was discovered by a

professor in 1963 at Massachusetts Institute of Technology

named Edward N. Lorenz, while he was working with the

problem of the weather modeling [4]. These equations that

exhibits chaos is also known as the „Lorenz equation‟ and the

attractor it produces are popularly known as the „Butterfly

Effect‟ [7]. Since then many chaotic systems described by three

dimensional autonomous equations were published in the

literature such as the Rössler systems [5], Chen systems [8], Lü

systems [9], Liu systems [10], and other systems.

Chaotic systems and its applications have been paid a lot of

attention by scientists, engineers, and researchers in recent

years. It has been thoroughly studied and applied in various

engineering, physical sciences, and scientific areas such as

electronic circuits [12 – 13], communication engineering [15 –

17], power electronics [19 – 20], chemical engineering [21],

control systems [22], electric motor drives [23], robotics [24],

economics [25], medical applications [26], ecology [14, 27],

computer science [28], power systems [29], and many more.

This paper has been organized as follows. In Section II, a

review of some of the previous chaotic attractors is presented.

Section III shows the results of a new chaotic attractor which is

based on the three dimensional non linear autonomous

differential equations with quadratic exponents. Section IV

summarizes the main results of this paper, and addresses some

possible research directions in the future.

II. Review of Some Chaotic Systems

A. Lorenz Systems and Strange Attractor

E. N. Lorenz, a Professor from Massachusets Institute of

Technology (MIT) found out the chaotic attractor in a three-

dimensional autonomous system that bears his name in his

study of weather pattern in 1963 [4].

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X X Y

Y XZ rX Y

Z XY bZ

(1)

which exhibits chaos when 10 , 8

3b , and 28r .

As we can see in Fig. 1, the topology looks like a shape of a

flying butterfly flipping its wings; thus, coining the term

„Butterfly Effect‟.

Fig. 1. Chaotic attractor of Lorenz equation in (x, y, z) space [4].

Commonly known as the Butterfly Effect [7].

B. Lü Systems and Strange Attractor

In 2002, Lü and Chen found another chaotic attractor which

is generated and produced by the following three dimensional

non linear autonomous systems [9].

( )x a y x

y xz cy (2)

z xy bz

which also exhibits chaotic attractor when 36a , 3b ,

and through varying the c parameter. See Fig. 4.

C. Liu Systems and Strange Attractor

In 2004, Liu et al. found another chaotic attractor, which is

not topologically equivalent to the Lorenz chaotic attractor.

The equations are also arranged in a three dimensional non

linear autonomous systems [10].

( )x a y x

y bx kxz (3)

2z cz hx

It also exhibits chaotic and strange attractor when 10a ,

40b , 1k , 2.5c , and 4h . Fig. 2. and Fig. 6.

illustrates the strange orbits

Fig. 2. A three-dimensional view of Liu chaotic attractor [10].

III. A New Lorenz Unlike Chaotic

Attractor A. System Description

A new chaotic attractor Lorenz unlike topology is proposed

in this paper. The three dimensional autonomous systems is

generated and described as follows:

3x a y x

3 3y x bx cz dy (4)

3

z m xy nz

where 10a , 96b , 1.7c , 2.9d ,

3.5m , and 9 . 3n . Results are shown in the

succeeding figures.

Fig. 3. Time series of system (4) [(x, y, z), t] loaded with

initial values (1, 2, 2.5).

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Fig. 4. Chaotic attractor of Lü equation with variations of c parameter when (a) c = 13, (b) c = 20, (c) c = 28, (d) c = 28.7 [9].

B. Numerical Simulations

Fig. 5. Time response of x (left figure) and y (right figure) loaded with (1, 2, 2.5) initial values in system (4).

(a)

(c) (d)

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Fig. 6. Phase portrait (above) of Liu autonomous systems (a) x – z plane, (b) x – y plane, (c) y – z plane [10].

Fig. 7. Time response of z loaded with (1, 2, 2.5) initial values in system (4).

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Fig. 8. Periodic orbit occurs at d = 10.5 in system (4).

(a)

(b)

Fig. 9. Chaotic attractor (state orbits) (a) and (b) of the proposed

system in (4) loaded with (1, 2, 2.5) initial values.

C. Chaotic attractor when the parameter d is increased

while the other parameters are fixed (bifurcations) in system

(4).

2.9d

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5.5d

10.5d (periodic orbit)

15.5d

25.5d

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35.5d

60.5d

System (4) describes three dimensional autonomous non

linear differential equations with six parameters involved: a, b,

c, d, m, and n. The equation has been loaded with initial

values (1, 2, 2.5) to demonstrate the behavior of the system. It

can be seen in Fig 3, Fig. 5, and Fig. 7, the time series of

system (4) when d = 2.9. The time response of the dependent

variables (x, y, z) are indeed chaotic, and the illustrations of

the waveform is very complex. It is interesting to note that in

system (4), periodicity occurs at d = 10.5, loaded with the

same initial values. This can be seen in Fig. 8.

Fig. 9 shows the chaotic attractor of the proposed system in

(4), where the topology of the non linear equation differs with

the Butterfly Effect of Lorenz systems, and of the other

chaotic attractors presented in the literature. Also in the same

figure (Fig. 9), noticed the two scroll generated by system (4).

It seems like a two black holes communicating with each

other.

In part C, the parameter d is varied in system (4) while the

other parameters are made fix. It is loaded with the same

initial conditions. Notice that as the parameter d in system (4)

is increasing, the chaotic attractor is changing its global

topological structure. This therefore signifies the bifurcations

of autonomous system in (4). Initial conditions can be varied,

changing the initial conditions with a different values even in

a very small difference, results an entirely different outcome

in the output. This shows the highly sensitivity of system (4).

IV. Conclusion

In this paper, a novel three dimensional autonomous non

linear differential equation and its corresponding Lorenz unlike

chaotic attractor is proposed and presented. This paper also

illustrates the time series in (x, y, z) space, phase portrait and

strange orbit of the proposed autonomous system in (4) and the

change of topology when the parameter d is varied. This

implies a change in a solution (bifurcations) to the non linear

differential equation autonomous system in (4) when the

parameter d is varied. Numerical simulations derived in this

paper are presented to demonstrate the chaotic behaviour on

the basis of the proposed system in (4). Results clearly show

the chaotic effectiveness of the proposed system in (4).

Dynamic analysis of the new chaotic system such as poincare

mapping, dissipation, eigenvalues & eigenvectors, invariance,

Lyapunov exponents, equilibrium, spectrum mapping shall be

reported in another forth coming research paper. The circuit

implementation, electrical, electronics, and communication

engineering applications of the proposed system in (4) are

highly encouraged for further research & development.

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