ISSN: 2277-3754

ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT)

Volume 4, Issue 10, April 2015

165

Abstract— In this paper we study various attractors of Duffing

map. We apply 0-1 test and Dynamic Lyapunov Indictor to

distinguish between periodic and chaotic behavior of various

attractors of Duffing map. For different set of values of

parameters of this map chaotic attractors are drawn and

corresponding plots of Lyapunov exponents and Dynamic

Lyapunov Indicator have been obtained. We evaluate 0-1 test

parameters in each case and compare the results obtained from

0-1 test parameter, Lyapunov exponent, and Dynamic Lyapunov

indicator.

Index Terms— 0-1 test, Duffing map, Dynamic Lyapunov

indicator, Lyapunov exponents.

I. INTRODUCTION

Since the discovery of chaotic dynamics in weather

systems by Lorenz in 1963 expansive interest by researchers

has demonstrated the presence of chaotic dynamics in

multitude of natural and man-made systems in almost all

sphere of life. A chaotic system is a highly complex dynamic

nonlinear system and its response exhibits sensitivity to the

initial conditions. The sensitive nature of chaotic systems is

commonly called as the butterfly effect. Chaos theory has

been applied to a variety of fields such as physical systems,

chemical reactors, secure communication etc.

To distinguish between chaotic and periodic motion there

are several methods. The most common tests are Lyapunov

exponent [12] and maximal Lyapunov exponent [10]. Fast

Lyapunov Indicator [6], Smaller Alignment Index [1] and

Dynamic Lyapunov Indicator [15] are some other tests that

have been used. The 0-1 test was first suggested by

Melbourne and Gottwald [7-8]. Gottwald and Melbourne [9]

have presented a theoretical justification of the test. The 0-1

test is universally applicable test which yields 0 for regular

motion and 1 for chaotic motion and which is easy to apply to

any continuous and discrete dynamical system. The test has

been applied to many systems like the two dimensional map

of a bouncing ball system by Litak, Budhraja and Saha [13],

where the authors confirmed the results by the calculation of

maximal Lyapunov exponent. Other systems where the test

has been applied are strange non-chaotic attractor by Dawes

and Freeland [4], where the authors concluded that the test

performs extremely well. Also the test has been applied on

nonlinear dynamical system including fractional order

dynamical system by Hui and Cong-Xu [11]. Plasma is a

highly complex system exhibiting a rich variety of nonlinear

dynamical phenomena. Chowdhury, Iyenger and Lahiri [3],

have applied the 0-1 test to the time series obtained from a

glow discharge plasma experiment, and it is found to be very

effective and simpler than the estimation of the largest

lyapunov exponent. The universal technique to examine the

nature of motion in deterministic systems is to calculate

maximal Lyapunov exponent but Falconer, Gottwald,

Melbourne and Wormnes [5] have analyzed data coming

from an experimental set up of a bipolar motor in an

alternating magnetic field and they investigated the

performance of 0-1 test. Budhraja [2] have also applied 0-1

test to Peter-de-Jong map and the author concluded that the

0-1 test can be regarded as a good indicator of chaotic or

periodic/quasi-periodic motion.

Dynamic Lyapunov Indicator (DLI) was suggested by

Saha and Budhraja [14]. The authors applied DLI to various

attractors of Gumowski Mira map and compared the results

with those obtained using fast Lyapunov Indicator (FLI),

Smaller Alignment Index (SALI). Yuasa and Saha [15]

studied Burger’s map, Chirikov map, and bouncing ball

dynamics using DLI. They have also compared the results

with FLI, SALI and 0-1 test. They concluded that DLI

provides a clear picture for identification of regular and

chaotic motion for all these maps. Deleanu [16 ] analyzed the

behavior of the 2-D Lozi map, the 2-D predator prey map and

the 3-D Lorentz BD map with the help of DLI and results

were found satisfactory. DLI was applied to Duffing map and

Ikeda map and results were compared to SALI and FLI by

Saha and Tehri [17] and DLI exhibits same results as SALI

and FLI. Saha and Sharma [18] applied DLI to the food chain

system and the results have been quite satisfactory.

