A discrete chotic multimodel sed on D Hénon mps Ameni Dridi, Rni Lind Filli, Mohmed Benreje 3 LA.R.A, Automtique, ENIT, BP 37, Le Belvédère, 00 Tunis, Tunisi. meni.dridi.ing@gmil.com rni_lind@hotmil.fr 3 mohmed.enreje@enit.rnu.tn Astrct In this pper, we propose to design new fmily of discrete-time chotic systems sed on the use of the multimodel pproch. This pproch is used to represent wy to generte new fmily of discrete-time chotic systems, from p discrete time chotic sis models. Our results re illustrted on the specific cse of new discrete-time chotic multimodel sed on two chotic D Hénon mps with two different sets of prmeters leding to different ehviors. The chos chrcteriztion of otined multimodels is performed using ifurction digrms. Inde Terms Chos, multimodels, discrete-time, Hénon mp, ifurction digrm I. INTRODUCTION In the recent yers, there is growing interest to the use of chos-sed techniques in the secure communiction field. Chotic systems proved tht they re efficient to uild roust cryptosystems due to their severl fetures especilly the noiselike time series nd the sensitive dependence on initil conditions [-4]. In order to hve n efficient cryptosystem, some rules, detiled in [5], need to e pplied where the compleity of the chotic used system is considered s fundmentl issue for ll types of cryptosystems. In prllel, glol pproch sed on multiple Liner Time Invrint (LTI) models defined round different operting point hs received significnt ttention. This multimodel pproch is conve polytopic representtion tht cn e otined y the interpoltion of LTI models. Every model represents vlid operting rnge. Three techniques re used to otin the mutimodel either y identifiction [6-9] when input nd output dt re ville or y lineriztion round different operting points or y polytopic trnsformtion[6-9], if we hve the nlytic model. Numerous works ws pulished concerning the mutimodel pproch nd its stility study [6-9]. In this pper, the mutimodel pproch is used to uild new clss of discrete-time chotic systems which constitutes n etension of previous results of continuous chotic processes using the mutimodel pproch. In [0] Cherrier nd Boutye hve proposed to use the definition of multimodel to interpolte continuous chotic susystems. It s proven tht the resulting system hs comple chotic ehvior. The pper is orgnized s follows: Section II presents the wy to design multimodel sed on p discrete time chotic susystems hving different prmeters nd interpolte them using the pproprite ctivtion function. The specific cse of discrete time multimodels sed on two D Hénon mps is presented in section III. It is lso tested in this section; the chotic ehviors through ifurction digrms nd concluding remrks re given. In section IV, cse of interpoltion of chotic nd non-chotic D Hénon mps is considered. II. BUILDING A NEW CHAOTIC DISCRETE-TIME MULTIMODEL: PROBLEM STATEMENT Consider the n-dimensionl discrete-time in Lurie systems s follows ( k ) A f ( ), i,,..., p () i i kt is the discrete-time, T smpling time, vector A i, i,,..., p, re n n n R is the stte constnt mtrices nd f i ( ), i,,..., p, nonliner vector. The mutimodel pproch proposed in [0] is etended to the cse of discrete-time chotic systems to interpolte p susystems hving different ehviors. The new multimodel, resulting from the interpoltion of systems () with different sets of prmeters, is descried s following p ( k ) ( )( A f ( ) i i i i y C ()
where y is the output vector, C constnt mtri with n pproprite size nd i, i,,..., p ctivtion functions modeling the weighting of the su-model i, chrcterized in the glol model, y A i, i,,..., p such us The prmeters chosen for the two sic models (4) nd (5) re such s =.4, = 0.3, =.5, =0.4 with initil vlues 0 0, 0, 0 [0-4]. p i i 0 i i, p (3) Since the multimodel is uilt in order to e integrted in cryptosystem nd for the purpose of incresing security, the ctivtion functions hve to e chosen such s they ensure kind of miing etween the different su-models. It doesn t hve to fvor model, ut llows rel trnsition etween them. This llows in one hnd, to enhnce the compleity of the system nd, secondly, to ensure continuous synchroniztion, in the sense tht there is no loss of synchroniztion [0]. In the net section, re proposed two multimodels corresponding to () uilt from two susystems hving two different ehviors using n pproprite ctivtion function. The first mutimodel is comintion of two chotic