ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 4, Issue 10, April 2015 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.

have applied the 0-1 test to the time series obtained from a Abstract— In this paper we study various attractors of Duffing glow discharge plasma experiment, and it is found to be very map. We apply 0-1 test and Dynamic Lyapunov Indictor to effective and simpler than the estimation of the largest distinguish between periodic and chaotic behavior of various . The universal technique to examine the attractors of Duffing map. For different set of values of parameters of this map chaotic attractors are drawn and nature of motion in deterministic systems is to calculate corresponding plots of Lyapunov exponents and Dynamic maximal Lyapunov exponent but Falconer, Gottwald, Lyapunov Indicator have been obtained. We evaluate 0-1 test Melbourne and Wormnes [5] have analyzed data coming parameters in each case and compare the results obtained from from an experimental set up of a bipolar motor in an 0-1 test parameter, Lyapunov exponent, and Dynamic Lyapunov alternating magnetic field and they investigated the indicator. performance of 0-1 test. Budhraja [2] have also applied 0-1

test to Peter-de-Jong map and the author concluded that the Index Terms— 0-1 test, Duffing map, Dynamic Lyapunov indicator, Lyapunov exponents. 0-1 test can be regarded as a good indicator of chaotic or periodic/quasi-periodic motion. I. INTRODUCTION Dynamic Lyapunov Indicator (DLI) was suggested by Saha and Budhraja [14]. The authors applied DLI to various Since the discovery of chaotic dynamics in weather attractors of Gumowski Mira map and compared the results systems by Lorenz in 1963 expansive interest by researchers with those obtained using fast Lyapunov Indicator (FLI), has demonstrated the presence of chaotic dynamics in Smaller Alignment Index (SALI). Yuasa and Saha [15] multitude of natural and man-made systems in almost all studied Burger’s map, Chirikov map, and bouncing ball sphere of life. A chaotic system is a highly complex dynamic dynamics using DLI. They have also compared the results nonlinear system and its response exhibits sensitivity to the with FLI, SALI and 0-1 test. They concluded that DLI initial conditions. The sensitive nature of chaotic systems is provides a clear picture for identification of regular and commonly called as the . has chaotic motion for all these maps. Deleanu [16 ] analyzed the been applied to a variety of fields such as physical systems, behavior of the 2-D Lozi map, the 2-D predator prey map and chemical reactors, secure communication etc. the 3-D Lorentz BD map with the help of DLI and results To distinguish between chaotic and periodic motion there were found satisfactory. DLI was applied to Duffing map and are several methods. The most common tests are Lyapunov and results were compared to SALI and FLI by exponent [12] and maximal Lyapunov exponent [10]. Fast Saha and Tehri [17] and DLI exhibits same results as SALI Lyapunov Indicator [6], Smaller Alignment Index [1] and and FLI. Saha and Sharma [18] applied DLI to the food chain Dynamic Lyapunov Indicator [15] are some other tests that system and the results have been quite satisfactory. have been used. The 0-1 test was first suggested by In this paper we study the application of DLI and 0-1 test to Melbourne and Gottwald [7-8]. Gottwald and Melbourne [9] various attractors of Duffing map. The scheme of the paper is have presented a theoretical justification of the test. The 0-1 as follows –in Section II we explain in detail the application test is universally applicable test which yields 0 for regular of 0-1 test, in Section III we explain the Dynamic Lyapunov motion and 1 for chaotic motion and which is easy to apply to Indicator .In Section IV we plot the various attractors of any continuous and discrete . The test has Duffing map, the plot of Lyapunov characteristic exponents, been applied to many systems like the two dimensional map the plots of DLI and obtain the 0-1 test parameters. of a bouncing ball system by Litak, Budhraja and Saha [13], where the authors confirmed the results by the calculation of II. THE 0-1 TEST [2] maximal Lyapunov exponent. Other systems where the test has been applied are strange non-chaotic by Dawes Consider a sequence of scalar output data Ø(n). and Freeland [4], where the authors concluded that the test Choose c >0 and define performs extremely well. Also the test has been applied on , n = 1, 2, 3, … (1) nonlinear dynamical system including fractional order Now calculate the total mean square displacement: dynamical system by Hui and Cong-Xu [11]. Plasma is a , highly complex system exhibiting a rich variety of nonlinear and the asymptotic growth rate dynamical phenomena. Chowdhury, Iyenger and Lahiri [3],

