R eceiv ed M a rch 10, 2008 ; R ev ised J u ly 31, 2008

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1 Letters International Journal of Bifurcation and Chaos, Vol. 19, No. 3 (009) c World Scientific Publishing Company THE DISCRETE HYPERCHAOTIC DOUBLE SCROLL Z E R AO U L IA E L H AD J Department of Mathematics, University of Tébessa, 1000, Algeria zeraoulia@mail.univ-tebessa.dz zelhadj1@yahoo.fr J. C. S P R O T T Department of Physics, University of Wisconsin, Madison, WI 53706, USA sprott@physics.wisc.edu R eceiv ed M a rch 10, 008 ; R ev ised J u ly 31, 008 In th is pa per w e presen t a n d a n a ly ze a n ew piecew ise lin ea r m a p o f th e pla n e ca pa b le o f g en e- ra tin g ch a o tic a ttra cto rs w ith o n e a n d tw o scro lls. D u e to th e sh a pe o f th e a ttra cto r a n d its h y perch a o ticity, w e ca ll it the discrete hyperchaotic double scroll. It h a s th e sa m e n o n lin ea rity a s u sed in th e w ell-k n o w n C h u a circu it. A rig o ro u s pro o f o f th e h y perch a o ticity o f th is a ttra cto r is g iv en a n d n u m erica lly ju stifi ed. Keywords: P iecew ise lin ea r m a p; b o rd er co llisio n b ifu rca tio n ; d iscrete h y perch a o tic d o u b le scro ll. 1. Introduction It is w ell-k n o w n th a t if tw o o r m o re L y a pu n o v ex po n en ts o f a d y n a m ica l sy stem a re po sitiv e th ro u g h o u t a ra n g e o f pa ra m eter spa ce, th en th e resu ltin g a ttra cto rs a re h y perch a o tic. T h e im po r- ta n ce o f th ese a ttra cto rs is th a t th ey a re less reg u - la r a n d a re seem in g ly a lm o st fu ll in spa ce, w h ich ex pla in s th eir im po rta n ce in fl u id m ix in g [S ch eizer & H a sler, 1996 ; Ab el et al., 1997 ; O ttin o, 198 9; O ttin o et al., 199]. O n th e o th er h a n d, th e a ttra c- to rs g en era ted b y C h u a s circu it [C h u a et al., ] g iv en b y x = α(y h(x)), y = x y + z, z = βy a re a sso cia ted w ith sa d d le-fo cu s h o m o clin ic lo o ps a n d a re n o t h y perch a o tic, w h ere h(x) = (m 1 x + (m 0 m 1 )( x + 1 x 1 ))/. T h e d o u - b le scro ll a ttra cto r fo r th is ca se is sh o w n in F ig. 1. T h e d o u b le scro ll is m o re co m plex th a n th e L o ren z-ty pe a n d th e h y perb o lic a ttra cto rs [M ira, 1997 ], a n d th u s it is n o t su ita b le fo r so m e po ten tia l a pplica tio n s o f ch a o s su ch a s secu re co m m u n ica tio n s a n d sig n a l m a sk in g [K a pita n ia k et al., 1994; T h a m ilm a ra n et al., 004]. H y perch a o tic a ttra c- to rs m a k e ro b u st to o ls fo r so m e a pplica tio n s, b u t th is circu it d o es n o t ex h ib it h y perch a o s b eca u se o f its lim ited d im en sio n a lity [C h u a et al., ]. T o reso lv e th is pro b lem, sev era l w o rk s h a v e fo cu sed o n th e h y perch a o tifi ca tio n o f C h u a s circu it u sin g sev - era l tech n iq u es su ch a s co u plin g m a n y C h u a circu its a s in [K a pita n ia k et al., 1994] w h ere a 15 -D d y n a m ica l sy stem is o b ta in ed. H o w ev er, th e resu ltin g sy stem is co m plica ted a n d d iffi cu lt to co n stru ct. A sim pler m eth o