In this paper we study the application of DLI and 0-1 test to

various attractors of Duffing map. The scheme of the paper is

as follows –in Section II we explain in detail the application

of 0-1 test, in Section III we explain the Dynamic Lyapunov

Indicator .In Section IV we plot the various attractors of

Duffing map, the plot of Lyapunov characteristic exponents,

the plots of DLI and obtain the 0-1 test parameters.

II. THE 0-1 TEST [2]

Consider a sequence of scalar output data Ø(n).

Choose c >0 and define

, n = 1, 2, 3, … (1)

Now calculate the total mean square displacement:

,

and the asymptotic growth rate

Attractors of Duffing Map: Application of DLI

and 0-1 Test Aysha Ibraheem, Narender Kumar

Department of Mathematics, University of Delhi, Delhi-110007, India.

Associate Professor, Department of Mathematics, Aryabhatta College, University of Delhi, New

Delhi-110021, India.

ISSN: 2277-3754

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Volume 4, Issue 10, April 2015

166

,

To avoid negative values of K , we may as well take

.

If the behavior of p(n) is asymptotically Brownian i.e. the

underlying dynamics is chaotic , then M(n) grows linearly in

time ; whereas if the behavior of p(n) is bounded(as in case of

periodic and Quasi periodic motion),then M(n) is also

bounded.

The asymptotic growth rate K of M(n) is then numerically

determined by means of linear regression of log(M(n)) versus

log(n). The main advantages of the test are:

1. The origin and nature of the data fed into the diagnostic

system (1) is irrelevant for the test.

2. The method is independent of the scalar observed and

almost any choice of c will serve. 3. The dimension of the underlying dynamical system does

not pose any practical limitations on the method as in the case

for traditional methods involving phase space reconstruction.

The only conditions which are necessary to be met while

working with the 0-1 test are:

1. Initial transients should have died out so that the

trajectories are on (or close to) the attractor at the time zero.

2. The time series is long enough to allow for asymptotic

behavior of p(n).

3. It is necessary that the data is essentially stationary as well

as deterministic.

III. DYNAMIC LYAPUNOV INDICATOR [15]

The dynamic Lyapunov indicator is defined by the largest

value estimated from all eigen value of Jacobian matrix J

such that

of the examined map for all discrete times. We plot the

largest eigen value at every time step of the evolving

Jacobian matrix and we observe that these eigenvalues form a

definite pattern for regular motion and are distributed

randomly for chaotic orbits.

IV. DUFFING MAP– APPLICATION OF DLI AND 0-1

TEST

There are two main types of dynamical systems:

differential equations and iterated maps(also called

difference equation). Differential equation describes the

continuous time evolution of the system, whereas difference

equation describes the discrete time evolution of the system.

The Duffing map is a discrete time dynamical system. It is an

example of a dynamical system that exhibits chaotic

behavior. The Duffing map takes a point (xn, yn) in the plane

and maps it to a new point given by,

=

Now, we will apply the 0-1 test and DLI to Duffing map.

For different values of parameters a and b, attractors of

Duffing map are drawn and corresponding Lyapunov

exponents have been obtained. Figure 1(a) shows the

attractor for parameters a = 2.77, b = 0.1. The value of K =

0.807058. Figure 1(b) shows the Lyapunov exponents are all

positive, indicating chaos. Figure 1(c) shows the plot of DLI

for this attractor. The value of K, DLI and positive Lyapunov

exponents are all in agreement.

1.5 1.0 0.5 0.0 0.5 1.0 1.5

1.5

1.0

0.5

0.0

0.5

1.0

1.5

1 (a)

1000 1050 1100 1150 1200 1250 1300

0.728

0.729

0.730

0.731

0.732

0.733

0.734

n

LC

E

LCE

1 (b)

0 200 400 600 800 1000

1

2

3

4

5

6

1 (c)

Figure 2(a) shows the attractor for parameters a = 2.77,

b = 0.3, and the value of K comes out to be 0.01042 which

shows regular motion. Figure 2(b) shows LCE and

Figure 2(c) shows DLI plot for this attractor. Negative

Lyapunov exponent, value of K near to 0 and regular DLI all

indicate regular motion.

