Volume 4, Issue 10, April 2015
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. THE0-1TEST[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.
Volume 4, Issue 10, April 2015 ,
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. DYNAMICLYAPUNOVINDICATOR[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. DUFFINGMAP–APPLICATIONOFDLIAND0-1 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
LCE
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.
Volume 4, Issue 10, April 2015
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
LCE
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
LCE
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
LCE
LCE
4 (b)
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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
LCE
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 1.5
2
1 0 1 2
6(a)
1000 1050 1100 1150 1200 1250 1300
0.874 0.876 0.878 0.880 0.882
n
LCE
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.
Volume 4, Issue 10, April 2015 REFERENCES
[1] Bountis, T., Application of the SALI method for Detecting Chaos and Order in Accelerator Mappings, downloaded from URL: http:// www.math.upatras. gr/ ̴ crans.
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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.
