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

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1 Attractors of Duffig Map: Applicatio of DLI ad - Test Aysha Ibraheem, Nareder Kumar Departmet of Mathematics, Uiversity of Delhi, Delhi-7, Idia. Associate Professor, Departmet of Mathematics, Aryabhatta College, Uiversity of Delhi, New Delhi-, Idia. Abstract I this paper we study various attractors of Duffig map. We apply - test ad Dyamic Lyapuov Idictor to distiguish betwee periodic ad chaotic behavior of various attractors of Duffig map. For differet set of values of parameters of this map chaotic attractors are draw ad correspodig plots of Lyapuov expoets ad Dyamic Lyapuov Idicator have bee obtaied. We evaluate - test parameters i each case ad compare the results obtaied from - test parameter, Lyapuov expoet, ad Dyamic Lyapuov idicator. Idex Terms - test, Duffig map, Dyamic Lyapuov idicator, Lyapuov expoets. I. INTRODUCTION Sice the discovery of chaotic dyamics i weather systems by Lorez i 93 expasive iterest by researchers has demostrated the presece of chaotic dyamics i multitude of atural ad ma-made systems i almost all sphere of life. A chaotic system is a highly complex dyamic oliear system ad its respose exhibits sesitivity to the iitial coditios. The sesitive ature of chaotic systems is commoly called as the butterfly effect. Chaos theory has bee applied to a variety of fields such as physical systems, chemical reactors, secure commuicatio etc. To distiguish betwee chaotic ad periodic motio there are several methods. The most commo tests are Lyapuov expoet [] ad maximal Lyapuov expoet []. Fast Lyapuov Idicator [], Smaller Aligmet Idex [] ad Dyamic Lyapuov Idicator [5] are some other tests that have bee used. The - test was first suggested by Melboure ad Gottwald [7-8]. Gottwald ad Melboure [9] have preseted a theoretical justificatio of the test. The - test is uiversally applicable test which yields for regular motio ad for chaotic motio ad which is easy to apply to ay cotiuous ad discrete dyamical system. The test has bee applied to may systems like the two dimesioal map of a boucig ball system by Litak, Budhraja ad Saha [3], where the authors cofirmed the results by the calculatio of maximal Lyapuov expoet. Other systems where the test has bee applied are strage o-chaotic attractor by Dawes ad Freelad [], where the authors cocluded that the test performs extremely well. Also the test has bee applied o oliear dyamical system icludig fractioal order dyamical system by Hui ad Cog-Xu []. Plasma is a highly complex system exhibitig a rich variety of oliear dyamical pheomea. Chowdhury, Iyeger ad Lahiri [3], have applied the - test to the time series obtaied from a glow discharge plasma experimet, ad it is foud to be very effective ad simpler tha the estimatio of the largest lyapuov expoet. The uiversal techique to examie the ature of motio i determiistic systems is to calculate maximal Lyapuov expoet but Falcoer, Gottwald, Melboure ad Wormes [5] have aalyzed data comig from a experimetal set up of a bipolar motor i a alteratig magetic field ad they ivestigated the performace of - test. Budhraja [] have also applied - test to Peter-de-Jog map ad the author cocluded that the - test ca be regarded as a good idicator of chaotic or periodic/quasi-periodic motio. Dyamic Lyapuov Idicator (DLI) was suggested by Saha ad Budhraja []. The authors applied DLI to various attractors of Gumowski Mira map ad compared the results with those obtaied usig fast Lyapuov Idicator (FLI), Smaller Aligmet Idex (SALI). Yuasa ad Saha [5] studied Burger s map, Chirikov map, ad boucig ball dyamics usig DLI. They have also compared the results with FLI, SALI ad - test. They cocluded that DLI provides a clear picture for idetificatio of regular ad chaotic motio for all these maps. Deleau [ ] aalyzed the behavior of the -D Lozi map, the -D predator prey map ad the 3-D Loretz BD map with the help of DLI ad results were foud satisfactory. DLI was applied to Duffig map ad Ikeda map ad results were compared to SALI ad FLI by Saha ad Tehri [7] ad DLI exhibits same results as SALI ad FLI. Saha ad Sharma [8] applied DLI to the food chai system ad the results have bee quite satisfactory. I this paper we study the applicatio of DLI ad - test to various attractors of Duffig map. The scheme of the paper is as follows i Sectio II we explai i detail the applicatio of - test, i Sectio III we explai the Dyamic Lyapuov Idicator.I Sectio IV we plot the various attractors of Duffig map, the plot of Lyapuov characteristic expoets, the plots of DLI ad obtai the - test parameters. II. THE - TEST [] Cosider a sequece of scalar output data Ø(). Choose c > ad defie, =,, 3, () Now calculate the total mea square displacemet:, ad the asymptotic growth rate 5