d in tro d u ces a n a d d itio n a l in d u cto r in th e ca n o n ica l C h u a circu it a s g iv en in [T h a m ilm a ra n et al., 004], w h ere a 4-D d y n a m ica l sy stem is o b ta in ed th a t co n v erg es to a h y perch a o tic a ttra c- to r b y a b o rd er co llisio n b ifu rca tio n [B a n erjee & G reb o g i, 1999]. O n th e o th er h a n d, th e stu d y o f piecew ise lin ea r m a ps [D ev a n ey, 198 4; C a o & L iu, 1998 ; Ah a ro n o v et al., 1997 ; Ash w in & F u, 00] 103

2 104 Z. Elhadj & J. C. Sprott F ig. 1. T he classic double scroll attractor obtained for α = 9.35, β = 14.79, m 0 = 1/7, m 1 = /7 [Chua et al., ]. ca n co n trib u te to th e d ev elo pm en t o f th e th eo ry o f d y n a m ica l sy stem s, especia lly in fi n d in g n ew ch a o tic a ttra cto rs w ith a pplica tio n s in scien ce a n d en g in eerin g [S ch eizer & H a sler, 1996 ; Ab el et al., 1997 ]. F u rth erm o re, th e tech n iq u es em plo y ed in th e circu it rea liza tio n o f sm o o th m a ps a re sim ple, a n d th e a ppro a ch ca n b e ex ten d ed to o th er sy stem s su ch a s piecew ise lin ea r o r piecew ise sm o o th m a ps [S u n eel, 006 ]. Also, it seem s th a t th e circu it rea liza tio n s o f lo w -d im en sio n a l m a ps is sim pler th a n w ith h ig h - d im en sio n a l co n tin u o u s sy stem s. F o r th is rea so n, w e presen t a d iscrete v ersio n o f C h u a s circu it a ttra cto r g o v ern ed b y a sim ple -D piecew ise lin ea r m a p th a t is ca pa b le o f pro d u cin g h y perch a o tic a ttra cto rs w ith th e sa m e sh a pe a s th e cla ssic d o u b le scro ll a ttra c- to r, w h ich is n o t h y perch a o tic. W e a n a ly tica lly sh o w th e h y perch a o ticity o f th e a ttra cto r a n d n u m erica lly sh o w th a t th e pro po sed m a p b eh a v es in a sim ila r w a y to th e 4-D d y n a m ica l sy stem g iv en in [T h a m ilm a ra n et al., 004], i.e. b o th h y perch a o tic a ttra cto rs a re o b ta in ed b y a b o rd er co llisio n b ifu rca tio n.. T h e D iscre te H y p e rch a otic D oub le S croll M a p In th is sectio n, w e presen t th e n ew m a p a n d sh o w so m e o f its b a sic pro perties. C o n sid er th e fo llo w in g -D piecew ise lin ea r m a p: ( ) x ah(y) f(x, y) = (1) bx w h ere a a n d b a re th e b ifu rca tio n pa ra m eters, h is g iv en a b o v e b y th e ch a ra cteristic fu n ctio n o f th e so -ca lled d o u b le scro ll a ttra cto r [C h u a et al., ], a n d m 0 a n d m 1 a re respectiv ely th e slo pes o f th e in n er a n d o u ter sets o f th e o rig in a l C h u a circu it. S y stem s su ch a s th e o n e in E q. (1) ty pica lly h a v e n o d irect a pplica tio n to pa rticu la r ph y sica l sy stem s, b u t th ey serv e to ex em plify th e k in d s o f d y n a m ica l b eh a v io rs, su ch a s ro u tes to ch a o s, th a t a re co m m o n in ph y sica l ch a o tic sy stem s. T h u s a n a n a ly tica l a n d n u m erica l stu d y is w a rra n ted. D u e to th e sh a pe o f th e n ew a ttra cto r a n d its h y perch a o ticity, w e ca ll it th e d iscrete h y perch a o tic d o u b le scro ll b eca u se o f its sim