ISSN: 2277-3754

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Volume 4, Issue 10, April 2015

167

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.4

0.6

0.8

1.0

1.2

1.4

2 (a)

1000 1050 1100 1150 1200 1250 1300

0.270

0.269

0.268

0.267

n

LC

E

LCE

2 (b)

0 200 400 600 800 1000

0.5

1.0

1.5

2.0

2.5

3.0

2 (c)

Figures 3(a), (b) and (c) are respectively plots of attractor

and Lyapunov exponent and DLI for parameters a = 2.77,

b = 0.1. Here, K = 0.9377 which is very near to 1. The value

of K, positive Lyapunov exponents and randomly distributed

DLI’s indicate chaos.

2 1 0 1 2

2

1

0

1

2

3 (a)

1000 1050 1100 1150 1200 1250 1300

0.820

0.822

0.824

0.826

0.828

n

LC

E

LCE

3 (b)

0 200 400 600 800 1000

2

4

6

8

3 (c)

Figures 4(a), (b) and (c) show respectively the attractor for

parameters a = 2.75, b = 0.4, the LCE and the DLIs. The value

of K = 0.01008, which shows regular motion, also well

indicated by negative LCE and regular DLIs.

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

4 (a)

1000 1050 1100 1150 1200 1250 1300

0.2200

0.2198

0.2196

0.2194

n

LC

E

LCE

4 (b)

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Volume 4, Issue 10, April 2015

168

0 200 400 600 800 1000

1.0

1.5

2.0

2.5

4 (c)

Figure 5(a) shows the attractor for parameters a = 2.77,

b = 0.01 for which the value of K = 0.89290 which is near to

1. Figure 5(c) shows Irregular pattern of DLI, 5(b) shows positive Lyapunov exponents which leads to same result.

1.5 1.0 0.5 0.0 0.5 1.0 1.5

1.5

1.0

0.5

0.0

0.5

1.0

1.5

5 (a)

1000 1050 1100 1150 1200 1250 1300

0.776

0.778

0.780

0.782

0.784

0.786

0.788

n

LC

E

LCE

5 (b)

0 200 400 600 800 1000

0

1

2

3

4

5

6

5 (c)

Figure 6(a), (b) and (c) shows the attractor for parameters

a = 2.88, b = 0.005, Lyapunov exponents and DLI

respectively. The value of K = 1.02742 and the results are in agreement

1.5 1.0 0.5 0.0 0.5 1.0 1.52

1

0

1

2

6(a)

1000 1050 1100 1150 1200 1250 1300

0.874

0.876

0.878

0.880

0.882

n

LC

E

LCE

6 (b)

0 200 400 600 800 1000

2

4

6

8

6(c)

V. CONCLUSION

We conclude that the DLI is quite efficient in analyzing

various types of motions in Duffing map and it can be

regarded as a good indicator of chaotic and periodic motion

with its prediction being comparable with that of Lyapunov

exponent and 0-1 test parameter K. It exhibits satisfactory

results for various attractors of Duffing map. As we see that

in cases (1), (3), (5), (6), irregular pattern of DLI shows

chaotic motion, and the same results are also obtained from

positive Lyapunov exponents and value of 0-1 test parameter

K which is very near to 1, and in case (2) and (4), definite

pattern of DLI shows regular motion and the results are in

agreement with negative Lyapunov exponents and value of K

which is very near to 0.It is important to verify this to other

discrete and continuous systems also.

ISSN: 2277-3754

ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT)

Volume 4, Issue 10, April 2015

169

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AUTHOR BIOGRAPHY

Aysha Ibraheem is a research scholar in the Department of Mathematics,

University of Delhi, Delhi-110007 doing her Ph.D. under the supervision of

Dr. Mridula Budhraja and Professor Ayub Khan (Co-supervisor). Her main

area of research is Non-Linear Dynamical System and Chaos Control.

Narender Kumar is presently working as an associate professor in the

Department of Mathematics, Aryabhatta College, University of Delhi, New

Delhi-110021. He has done his Ph.D. in September, 2008 from University of

Delhi, Delhi under the supervision of Prof. (Mrs.) Davinder Bhatia and

Prof. S.C. Arora. His title of Ph.D. thesis is “Vector Optimization Involving

n-Set Functions” He has published a total of 7 research papers.