2 , To avoid egative values of K, we may as well take. If the behavior of p() is asymptotically Browia i.e. the uderlyig dyamics is chaotic, the M() grows liearly i time ; whereas if the behavior of p() is bouded(as i case of periodic ad Quasi periodic motio),the M() is also bouded. The asymptotic growth rate K of M() is the umerically determied by meas of liear regressio of log(m()) versus log(). The mai advatages of the test are:. The origi ad ature of the data fed ito the diagostic system () is irrelevat for the test.. The method is idepedet of the scalar observed ad almost ay choice of c will serve. 3. The dimesio of the uderlyig dyamical system does ot pose ay practical limitatios o the method as i the case for traditioal methods ivolvig phase space recostructio. The oly coditios which are ecessary to be met while workig with the - test are:. Iitial trasiets should have died out so that the trajectories are o (or close to) the attractor at the time zero.. The time series is log eough to allow for asymptotic behavior of p(). 3. It is ecessary that the data is essetially statioary as well as determiistic. III. DYNAMIC LYAPUNOV INDICATOR [5] The dyamic Lyapuov idicator is defied by the largest value estimated from all eige value such that of Jacobia matrix J Duffig map are draw ad correspodig Lyapuov expoets have bee obtaied. Figure (a) shows the attractor for parameters a =.77, b =.. The value of K = Figure (b) shows the Lyapuov expoets are all positive, idicatig chaos. Figure (c) shows the plot of DLI for this attractor. The value of K, DLI ad positive Lyapuov expoets are all i agreemet (a) of the examied map for all discrete times. We plot the largest eige value at every time step of the evolvig Jacobia matrix ad we observe that these eigevalues form a defiite patter for regular motio ad are distributed radomly for chaotic orbits. IV. DUFFING MAP APPLICATION OF DLI AND - TEST There are two mai types of dyamical systems: differetial equatios ad iterated maps(also called differece equatio). Differetial equatio describes the cotiuous time evolutio of the system, whereas differece equatio describes the discrete time evolutio of the system. The Duffig map is a discrete time dyamical system. It is a example of a dyamical system that exhibits chaotic behavior. The Duffig map takes a poit (x, y ) i the plae ad maps it to a ew poit give by, = Now, we will apply the - test ad DLI to Duffig map. For differet values of parameters a ad b, attractors of (b) 8 (c) Figure (a) shows the attractor for parameters a =.77, b =.3, ad the value of K comes out to be which shows regular motio. Figure (b) shows ad Figure (c) shows DLI plot for this attractor. Negative Lyapuov expoet, value of K ear to ad regular DLI all idicate regular motio.

3 (a) (b) (b) (c) Figures (a), (b) ad (c) show respectively the attractor for parameters a =.75, b =., the ad the DLIs. The value of K = 8, which shows regular motio, also well idicated by egative ad regular DLIs. 8 (c) Figures 3(a), (b) ad (c) are respectively plots of attractor ad Lyapuov expoet ad DLI for parameters a =.77, b =.. Here, K =.9377 which is very ear to. The value of K, positive Lyapuov expoets ad radomly distributed DLI s idicate chaos... (a) (a) (b) 7

4 Figure (a), (b) ad (c) shows the attractor for parameters a =.88, b = 5, Lyapuov expoets ad DLI respectively. The value of K = 7 ad the results are i agreemet.5. 8 (c) Figure 5(a) shows the attractor for parameters a =.77, b = for which the value of K =.899 which is ear to. Figure 5(c) shows Irregular patter of DLI, 5(b) shows positive Lyapuov expoets which leads to same result. (a) LC E (a) (b) LC E (c) 3 5 (b) V. CONCLUSION 5 (c) We coclude that the DLI is quite efficiet i aalyzig various types of motios i Duffig map ad it ca be regarded as a good idicator of chaotic ad periodic motio with its predictio beig comparable with that of Lyapuov expoet ad - test parameter K. It exhibits satisfactory results for various attractors of Duffig map. As we see that i cases (), (3), (5), (), irregular patter of DLI shows chaotic motio, ad the same results are also obtaied from positive Lyapuov expoets ad value of - test parameter K which is very ear to, ad i case () ad (), defiite patter of DLI shows regular motio ad the results are i agreemet with egative Lyapuov expoets ad value of K which is very ear to.it is importat to verify this to other discrete ad cotiuous systems also

