COEXISTENCE OF POINT, PERIODIC AND STRANGE ATTRACTORS

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1 Interntionl Journl of Bifurction nd Chos, Vol., No. 5 () 59 (5 pges) c World Scientific Publishing Compn DOI:./S8759 COEISTENCE OF POINT, PERIODIC AND STRANGE ATTRACTORS JULIEN CLINTON SPROTT Deprtment of Phsics, Universit of Wisconsin, Mdison, WI 576-9, USA IONG WANG nd GUANRONG CHEN Deprtment of Electronic Engineering, Cit Universit of Hong Kong, Kowloon, Hong Kong wngxiong8686@gmil.com Received November, For dnmicl sstem described b set of utonomous ordinr differentil equtions, n ttrctor cn be point, periodic ccle, or even strnge ttrctor. Recentl, new chotic sstem with onl one stble equilibrium ws described, which locll converges to the stble equilibrium but is globll chotic. This pper further shows tht for certin prmeters, besides the point ttrctor nd chotic ttrctor, this sstem lso hs coexisting stble limit ccle, demonstrting tht this new sstem is trul complicted nd interesting. Kewords: Stble equilibrium; periodic solution; strnge ttrctor.. Introduction Mn dnmicl sstems in the phsicl world re b nture dissiptive. Such dissiption m come from internl friction, thermodnmic loss, energ or mteril loss, mong mn cuses. Orbits of dissiptive dnmicl sstem will shrink into zerovolume subsets in the stte spce s time goes to infinit. For dnmicl sstem described b set of utonomous ordinr differentil equtions (ODEs), ẋ = f(x), x R n,iff(x e ) = hs rel solution, then x e is clled n equilibrium of the dnmicl sstem. An equilibrium is sid to be hperbolic if ll eigenvlues of the sstem s Jcobin mtrix evluted t the equilibrium hve nonzero rel prts. The signs of these rel prts of the eigenvlues determine the stbilit of the equilibrium. The Hrtmn Grobmn Theorem [Teschl, ] sttes tht the behvior of dnmicl sstem ner hperbolic equilibrium is topologicll equivlent to (i.e. qulittivel the sme s) the behvior of its linerized model in neighborhood of the equilibrium. Thus, if the equilibrium is stble, it will be point ttrctor of the sstem, which ttrcts ll nerb orbits. Besides the zero-dimensionl point ttrctors, there re lso one-dimensionl periodic-ccle ttrctors, clled limit ccles, in which n orbit circles round in the stte spce. Although point ttrctors nd limit ccles re the most common ttrctors with integer dimension nd regulr structure, ttrctors cn lso be complicted point sets with frctl structure. An ttrctor is sid to be strnge if it hs noninteger dimension, nd exmples of such strnge ttrctors re mnifest [Sprott, 99, 99, 997; Sprott & Linz, ; Lorenz, 96; Chen & Uet, 999; Uet & Chen, ]. Author for correspondence 59-

2 J. C. Sprott et l. Most sstems with strnge ttrctors hve t lest one unstble equilibrium. However, in ddition to the forementioned Sprott sstems [Sprott, 99, 99, 997; Sprott & Linz, ], thelorenzsstem [Lorenz, 96] nd the Chen sstem [Chen & Uet, 999; Uet & Chen, ] both hve two unstble sddle-foci nd one unstble node, which cn generte two-wing butterfl-shped strnge ttrctor, usull referred to s chotic ttrctor, forits specil properties chrcterized b sensitive dependence on initil conditions. Of course, it is known tht there re lso strnge but nonchotic ttrctors, depending on the definitions used. An interesting question is whether simple sstem (s, one tht is three-dimensionl nd utonomous with onl qudrtic nonlinerities) cn hve ll three of these ttrctors concurrentl. Wht follows is just such n exmple.. Coexistence of Point, Periodic nd Strnge Attrctors A chotic sstem with onl one equilibrium, stble node-focus, ws introduced in [Wng & Chen, ]. This sstem ws found b dding nonzero Three dimensionl view x z phse plne.5.5 constnt to cse E in [Sprott, 99] s follows: ẋ = z + ẏ = x () ż = x, when, the stbilit of the single equilibrium is fundmentll ltered. Specificll, when >, sstem () possesses onl one stble equilibrium: ( P (x E, E,z E )=, ) 6, 6. () Interestingl, this stble equilibrium cn coexist pecefull with strnge ttrctor, s reported in [Wng & Chen, ]. This mens tht both point ttrctor nd strnge ttrctor dominte the sstem dnmics in region of the stte spce, so it is es to imgine tht there should be n unstble boundr between the two ttrctors. Will these two bsins of ttrction hve smooth boundr, or will the be intertwined in frctl or other tpe of complicted mnner? The following discover mkes this question even more fscinting nd hrder to nswer..5.5 x phse plne z phse plne.5.5 Fig.. Coexistence of point, periodic, nd strnge ttrctors of sstem () with =.; the point ttrctor (green) is generted from initil conditions (.,, ), the periodic ttrctor (red) from initil conditions (,, ), nd the strnge ttrctor (blue) from initil conditions (,, ). 59-