ila rity to th e w ell-k n o w n C h u a circu it [C h u a et al., ]. O n e o f th e a d v a n ta g es o f th e m a p (1) is its ex trem e sim plicity a n d m in im a lity in v iew o f th e n u m b er o f term s a n d co n serv a tio n o f so m e im po r- ta n t pro perties o f th e cla ssic d o u b le scro ll. F irstly, th e a sso cia ted fu n ctio n f(x, y) is co n tin u o u s in R, b u t it is n o t d iff eren tia b le a t th e po in ts (x, 1) a n d (x, 1) fo r a ll x R. S eco n d ly, th e m a p (1) is a d iff eo - m o rph ism ex cept a t po in ts (x, 1) a n d (x, 1) w h en abm 1 m 0 0, sin ce th e d eterm in a n t o f its J a co b ia n is n o n zero if a n d o n ly if abm 1 0 o r abm 0 0, b u t it d o es n o t preserv e a rea a n d it is n o t a rev ersin g tw ist m a p fo r a ll v a lu es o f th e sy stem pa ra m eters. T h ird ly, th e m a p (1) is sy m m etric u n d er th e co o r- d in a te tra n sfo rm io n (x, y) ( x, y), a n d th is tra n sfo rm a tio n persists fo r a ll v a lu es o f th e sy s- tem pa ra m eters. T h erefo re, th e ch a o tic a ttra cto r o b ta in ed fo r m a p (1) is sy m m etric ju st lik e th e cla s- sic d o u b le scro ll [C h u a et al., ]. O n th e o th er h a n d, a n d d u e to th e sh a pe o f th e v ecto r fi eld f o f th e m a p (1), th e pla n e ca n b e d iv id ed in to th ree lin - ea r reg io n s d en o ted b y : R 1 = {(x, y) R /y 1}, R = {(x, y) R / y 1}, R 3 = {(x, y) R /y 1}, w h ere in ea ch o f th ese reg io n s th e m a p (1) is lin ea r. H o w ev er, it is ea sy to v erify th a t fo r a ll v a lu es o f th e pa ra m eters m 0, m 1 su ch th a t m 0 m 1 > 0, th e m a p (1) h a s a sin g le fi x ed po in t (0, 0), w h ile if m 0 m 1 < 0, th e m a p (1) h a s th ree fi x ed po in ts, a n d th ey a re g iv en b y P 1 = ((m 1 m 0 )/bm 1, (m 1 m 0 )/m 1 ), P = (0, 0), P 3 = ((m 0 m 1 )/bm 1, (m 0 m 1 )/m 1 ). O b v io u sly, th e J a co b ia n m a trix o f th e m a p (1) ev a lu a ted a t th e fi x ed po in ts( P 1 a n d ) P 3 is th e sa m e a n d is g iv en 1 abm1 b y J 1,3 =. T h erefo re, th e tw o eq u ilib riu m po in ts P 1 a n d P 3 h a v e th e sa m e sta b ility ty pe. T h e J a co b ia n m a trix o f th e m a p (1) ev( a lu a ted a) t 1 abm0 th e fi x ed po in t P is g iv en b y J =, 1 0 a n d th e ch a ra cteristic po ly n o m ia ls fo r J 1,3 a n d J a re g iv en respectiv ely b y λ λ + abm 1 = 0 a n d

3 T he D iscrete H y perchaotic D ou ble Scroll 105 λ λ + abm 0 = 0, w h ere th e lo ca l sta b ility o f th ese eq u ilib ria ca n b e stu d ied b y ev a lu a tin g th e eig en v a lu es o f th e co rrespo n d in g J a co b ia n m a trices g iv en b y th e so lu tio n o f th eir ch a ra cteristic po ly n o m ia ls. 3. T h e H y p e rch a oticity of th e A ttra ctor In th is sectio n, w e g iv e su ffi cien t co n d itio n s fo r th e h y perch a o ticity o f th e d iscrete h y perch a o tic d o u b le scro ll g iv en b y th e m a p (1). N o te th a t th is pro perty is a b sen t fo r th e cla ssic d o u b le scro ll [K a pita n ia k et al., 1994; T h a m ilm a ra n et al., 004]. It is sh o w n in [L i & C h en, 004] th a t if w e co n - sid er a sy stem x k+ 1 = f(x k ), x k Ω R n, su ch th a t f (x) N < + () w ith a sm a llest eig en v a lu e o f f (x) T f (x) th a t sa tisfi es λ m in (f (x) T (f (x))) θ > 0, (3) w h ere N θ, th en, fo r a n y x 0 Ω, a ll th e L y a pu n o v ex po n en ts a t x 0 a re lo ca ted in sid e [ln θ/, ln N], th a t is, ln θ l i(x 0 ) ln N, i = 1,,..., n, (4) w h ere l i (x 0 ) a re th e L y a pu n o v ex po n en ts fo r th e m a p f. F o r th e m a p (1), o n e h a s th a t a n d b + a m 1 + b + b 4 + a m 1 + a4 m 4 1 a b m , if y 1 f (x, y) = b + a m 0 + b + b 4 + a m 0 + a4 m 4 0 a b m , if y 1 λ m in (f (x) T (f (x))) = b + a m 1 b + b 4 + a m 1 + a4 m 4 1 a b m , if y 1 < + (5 ) b + a m 0 b + b 4 + a m 0 + a4 m 4 0 a b m , if y 1. (6 ) If ( ) 1 a > m a x m 1, 1 m 0 ( ) (7 ) am 1 am b > m a x 0 a m 1 1, a m 0 1 th en b o th L y a pu n o v ex po n en ts o f th e m a p (1) a re po sitiv e fo r a ll in itia l co n d itio n s (x 0, y 0 ) R, a n d h en ce th e co rrespo n d in g a ttra cto r is h y perch a o tic. F o r m 0 = 0.43 a n d m 1 = 0.41, o n e h a s th a t a >. 439, a n d fo r b = 1.4, o n e h a s th a t a > As a test o f th e prev io u s a n a ly sis, F ig. sh o w s th e L y a - pu n o v ex po n en t spectru m fo r th e m a p (1) fo r m 0 = 0.43, m 1 = 0.41, b = 1.4, a n d a T h e reg io n s o f h y perch a o s a re a 3.33 a n d 3.33 a O n th e o th er h a n d, th e d iscrete h y perch a o tic d o u b le scro ll sh o w n in F ig. 3 resu lts fro m a sta - b le perio d -3 o rb it tra n sitio n in g to a fu lly d ev elo ped ch a o tic reg im e. T h is pa rticu la r ty pe o f b ifu rca tio n is ca lled a b o rd er-co llisio n b ifu rca tio n a s sh o w n in F ig. 4, a n d it is th e o n ly o b serv ed scen a rio. If w e fi x pa ra m eters b = 1.4, m 0 = 0.43, a n d m 1 = 0.41 a n d v a ry a R, th en th e m a p (1) ex h ib its th e fo l- lo w in g d y n a m ica l b eh a v io rs a s sh o w n in F ig. 4. In th e in terv a l a < , th e m a p (1) d o es n o t co n v erg e. F o r a , th e m a p (1) b eg in s w ith a rev erse b o rd er-co llisio n b ifu rca tio n, lea d in g to a sta b le perio d -3 o rb it, a n d th en co lla pses to a po in t th a t is reb o rn a s a sta b le perio d -3 o rb it lea d in g to fu lly d ev elo ped ch a o s. F o r a > , th e m a p (1) d o es n o t co n v erg e. H o w ev er, it seem s th a t th e pro po sed m a p b eh a v es in a sim ila r w a y to th e 4-D d y n a m ica l sy stem g iv en in [T h a m ilm a ra n et al., 004], i.e. b o th h y perch a o tic a ttra cto rs a re o b ta in ed b y a b o rd er-co llisio n b ifu rca tio n [B a n erjee & G reb o g i, 1999]. F ig u re 5 sh o w s reg io n s in th e ab-pla n e g iv en b y (a, b) [ , ] [, ] o f u n b o u n d ed