systems nd the second comintion of chotic nd non-chotic system. Bifurction digrms of the otined multimodels re used to show if they re chotic or not. Fig. The chotic ttrctor of the Hénon mp for nd 00,0 =.4, = 0.3 III. IMPLEMENTATION OF THE CHAOTIC D HÉNON MAPS In this section, for this first emple, we hve chosen s se models two systems of D Hénon mps, with two different sets of prmeters. Considered first discrete-time D Hénon susystem, which is descried s follows [0-4] where k, k ( k ) ( k ) (4) is the stte vector nd nd re ifurction prmeters of Hénon mp. To uild the multimmodel, the system (4) is interpolted with the following Hénon mp using two different sets of prmeters chrcterizing y two different chotic ehviors. ( k ) ( k ) The corresponding ttrctors re found respectively in Fig. nd Fig.. (5) Fig. Chotic ttrctor of Hénon mp for 0 0, 0 =.5, =0.4 nd Once cn note tht the first susystem Hénon mp does not hve strnge ttrctor for ll vlues of the prmeters nd. For emple, y keeping fied t 0.3, the ifurction digrm of Fig. 3 shows tht for 0.4. the Hénon mp hs stle periodic orit. Besides, s presented in Fig. 4, for 0.4 nd, the second susystem discrete-time Hénon mp (5) hs chotic ehvior, illustrted y the ifurction digrm of Fig. 4. The chosen ctivtion function is descried s following [0] ( ) ( tnh( )) / (6)
where is prmeter set so tht the μ function performs rel trnsition etween the two Hénon susystems. The resulting multi-model simultions re shown in Fig. 5, ws set t the vlue of 0.5. 0.65,.. The mutlimodel otined from the comintion of two chotic systems (4) nd (5) gives us lrger intervl of prmeters vlues which is dvntgeous to the security of the encrypting scheme [5]. Fig 3. The ifurction digrmmen of the Hénon mpfor =0.3, nd 0 0, 0 vrile, Fig 5. The chotic ttrctor of the multimodel for =.5, = 0.4 nd 0 0, 0 =.4, =0.3, Fig 4. Bifurction digrmme Hénon mp for vrile, 0 0, 0 =0.4 nd Bifurction digrm Fig. 6 shows tht the chotic ehvior of the multimodel is otined for 0.9,.4, such s 0.3, =0.4 nd.5 while, for the sme fied vlues, the chotic ehvior of (4) is otined for.5,.4 s shown in Fig.3. The intervl size of the multimodel s vlues originting chos is lrger thn those of system (4). The sme pplies is otined for the multimodel y vrying the prmeter nd for the sme fied vlues. In fct, s it is shown in Fig.6 the chotic ehvior of the multimodel is otined for.,.. While the chotic ehvior of (5) is otined for Fig 6.Bifurction digrm of the discrete-time multimodel for =0.3, =.5, =0.4 nd 0 0, 0 Fig 6.Bifurction digrm of the discrete-time multimodel for =.4, = 0.3, =0.4 nd 0 0, 0 vrile, vrile,
IV. INTERPOLATION OF A CHAOTIC AND A NON- CHAOTIC D HÉNON MAPS In this section, for this second emple, we hve chosen s se models two systems: chotic D Hénon susystem (4) with fied prmeters Hénon susystem (5). =.4, For the set prmeter 0.3, =0.3, nd non-chotic D 0.9 with initil vlues 0 0, 0, the susystem doesn t present chotic ehvior s it is shown in Fig. 7. In fct, the figure doesn t illustrte strnge ttrctor nd the ifurction digrm Fig. 8 shows tht chosen prmeters doesn t led to chos. However, ifurction digrm of Fig. 0 shows tht for the chosen prmeter the multimodel doesn t hve chotic ehvior. Figure 9. Attrctor of the otined multimodel for 0.3, 0.9 nd 0 0, 0 =.4, = 0.3, Fig 7. Attrctor of the non chotic Hénon mp for 0.3, nd 0 0, 0 0.9 Figure 0. Bifurction digrm of the multimodel for 0.3, 0.9 nd 0 0, 0 vrile, = 0.3, V. CONCLUSION Fig 8. Bifurction digrm of the Hénon mp for vrile, 0.9 nd 0 0, 0 The simultion results of Fig. 9 don t give cler illustrtion of the multimodel ttrctor otined from the interpoltion of the first susystem (4) chrcterized y =.4, = 0.3 nd the second susystem (5) chrcterized y 0.3, 0.9 nd for the chosen ctivtion function (6). The multimodel pproch is used in this pper to uild discrete-time chotic multimodels. It hs een shown, y the use of discrete time Hénon mp D, tht the interpoltion of two chotic systems cn enhnce the compleity of the chos however miing non-chotic system with chotic one doesn t led necessry to chos. Bifurction digrms illustrte the rnge of possile prmeters, giving to the multimodel, chotic ehvior. REFERENCES [] A. V. Oppenheim, K. M. Cuomo nd S. H. Strogtz, Synchroniztion of Lorenz-sed chotic circuits with pplictions to communictions, IEEE Trns. on Circ. Syst. II, vol.40, no.0, pp. 66 633, 993.
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