Broken symmetry in modified Lorenz model

Transcription

1 Int. J. Dnamical Sstems and Differential Equations, Vol., No., 1 Broken smmetr in modified Lorenz model Ilham Djellit*, Brahim Kilani Department of Mathematics, Universit Badji Mokhtar, Laborator of Mathematics, Dnamics and Modelization, Facult of Sciences, Annaba 3, Algeria ilhem.djelit@univ-annaba.org kilbra@ahoo.fr *Corresponding author Julien Clinton Sprott Department of Phsics, Universit of Wisconsin, 115 Universit Avenue, Madison WI 537 USA sprott@phsics.wisc.edu Abstract: The Lorenz model is of interest because of its abundant bifurcations and dnamical phenomena, due largel to the presence of critical sets or nondefinition sets. The model is investigated as a three-parameter quadratic famil. This article further develops and refines a stud of its basins of attraction, and it is eplained b using two tpes of nonclassical singularit sets. This has an important impact on the number of preimages and shows the essential role plaed b the vanishing denominator in the inverses. A deeper analsis of the global dnamic properties of the model in the parameter ranges where three stead states eist, reveals the role of smmetr with an interesting and comple dnamic structure. Kewords: bifurcation, smmetr, critical sets, vanishing denominator. Reference to this paper should be made as follows: Djellit, I., Kilani, B. and Sprott, J.C. () Broken smmetr in modified Lorenz Model, Int. J. Dnamical Sstems and Differential Equations, Vol., No., pp.. Biographical notes: Ilham Djellit received her Ph.D in Applied Mathematics from Paul Sabatier Universit - Toulouse (France) in Since then, she has been with the Department of Mathematics in Universit Badji Mokhtar - Annaba (Algeria), and as a Professor since 5. Her research topic is mainl in dnamical sstems and chaos. Brahim Kilani received his Ph.D in Mathematics from Universit Badji Mokhtar - Annaba(Algeria) in 1. He is currentl the head of the department of Computing and Mathematics since 11. His research interests include chaotic sstems and bifurcations. Julien Clinton Sprott earned his bachelor s degree from MIT in 19 and his Ph.D in phsics from the Universit of Wisconsin - Madison in 199. His professional interests are in eperimental plasma phsics and nonlinear dnamics. Copright Inderscience Enterprises Ltd.

2 I. Djellit et al. 1 Introduction The Burgers mapping epressed b { Tε,µ,ν 1 : = (1 εν) ε = (1+εµ+ε) (1) provides an etraordinaril rich repertoire of mathematical eamples illustrating relations occurring in the theor of turbulent fluid motion, and obtained b a drastic change of the Navier-Stokes equations. The mapping(1) is smooth and noninvertible. For the parameter values considered in Burgers (1939), the behavior of the solutions is mainl determined b the three fied points, (,) and ( µ,± µν), and the singular line = µ 1/ε, such that ever point maps onto the-ais, which is the stable manifold of the hperbolic point (,) so long asεν is less than. The other fied points are stable forε less than1/µ. Identical behavior is observed in certain other mappings, such as the Lorenz model given b T a,b,c : { = (1+ab) b = (1 b) +c+b () Lorenz (199) analzed this sstem with c =, which was obtained as an approimation to an ordinar differential equation via Euler s forward differencing scheme. He was interested in the chaotic behavior when b is etremel large. He illustrated the pertinence of the concept of computational chaos. The authors considered these same mappings for detecting fractal sets in Djellit et al. (1). The fractal structure was revealed b fractal basin boundaries and b the patterns of self-similarit. Moreover, Whitehead and MacDonald (19) considered a mapping derived from a differential equation model of turbulence given b T ε,µ,ν : { = (1 εν) ε = (1+εµ+ε) (3) Like the other models, this mapping ehibits chaos. Elabbas et al. (7) gave the theoretical analsis of (3) for ε = 1, and comple behavior including chaos was observed using a numerical method. The map () is not smmetric in the equation, but the attractors are smmetric about = for c =. The standard analsis of local stabilit and bifurcations suggests that oscillations occur in the presence of onl one unstable equilibrium, whereas the coeistence of three equilibria is characterized b bi-stabilit, the central equilibrium being on the boundar which separates the basins of the two stable ones. All these maps give identical behavior and displa homoclinic structure associated with the basin bifurcation bounded-nonbounded. The main topic of this paper is to investigate the basic patterns of comple nonuniqueness of the dnamic behavior of a class of Lorenz mappings proposed in Lorenz (199) and given b Eq. (). It is mainl focused on a new research area to identif and verif some properties of focal points on such maps. The eistence of invariant sets and weak attractors in the sense of Tsbulin and Yudovich is rigourousl proved in Djellit and Hachemi (11). We conclude that nonunique dnamics, associated with etremel comple structures of the basin boundaries, can have a profound effect on our understanding of the dnamical behavior. The model is first investigated as a two-parameter quadratic famil, and its analsis is eplained b using two tpes of nonclassical singularit sets. The first one, the critical curve, separates the plane into two regions having a different number of real inverses (here one and three). The