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ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 4, Issue 10, April 2015 , Duffing map are drawn and corresponding Lyapunov exponents have been obtained. Figure 1(a) shows the To avoid negative values of K , we may as well take attractor for parameters a = 2.77, b = 0.1. The value of K = . 0.807058. Figure 1(b) shows the Lyapunov exponents are all If the behavior of p(n) is asymptotically Brownian i.e. the positive, indicating chaos. Figure 1(c) shows the plot of DLI underlying dynamics is chaotic , then M(n) grows linearly in for this attractor. The value of K, DLI and positive Lyapunov time ; whereas if the behavior of p(n) is bounded(as in case of exponents are all in agreement. periodic and Quasi periodic motion),then M(n) is also bounded. The asymptotic growth rate K of M(n) is then numerically 1.5 determined by means of linear regression of log(M(n)) versus log(n). The main advantages of the test are: 1.0 1. The origin and nature of the data fed into the diagnostic 0.5 system (1) is irrelevant for the test. 2. The method is independent of the scalar observed and 0.0 almost any choice of c will serve. 0.5 3. The dimension of the underlying dynamical system does not pose any practical limitations on the method as in the case 1.0 for traditional methods involving reconstruction. The only conditions which are necessary to be met while 1.5 working with the 0-1 test are: 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1. Initial transients should have died out so that the 1 (a) trajectories are on (or close to) the attractor at the time zero. LCE 2. The time series is long enough to allow for asymptotic 0.734 behavior of p(n). 3. It is necessary that the data is essentially stationary as well 0.733 as deterministic. 0.732

III. DYNAMIC LYAPUNOV INDICATOR [15] 0.731 LCE The dynamic Lyapunov indicator is defined by the largest 0.730 value estimated from all eigen value of Jacobian matrix J 0.729 such that 0.728

1000 1050 1100 1150 1200 1250 1300 of the examined map for all discrete times. We plot the n largest eigen value at every time step of the evolving 1 (b) Jacobian matrix and we observe that these eigenvalues form a definite pattern for regular motion and are distributed 6 randomly for chaotic orbits. 5 IV. DUFFING MAP– APPLICATION OF DLI AND 0-1 TEST 4 There are two main types of dynamical systems: differential equations and iterated maps(also called 3 difference equation). Differential equation describes the continuous time evolution of the system, whereas difference 2 equation describes the discrete time evolution of the system. The Duffing map is a discrete time dynamical system. It is an 1 example of a dynamical system that exhibits chaotic 0 200 400 600 800 1000 behavior. The Duffing map takes a point (xn, yn) in the plane and maps it to a new point given by, 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 Now, we will apply the 0-1 test and DLI to Duffing map. Lyapunov exponent, value of K near to 0 and regular DLI all For different values of parameters a and b, attractors of indicate regular motion.

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ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 4, Issue 10, April 2015 1.4 LCE

1.2 0.828

1.0

0.826

0.8 LCE 0.6 0.824

0.4 0.822

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.820 2 (a) 1000 1050 1100 1150 1200 1250 1300 LCE n 3 (b)

0.267 8

0.268

6 LCE

0.269

4 0.270

1000 1050 1100 1150 1200 1250 1300 2 n 2 (b) 0 200 400 600 800 1000 3.0 3 (c) Figures 4(a), (b) and (c) show respectively the attractor for 2.5 parameters a = 2.75, b = 0.4, the LCE and the DLIs. The value

2.0 of K = 0.01008, which shows regular motion, also well indicated by negative LCE and regular DLIs. 1.5

1.0 2.0

0.5 0 200 400 600 800 1000 1.5 2 (c)

Figures 3(a), (b) and (c) are respectively plots of attractor 1.0 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 0.5 of K, positive Lyapunov exponents and randomly distributed DLI’s indicate chaos. 0.0 0.0 0.5 1.0 1.5 2.0 2 4 (a) LCE

0.2194 1

0.2196

0 LCE 0.2198

0.2200 1

1000 1050 1100 1150 1200 1250 1300 n 2 4 (b) 2 1 0 1 2 3 (a)

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ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 4, Issue 10, April 2015

2.5 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 2.0 agreement 2

1.5

1

1.0

0

0 200 400 600 800 1000 4 (c) 1

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 2 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1. Figure 5(c) shows Irregular pattern of DLI, 5(b) shows 6(a) positive Lyapunov exponents which leads to same result. LCE

0.882 1.5

1.0 0.880

0.5

0.878 LCE 0.0

0.876 0.5

1.0 0.874

1.5

1000 1050 1100 1150 1200 1250 1300 1.5 1.0 0.5 0.0 0.5 1.0 1.5 5 (a) n 6 (b) LCE 8 0.788

0.786 6 0.784

0.782 4 LCE

0.780

2 0.778

0.776 0 200 400 600 800 1000 1000 1050 1100 1150 1200 1250 1300 6(c) n 5 (b) V. CONCLUSION