4 106 Z. Elhadj & J. C. Sprott F ig.. Variation of the L yapunov ex ponents of map (1) v ersus the parameter a w ith b = 1.4, m 0 = 0.43, and m 1 = F ig. 5. R egions of dynamical behav iors in the ab-plane for the map (1). (w h ite), perio d ic o rb its o f perio d s 1 a n d 3 (b lu e), a n d ch a o tic (in clu d in g h y perch a o tic a ttra cto rs) (red ) so lu tio n s in th e ab-pla n e fo r th e m a p (1), w ith 10 6 itera tio n s fo r ea ch po in t. F ig. 3. T he discrete hyperchaotic double scroll attractor obtained from the map (1) for a = 3.36, b = 1.4, m 0 = 0.43, and m 1 = 0.41 w ith initial conditions x = y = 0.1. F ig. 4. T he border collision bifurcation route to chaos of map (1) v ersus the parameter a w ith b = 1.4, m 0 = 0.43, and m 1 = C onclusion W e h a v e d escrib ed a n ew sim ple -D d iscrete piecew ise lin ea r ch a o tic m a p th a t is ca pa b le o f g en era tin g a h y perch a o tic d o u b le scro ll a ttra cto r. S o m e im po r- ta n t d eta iled d y n a m ica l b eh a v io rs o f th is m a p w ere fu rth er in v estig a ted. R e fe re nce s Ab el, A., B a u er, A., K erb er, K. & S ch w a rz, W. [1997 ] C h a o tic co d es fo r C D M A a pplica tio n, Proc. E C C TD 9 7 1, Ah a ro n o v, D., D ev a n ey, R. L. & E lia s, U. [1997 ] T h e d y n a m ics o f a piecew ise lin ea r m a p a n d its sm o o th a ppro x im a tio n, Int. J. B ifurcation and C haos 7, Ash w in, P. & F u, X. C. [00] O n th e d y n a m ics o f so m e n o n h y perb o lic a rea -preserv in g piecew ise lin - ea r m a ps, in Mathematics in Signal Processing V, O x fo rd U n iv. P ress IM A C o n feren ce S eries. B a n erjee, S. & G reb o g i, C. [1999] B o rd er co llisio n b ifu r- ca tio n s in tw o -d im en sio n a l piecew ise sm o o th m a ps, Phys. R ev. E. 5 9, C a o, Y. & L iu, Z. [1998 ] S tra n g e a ttra cto rs in th e o rien ta tio n -preserv in g L o zi m a p, C haos Solit. F ract. 9, C h u a, L. O., K o m u ro, M. & M a tsu m o to, T. [198 6 ] T h e d o u b le scro ll fa m ily, P a rts I a n d II, IE E E Trans. C ircuits Syst. CAS-33,

5 T he D iscrete H y perchaotic D ou ble Scroll 107 D ev a n ey, R. L. [198 4] A piecew ise lin ea r m o d el fo r th e zo n es o f in sta b ility o f a n a rea -preserv in g m a p, Physica D 10, K a pita n ia k, T., C h u a, L. O. & Z h o n g, G.-Q. [1994] E x perim en ta l h y perch a o s in co u pled C h u a s circu its, C ircuits Syst. I: F und. Th. Appl. 4 1, L i, C. & C h en, G. [004] E stim a tin g th e L y a pu n o v ex po n en ts o f d iscrete sy stem s, C haos 14, M ira, C. [1997 ] C h u a s circu it a n d th e q u a lita tiv e th e- o ry o f d y n a m ica l sy stem s, Int. J. B ifurcation and C haos 7, O ttin o, J. M. [198 9] The Kinematics of Mixing: Stretching, C haos, and Transport (C a m b rid g e U n iv ersity P ress, C a m b rid g e). O ttin o, J. M., M u zzio n, F. J., T ja h ja d i, M., F ra n jio n e, J. G., J a n a, S. C. & K u sch, H. A. [199] C h a o s, sy m m e- try, a n d self-sim ila rity : E x plo rin g o rd er a n d d iso rd er in m ix in g pro cesses, Sci. 5 7, S ch eizer, J. & H a sler, M. [1996 ] M u ltiple a ccess co m - m u n ica tio n u sin g ch a o tic sig n a ls, Proc. IE E E ISC AS 9 6, Atla n ta, U S A, 3, 108. S u n eel, M. [006 ] E lectro n ic circu it rea liza tio n o f th e lo g istic m a p, Sadhana Acad. Proc. E ngin. Sci. IAS 31, T h a m ilm a ra n, K., L a k sh m a n a n, M. & V en k a tesa n, A. [004] H y perch a o s in a m o d ifi ed ca n o n ica l C h u a s circu it, Int. J. B ifurcation and C haos 14, 1 44.