5 REFERENCES [] Boutis, T., Applicatio of the SALI method for Detectig Chaos ad Order i Accelerator Mappigs, dowloaded from URL: gr/ cras. [] Budhraja, M., Kumar, N. ad Saha, L.M, The - test Applied to Peter De Jog Map,() Iteratioal Joural of Egieerig ad Iovative Techology, Volume, [3] Chaowdhury, D.R., Iyeger, A.N.S. ad Lahiri, S., () Gottwald Melboure (-) test for chaos i a plasma, Noliear Process i Geophysics, 9, [] Dawes, J.H.P ad Freelad, M.C., The - test for chaos ad strage o chaotic attractors, Preprit. [5] Falcoer,I., Gottwald, G.A., Melboure, I. ad Wormes, K. (7) Applicatio of the - test for chaos to experimetal data, SIAM J. Appl. Dy. Sys. (), [] Froeschle, C., Goczi,R., ad Lega, E. (997 b) The Fast Lyapuov Idicators: A Simple Tool to Detect Weak Chaos,Applicatio to the Structure of the Mai Asteroidel Belt, Plaetry ad Space Sciece, 5, [7] Gottwald, G.A. ad Melboure, I. (). A ew test for chaos i determiistic systems, Proc. Roy. Soc. A,, 3. [8] Gottwald, G.A. ad Melboure, I. (5) Testig for chaos i determiistic systems with oise, Physica D,,. [9] Gottwald, G.A. ad Melboure,I. (9) O the validity of the - test for chaos, No liearity, (), 37. [] Katz, H., A robust method to estimate the maximal Lyapuov expoet of a time series, (99) Physics Letters A, 85 (), [] Ke-Hui, S., Xua, L. ad Cog Xu, Z. () The - test algorithm for chaos ad its applicatio, Chiese Physics B, 9 (),. [] Korsch, H.J. ad Jodl. H. J., Chaos: A Program Collectio for the P C, d Editio, Spriger, New York. [3] Litak, G., Syta, A., Budhraja, M. ad Saha, L.M. (9) Detectio of the chaotic behavior of a boucig ball by - test, Chaos, Solitos ad Fractals (3), [] Saha, L.M., Budhraja. M. (7) The Largest eigevalue: A Idicator of Chaos? It. J. Appl. Math ad Mech. 3 (), -7. [5] Yuasa, M. ad Saha, L.M.(7), Idicators of Chaos, Preprit. [] Deleau, D. () Dyamic Lyapuov Idicator: A Practical Tool for Distiguishig betwee Ordered ad Chaotic Orbits i Discrete Dyamical Systems, Proceedigs of the th WSES Iteratioal Coferece o No-Liear Aalysis No-Liear Systems ad Chaos (NOLSAC' Iasi, Romaia, 7-. [7] Saha, L. M., Tehri, R. () Applicatios of recet Idicators of regularity ad chaos to discrete maps, It. J. Appl. Math ad Mech. (): [8] Saha, L. M., Sharma, R. (3) Complexity Measure i simple type food chai system, Joural of Advaces i Mathematics vol 5, AUTHOR BIOGRAPHY Aysha Ibraheem is a research scholar i the Departmet of Mathematics, Uiversity of Delhi, Delhi-7 doig her Ph.D. uder the supervisio of Dr. Mridula Budhraja ad Professor Ayub Kha (Co-supervisor). Her mai area of research is No-Liear Dyamical System ad Chaos Cotrol. Nareder Kumar is presetly workig as a associate professor i the Departmet of Mathematics, Aryabhatta College, Uiversity of Delhi, New Delhi-. He has doe his Ph.D. i September, 8 from Uiversity of Delhi, Delhi uder the supervisio of Prof. (Mrs.) Davider Bhatia ad Prof. S.C. Arora. His title of Ph.D. thesis is Vector Optimizatio Ivolvig -Set Fuctios He has published a total of 7 research papers. 9