We conclude that the DLI is quite efficient in analyzing

6 various types of motions in Duffing map and it can be regarded as a good indicator of chaotic and periodic motion 5 with its prediction being comparable with that of Lyapunov exponent and 0-1 test parameter K. It exhibits satisfactory 4 results for various attractors of Duffing map. As we see that

3 in cases (1), (3), (5), (6), irregular pattern of DLI shows chaotic motion, and the same results are also obtained from 2 positive Lyapunov exponents and value of 0-1 test parameter K which is very near to 1, and in case (2) and (4), definite 1 pattern of DLI shows regular motion and the results are in agreement with negative Lyapunov exponents and value of K 0 0 200 400 600 800 1000 which is very near to 0.It is important to verify this to other 5 (c) discrete and continuous systems also.

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ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 4, Issue 10, April 2015 REFERENCES AUTHOR BIOGRAPHY [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. [2] Budhraja, M., Kumar, N. and Saha, L.M, The 0-1 test Applied to Peter De Jong Map,(2012) International Journal of Engineering and Innovative Technology, Volume 2, 253-257. [3] Chaowdhury, D.R., Iyenger, A.N.S. and Lahiri, S., (2012)

Gottwald Melbourne (0-1) test for chaos in a plasma, Nonlinear Process in Geophysics, 19, 53-56. Aysha Ibraheem is a research scholar in the Department of Mathematics, [4] Dawes, J.H.P and Freeland, M.C., The 0-1 test for chaos and University of Delhi, Delhi-110007 doing her Ph.D. under the supervision of strange non chaotic attractors, Preprint. Dr. Mridula Budhraja and Professor Ayub Khan (Co-supervisor). Her main [5] Falconer,I., Gottwald, G.A., Melbourne, I. and Wormnes, K. area of research is Non-Linear Dynamical System and Chaos Control. (2007) Application of the 0-1 test for chaos to experimental data”, SIAM J. Appl. Dyn. Sys. 6 (2), 395-402. [6] Froeschle, C., Gonczi,R., and Lega, E. (1997 b) The Fast Lyapunov Indicators: A Simple Tool to Detect Weak Chaos ,Application to the Structure of the Main Asteroidel Belt, Planetry and Space Science, 45, 881-886. [7] Gottwald, G.A. and Melbourne, I. (2004). A new test for chaos in deterministic systems, Proc. Roy. Soc. A, 460, 603 – 611.

[8] Gottwald, G.A. and Melbourne, I. (2005) Testing for chaos in Narender Kumar is presently working as an associate professor in the deterministic systems with noise, Physica D, 212,100 – 110. Department of Mathematics, Aryabhatta College, University of Delhi, New [9] Gottwald , G.A. and Melbourne ,I. (2009) On the validity of the Delhi-110021. He has done his Ph.D. in September, 2008 from University of 0 -1 test for chaos, Non linearity, 22 (6), 1367. Delhi, Delhi under the supervision of Prof. (Mrs.) Davinder Bhatia and [10] Kantz, H., A robust method to estimate the maximal Lyapunov exponent of a time series, (1994) Physics Letters A, 185 (1), Prof. S.C. Arora. His title of Ph.D. thesis is “Vector Optimization Involving 77 – 87. n-Set Functions” He has published a total of 7 research papers. [11] Ke-Hui , S., Xuan, L. and Cong – Xu, Z. (2010) The 0-1 test algorithm for chaos and its application, Chinese Physics B, 19 (11), 2010. [12] Korsch, H.J. and Jodl. H. J., Chaos: A Program Collection for the P C, 2nd Edition, Springer, New York. [13] Litak, G., Syta, A., Budhraja, M. and Saha, L.M. (2009) Detection of the chaotic behavior of a bouncing ball by 0-1 test, Chaos, Solitons and 42(3), 1511-1517. [14] Saha, L.M., Budhraja. M. (2007) The Largest eigenvalue: An Indicator of Chaos? Int. J. Appl. Math and Mech. 3 (1) , 61-71. [15] Yuasa, M. and Saha, L.M.(2007), Indicators of Chaos, Preprint. [16] Deleanu, D. (2011) Dynamic Lyapunov Indicator: A Practical Tool for Distinguishing between Ordered and Chaotic Orbits in Discrete Dynamical Systems, Proceedings of the 10th WSES International Conference on Non-Linear Analysis Non-Linear Systems and Chaos (NOLSAC' Iasi, Romania, 117-122. [17] Saha, L. M., Tehri, R. (2010) Applications of recent Indicators of regularity and chaos to discrete maps, Int. J. Appl. Math and Mech. 6 (1): 86-93. [18] Saha, L. M., Sharma, R. (2013) Measure in simple type food chain system, Journal of Advances in Mathematics vol 5, 590-598.

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