3 Broken smmetr in modified Lorenz model 3 second one is a curve of non-definition for two of the three inverses of the map, i.e., these inverses have a vanishing denominator on this line. We also present interesting results and tpical scenarios between the map and its inverses. Several papers have shown the importance of the critical curves in the bifurcations associated with invariant chaotic areas in noninvertible maps, and we consider this paper as an interaction between analtical methods and numerical methods. The rest of the paper is organized as follows. Section describes some peculiar properties of the Lorenz map. We discuss some cases where bifurcation can cause qualitative changes in the structure of the domain as some parameters are varied. Also, critical and prefocal curves are considered and have been used to eamine the structure of the basins. In Section 3, the peculiar case of the embedding parameter b = 1 is considered, the inverses are determined and analzed, and the essential role plaed b the vanishing denominator of these inverses is evidenced. Finall, we consider the modified Lorenz map and the effects of this perturbation on the smmetr in Section. The Lorenz model Consider the dnamical sstem generated b a famil of two-dimensional continuous noninvertible mapst a,b defined b T a,b : { = (1+ab) b = () where a, b are real parameters, and the functions f (, ) = (1 + ab) b and g(,) = (1 b) +b are continuous and differentiable. The mapt a,b is noninvertible. There are regions in the phase plane where we can have different numbers of preimages. Indeed, a point(, ) R ma have up to three rank-one preimages that can be computed b solving the third-degree algebraic sstem () with respect to and. The locus, where the number of preimages changes, is made up b the critical curves. The critical curves are computed as follows: 1 coincides with the set of points in which the Jacobian determinant vanishes, i.e.,detdt a,b =, and the critical curve = T a,b ( 1 ) divides the plane in two zonesz 1 andz 3. ThenT a,b is of tpez 1 < Z 3 if we adopt the notation of Mira et al. (199), where points on one side of the critical curve have one preimage, and points on the other side of have three preimages. The map (), which ehibits pitchfork and Neimark-Sacker bifurcations, has an important feature in that their inverses have denominators that vanish along a line in the plane for b = 1. This has a great consequence on the chaotic attractor structure. This is due to the fact that the attractor contains the focal point of the inverse. It is evident that the origin O = (,) is a fied point of T a,b (not unique), and that the whole line = is mapped into the origin in one iteration. This kind of noninvertible map of the plane is structurall well-known. Its embedding into a larger structurall stable famil T a,b, for b < 1 and b > 1, gives regionsz k ; (k = 1,3) being the number of real preimages) which eplains the comple nature when the embedding parameter takes as value 1, leading to a structurall different map. Recent works dealing with cases of multiple attractors in noninvertible maps have highlighted how noninvertibilit can be a source of bifurcations and comple structures of the basins of attraction (see references cited in Djellit et al. (1), Djellit and Hachemi (11), and Elabbas et al. (7)). However, the global phenomena and the comple structures of the basins shown here are due to the dual situation which occurs when the inverse map has two prefocal curves and is itself undefined in the whole plane.