3 M 7, 8:6 WSPC/S Coexistence of Point, Periodic nd Strnge Attrctors In ddition to hving point ttrctor nd strnge ttrctor, sstem () is now found to hve periodic ttrctor s well when is in the vicinit of., s shown in Fig., giving rise to the coexistence of point, periodic nd strnge ttrctors. The point ttrctor (green) is generted from () Tble. Lpunov exponents with different initil vlues. Initil Conditions Lpunov Exponents (.,, ) (,, ) (,, ).;.;.9.;.7;.99.6;.;.6 Dimension.57 Bifurction digrm with initil vlues (.,, ) (b) Bifurction digrm with initil vlues (,, ) 5. (c) Bifurction digrm with initil vlues (,, ). Fig Bifurction digrm versus prmeter, with different initil conditions, showing period-doubling route to chos. 59-

4 J. C. Sprott et l. initil conditions (x,,z )=(.,, ), the periodic ttrctor (red) from initil conditions (,, ), nd the strnge ttrctor (blue) from initil conditions (,, ). The Lpunov exponents for =. ccurte to three digits re shown in Tble. Figure shows the bifurction digrms versus prmeter with different initil conditions, demonstrting period-doubling route to chos. These digrms lso show tht, t =., the three different initil conditions led to three different ttrctors. () (b) (c) Fig.. Bsins of ttrction of the point, periodic, nd strnge ttrctors of sstem (), ll with =. on three crosssections in the plne contining the equilibrium point, mrked b green, red, nd blue, respectivel. The strnge ttrctor resides in the blue bsin; the periodic ccle hs severl points in the red bsin, nd the equilibrium point is single point in the green region. The blck points re cross-sections of the ttrctors. 59-

5 . Bsins of Attrction The three different tpes of ttrctors coexist pecefull in this simple sstem, with ech dominting the dnmics in different prt of the stte spce. Their bsins of ttrction represent mthemticll-involved subtle issue, becuse it is well known tht even for multiple point ttrctors, the bsin boundries cn be frctl. It turns out tht the boundries of the bsins of ttrction of sstem () do hve frctl structure, three cross-sections of which in the plnes contining the equilibrium point re shown in Fig.. On these sections, the bsins of ttrction of the point, periodic, nd strnge ttrctors of sstem () with =. re indicted b green, red, nd blue, respectivel. The strnge ttrctor resides in the blue bsin; the periodic ccle hs severl points in the red bsin, nd the equilibrium hs single point in the green region. In this figure, the blck points re cross-sections of the respective ttrctors. As the prmeter is grdull chnged, the bsins of ttrction lso grdull chnge, which mkes the estimte of the bsin boundries difficult but interesting.. Discussions An ttrctor is defined s the smllest ttrcting point set tht cnnot be itself decomposed into two or more subsets with distinct bsins of ttrction. This restriction is necessr since dnmicl sstem m hve different tpes of multiple ttrctors, ech with its own bsin of ttrction. Most sstems hve onl one ttrctor or one single tpe of ttrctor. Others m hve two different tpes of coexisting ttrctors, most likel strnge ttrctors nd periodic ccles. It is interesting nd striking to see tht the simple sstem reported here hs ll three different common tpes of ttrctors coexisting side b side. We do not Coexistence of Point, Periodic nd Strnge Attrctors hve definite nswer to the question bout the mechnism for the birth nd deth of these different tpes of ttrctors, except to note tht clssicl locl nltic theor does not ppl becuse the unique equilibrium point of the sstem is not hperbolic. One must then resort to the theories of globl bifurction nd chos [Wiggins, 988], which leves n importnt et chllenging theoreticl s well s technicl problem for future reserch. Acknowledgment This reserch ws supported b the Hong Kong Reserch Grnts Council under the GRF Grnt CitU9/. References Chen, G. & Uet, T. [999] et nother chotic ttrctor, Int. J. Bifurction nd Chos 9, Lorenz, E. N. [96] Deterministic nonperiodic flow, J. Atmosph. Sci.,. Sprott, J. C. [99] Automtic genertion of strnge ttrctors, Comput. Grph. 7, 5. Sprott, J. C. [99] Some simple chotic flows, Phs. Rev. E 5, Sprott, J. C. [997] Simplest dissiptive chotic flow, Phs. Lett. A 8, 7 7. Sprott, J. C. & Linz, S. J. [] Algebricll simple chotic flows, J. Chos Th. Appl. 5,. Teschl, G. [] Ordinr Differentil Equtions nd Dnmicl Sstems (Americn Mthemticl Societ, Providence). Uet, T. & Chen, G. [] Bifurction nlsis of Chen s eqution, Int. J. Bifurction nd Chos, Wng,. & Chen, G. [] A chotic sstem with onl one stble equilibrium, Commun. Nonlin. Sci. Numer. Simul. 7, 6 7. Wiggins, S. [988] Globl Bifurction nd Chos (Springer-Verlg, N). 59-5