4 I. Djellit et al..1 Fied points and prefocal curves The fied points of T a,b for the sstem () are solutions obtained b a trivial manipulation of () with = and =. Besides the solution (,), two additional fied points are guaranteed to eist if a >, and we can easil verif that if a <, then the origin (,) is the unique fied point of the map T a,b defined b (). If a >, then two further fied points,p 1 andp, eist, smmetric with respect to the-ais, with = ± a; = a. From the Jacobian matri, we can see that J = detdt a,b (,) = (1 b)(1+ab b)+b vanishes on two curves given b = 1 b b (1+ab) + (1 b) forb and b 1 or = for b = 1. The latter curve is focused b T a,b into a single point (following the terminolog used in Mira et al. (199) and in Ferchichi and Djellit (9)). Since T a,b ( = ) = (,), we epect that the map T a,b has at least one inverse with the denominator, and such that O(,) is a focal point with respect to the -ais as a corresponding prefocal curveδ O. In fact, the inverses T 1 1 : { = = +(1+a) (5) and T 1 : { = = +(1+a) () are such that the have a focal point in O(,) with = as a prefocal curve, where >. The non-definition set δ s is given b { δ s = (,) R } = and δ s is a smooth curve in the phase plane. The two-dimensional dnamical sstem obtained b successive iteration oft1, 1 is well defined, provided that the initial conditions belong to E, given b E = R \ k= T k a,b, (δ s) The second curve given b 1 : = 1 b b (1+ab) + (1 b) (obtained b setting J = ) is a curve of merging two preimages. For a >,b > and b 1, the phase plane includes a region of noninvertibilit of the map (). The noninvertibilit region is an unbounded set defined b 7(1 b) b( (1+ab)(1 b)/b) 3 <. This curve has a pointing cusp on the-ais where =. The antecedents have coordinates(,), such that from (),satisfiesb 3 b( (1+ab)(1 b))/b) (1 b) =, and = ((1+ab) )/b. Moreover, the mapta,b 1 is of tpez 1 < Z 3 whose critical curve : 7((1 b) b( (1+ab)(1 b)/b) 3 =, image of 1, separates the plane into two areas Z 1 andz 3 where there eists one antecedent and three antecedents, respectivel. To understand the behavior of the map when b 1, we follow the evolution of the phase plane when we var the parameters a and b. We made a surve in the phase plane and plotted the different singularities, their basins of attraction, and the critical curves for the sstem. B analzing the figures, we show how the particular feature is involved in eplaining some properties of the dnamic behaviors of the map associated with the basin boundaries and attracting sets. Consider the case a =.1, and var the parameterb. There are two stable nodes. Both of the basins are simpl connected sets and are located smmetricall with respect to the origin.

5 Broken smmetr in modified Lorenz model 5 For b = 1., the points above the curve formed b with a pointing cusp given b 7(1 b) b( (1+ab)(1 b)/b) 3 < have three preimages, one in the area R 1, the second in R, and the third in R 3. The map T a,b is a Z 1 < Z 3 tpe according to Mira et al. (199) (see Figure 1). For b = 1.5, the points above the curve formed b, which loses its pointing cusp and is given for this b-value b =, have two preimages, one in the area R 1, and the other inr. Here, we are concerned with the question of how the number of preimages is reduced as b 1. The area R 3 disappears, and we obtain instead the line =. This line corresponds to the prefocal curve, representing a transition curve as seen in Figure. The mapt a,b changes, losing one root on each side of the plane, and becoming a map with either two real preimages or none. 3 Peculiar case b=1 We consider the case b = 1 where the map T a,1 admits two inverses given in Eqs. (5 ). One of these components is epressed as a quotient whose denominator depends on. If we fib = 1 then the Eq. () becomes { T a,1 : = (1+a) = (7) Then if >, we have two solutions, and no solutions if <. So, T a,1 is a Z Z noninvertible map, wherez (the region whose points have no preimages) is the half plane {(, )/ < }, and Z (the region whose points have two distinct rank-1 preimages) is the half plane{(, )/ > }. The Jacobian vanishes on the line =. The image of this line bt a,1 is reduced to a point (,). Therefore, according to the definition, this point is a focal point of Ta,1 1, and the line = is associated with the prefocal curve. A specific class of map, with at least one of the components defined b a fractional rational function, has been studied in Ferchichi and Djellit (9) and has ver interesting properties. Some particular dnamical properties have been observed in iterated maps (or in one of the inverses) where the denominator vanishes, or where a component has the indeterminate form / at a point Ta,1 1. This characteristic has revealed new tpes of singularities in the phase plane, such as focal points and prefocal curves. The presence of these sets ma cause new kinds of bifurcations generated b contact between them and other singularities, which gives rise to new dnamical phenomena and new basin structures and invariant sets. In the present case, the inverse Ta,1 1 = T 1 1 T 1 of T a,1 has a vanishing denominator. Indeed, we consider a smooth arc γ transverse to δ s, and we stud the shape of its image under T1, 1, i.e., T 1(γ) with the condition that. We assume that γ is 1, deprived of the point where it crossesδ s. First we consider arcs γ(t) in the positive half-plane with the following parametric form: { (t) = ξ1 t+ξ γ(t) : t +ξ 3 t 3 (t) = η 1 t+η t +η 3 t 3 where η 1,η, η 3 are such that (t) = η 1 t+η t +η 3 t 3 when t + and (t) whent. Proceeding in this fashion we can (in principle) generate the desired limit. However, the use of a truncated power series does not effect calculations when we take (t) of degree less than 3 with respect to t, since we can have all possible cases of T 1 i (γ(t)), i = 1, : lim t > + lim T 1 t + 1, (γ(t)) = lim ( η 1 t+η t +η 3 t 3, (1+a) η1 t+η t +η 3 t 3 (ξ 1 t+ξ t +ξ 3 t 3 ) ) t + η1 t+η t +η 3 t 3

6 I. Djellit et al. = (,1+a) ifη 1, the slope of the half-tangent of (t) int = ism = η1 ξ 1, then we have a transversal contact withδ s. Ifξ 1 = the slope is vertical, but the contact remains transversal. lim T 1 t + 1, (γ(t)) = (, ) if η 1=η = (m =, then a tangential contact with δ s ). ξ 1 = (,1+a) ifη 1 = ξ 1 = (m = ξ 1, with a transversal contact withδ s ). = (,1+a) ifξ 1 = η 1 = η = (the slopem = ξ, with a tangential contact withδ s ). = (,1+a ξ 1 η ) ifη 1 = (m = ξ 1, and a tangential contact withδ s ). = (,1+a+ ξ 1 η ). Thus varingξ 1 andη, the curvature of such arcs tangent to = at the origin can be obtained, which satisfies the definition of prefocal sets of the inverse (see Whitehead and MacDonald (19), Mira et al. (199)). This propert clarifies the geometric structure of the attracting set occurring for particular choices of the parameters. Arcs crossing tangentiall through the origin have two distinct rank-one preimages crossing throughδ O, and all transverse directions tend to the single point(,1+a). Besides the elements seen up to now, there is also another particularit in the dnamics of T a,1. The second iterate of the line = 1+a is the fied point O, Ta,1 ( = 1+a) = (, ), and the significance of this computation lies in the requirement that this line be the prefocal curve ofta,1 associated with the origin (see Figure 3). Note that the eistence of two curves, the lines = and = 1+a, mapped into the origin make the iteration properties of T a,1 ver different from that of maps with a unique inverse. An arc crossing these lines twice is mapped into an arc with a loop at the origin as seen in Ferchichi and Djellit (9). Since the origin is a saddle fied point, this requires that these curves belong to the stable set of the point. Their role is important in understanding the homoclinic bifurcation of the point (, ), giving rise to a unique chaotic attractor which intersects the line = 1+a in infinitel man arcs with self-similar structure and loops issuing from the origin. Broken smmetr in the modified Lorenz model Now consider the mapt a,b,c given in Eq.(). Given and, if we tr to solve the algebraic sstem with respect to the unknowns and, we get three solutions from () in which satisfies b 3 +cb b( (1+ ab)(1 b))/b) (1 b) = and = ((1+ab) )/b. The Jacobian determinant is given b detdt a,b,c (,) = (1 b)[(1+ab) b]+ b +cb. It vanishes on the curve 1 : = 1 b (1+ab) + c+b (1 b). The noninvertibilit region is an unbounded set defined b 7[(1 b) +c 3 /7b+c( (1+ab)(1 b)/b)] [b( (1+ab)(1 b)/b+c/3)] 3 <. This curve also possesses a pointing cusp. When c =, the boundar that separates the basin of the two fied points, smmetric with respect to the saddle fied point O = (,), is still formed b the whole stable manifoldw s (O).

7 Broken smmetr in modified Lorenz model 7 When b = 1, for c =,(, ) is a focal point for the two inverse determinations. T1 1 (,) = (( c+ (c +))/,((1+a)(c c ++)/(c c +)) T 1 (,) = (( c c +)/,((1+a)(c+ c + +)/(c+ c +)) - Forc >,(,) is a focal point for the inverse determinationt Forc <,(,) is a focal point for the inverse determinationt 1. Figure a, obtained with a = 3., b =.1, and c =.1, shows the basins of the two fied points P 1 = ( c+ c +b a b,a) and P = ( c c +b a b,a), which are stable foci. For c =.5, P becomes unstable with the appearance of a closed curve via a Neimark- Sacker bifurcation in Figure b, where the basins of the two fied points P 1 and P are represented b blue and orange respectivel, and the are simpl connected sets. The basins do not maintain the same qualitative structure, and the smmetr has broken. This asmmetr has a real effect on the local stabilit properties of the equilibria, and it results in an evident asmmetr in the basins of attraction. As shown in Figure b, whenc, the etension of the basin of the fied pointb(p ) located on the left side of the origin is less than the etension of B(P 1 ). The origin O is still a saddle point without an homoclinic orbit. We observe that small changes in the parameter b have some effects on the properties of the attractors, and the ma cause remarkable asmmetries in the structure of the basins, which can onl be detected from the global properties of the studied model. In Figures b, c, and Figure 5, we show that the attracting sets are quite close to their boundar, and as b is increased, the homoclinic bifurcation of the saddle point O occurs when the two attractors contact their boundaries. In Figure 5, the attracting set is uniquel tangent to. In Figure 5, we show in blue region the unstable manifold of the saddle point O and the points that visit the attractor and the critical curve. For the special case b = 1 (see Figure ), we find a similar situation using the two branches of the unstable manifold of the saddle point O which converge to this homoclinic closed curve. The focus point is internal to this closed structure which remains bounded and tangential to both branches of the invariant manifold at (,). In Figure 7, the ver complicated unstable manifold has degenerated from a simple closed curve to a chaotic comple curve with loops associated with the focal point (, ). This chaotic behavior is created b the unstable manifold. When the parameter c is increased, the structure of the homoclinic closed curve epands graduall with loops created from the focuso around the focusp. 5 Conclusion Lorenz mappings have an etraordinaril rich repertoire of behaviors, giving rise to eamples of man kinds of regular and irregular features. The are smooth and noninvertible with nonconstant Jacobian. For the time-increment parameter value b = 1 andc =, the behavior of the solutions is mainl determined b the inverse maps having a vanishing denominator and undefined in the whole plane but onl in the positive half plane with respect to. The structure of the chaotic attractor is strongl influenced b the fact that the focus (which can also be a saddle point) is contained in the attractor. Whenc, comple and degenerate behaviors can be seen in the phase plane. Acknowledgements The authors would like to thank M. R. Ferchichi, and M. L. Sahari for help with programming and numerous stimulations.

8 I. Djellit et al. a =.1, b = 1., c =.5 1 Z δq 3 δs R R 1 R Z Figure 1: Critical curves of the map which is of tpez 1 < Z 3 References Burgers, J. M. (1939) Mathematical eamples illustrating relations occurring in the theor of turbulent fluid motion, Transactions of the Roal Netherlands Academ of Sciences, Amsterdam, Vol. 17, pp Lorenz, E. N. (7) Computational Chaos: a perlude to computational instabilit, Phsica D, Vol. 35, No. 3, pp Djellit, I., Sprott, J.C., and Ferchichi, M.R. (1) Fractal Basins in Lorenz model, Chinese Phsics Letters, Vol., No., 51. Djellit, I., Hachemi-Kara, A. (11) Weak Attractors and Invariant Sets in Lorenz Model, Facta Universitat., Facta Universitat.,SER.: ELE.C. ENERG., Vol., No., pp.71. Whitehead, R.R., MacDonald, N. (19) A chaotic map that displas its own homoclinic structure, Phsica D, Vol. 31, pp Elabbas, E. M., Agiza, H. N., El-Metwall, H., and Elsadan, A. A. (7) Bifurcation analsis, Chaos and control in the Burgers mapping, International Journal of Nonlinear Sciences, Vol., No. 3, pp Mira, C., Gardini, L., Barugola, A., and Cathala, J. C. (199) Chaotic Dnamics in Twodimensional Noninvertible Maps, World Scientific, Singapore. Ferchichi, M.R., Djellit, I. (9) On Some Properties of Focal Points,Discrete Dnamics in Nature and Societ, Vol. 9, Art.ID 5.

9 Broken smmetr in modified Lorenz model 9 3 a =.1, b = 1.1, c = 1 Z 3 1 R R R 3 Z Figure : When the number of preimages changes, the map becomes of tpez Z 3 T 1 (γ ) T 1 (γ1 ) T 1 (γ3 ) a =.1, b = 1, c = γ γ 1 δq δs 1 γ 3 1 T 1 1 (γ ) T 1 1 (γ1 ) T 1 1 (γ3 ) Figure 3: Preimages of arcs issuing from the origin and belonging to the positive half-plane with respect to.

10 1 I. Djellit et al. a = 3, b =.1, c = (a) a = 3, b =.1, c = (b) 5 a = 3, b =.1, c = (c) Figure : The blue region and orange points denote the basins of P 1 and P for the map T a,b,c

11 Broken smmetr in modified Lorenz model 11 5 a = 3, b =.1, c = Figure 5: The blue and orange points denote the basins ofp 1 andp for the mapt a,b,c a =.5, b = 1, c =.7 1 δq δs Figure : The homoclinic bifurcation and closed invariant curve around the focusp a =.5, b = 1, c =.7 1 δq δs Figure 7: The homoclinic closed curve and loops issuing from the origin