arXiv:1412.7667v1 [nlin.CD] 24 Dec 2014 eprtr ieec sgnrtdaogteaxis the along generated is difference temperature osdrtecs hnteelpodrttsaon the around rotates field. ellipsoid neglected gravitational the axis of be when case can influence the forces the Consider centrifugal with the comparison that in such be to constant a rudisai.Teai a osatangle constant a has axis The axis. its around pc oteelpodmto.Tevalue The motion. ellipsoid the to spect Here h rvt vector gravity the oe fcneto)wsotie yGuhvk and form Glukhovsky following the by in obtained [1] was Dolzhansky convection) of model a etn.I sasmdta h lisi together ellipsoid velocity constant the the that with assumed rotates sources is heat It with exter- inhomogeneous heating. stationary nal of condition the under ino icu nopesbefli o nieteellip- the inside flow fluid soid incompressible viscous of tion ∗ A σ x z orsodn uhr [email protected] author: Corresponding orsodn ahmtclmdl(three-mode model mathematical Corresponding osdrtefloigpyia rbe:teconvec- the problem: physical following the Consider = = = = x a x 1 1 3 µ a 2 a µ λ − a 1 2 1 2 n h vector the and a T , gβa 2 1 1 1 − + aut fMteaisadMcais t eesugState Petersburg St. Mechanics, and Mathematics of Faculty a + 2 q a a 3 ω λµ 0 2 2 2 2 2 rd h rsn okdmntae ueial htti s this that numerically a demonstrates possesses work system, present The ered. rdsse,ulk h lsia oezoe ln ihsel with along one, Lorenz classical the unlike system, ered elclzd nltclnmrcllclzto fhidde of localization Analytical-numerical localized. be eateto ahmtclIfrainTcnlg,Unive Technology, Information Mathematical of Department 3 sagain fti eprtr.(Fig. temperature. this of gradient a is x a cos nti ae oezlk ytm eciigtepoeso process the describing system, Lorenz-like a paper this In oolncobtadhde trco nteLrn-iesy Lorenz-like the in attractor hidden and orbit Homoclinic + q 2 2 .INTRODUCTION I. 2 , = 2 gβa eciigtefli ovcinmto nterttn cavit rotating the in motion convection fluid the describing z y x ˙ ˙ ˙ 2 2 a T α g a = = = Ω + λ hsvco ssainr ihre- with stationary is vector This . 1 2 3 0 2 a xy − Ayz g cos Ra , 2 − Ω xz x a spae nteplane the in placed is 1 − 3 3 0 C , α + + .A Leonov, A. G. z. q 2 3 Cz Ra = 1 = homoclinic = y , 2 − − gβa a a , a 1 σx, y, 3 a ( 3 2 = 2 a q λµ ˆ 1 a 1 2 1 1 2 a > 2 + a , a gβa Ω rjcoyadachaotic a and trajectory 2 1 1 a a 2 0 2 2 .V Kuznetsov, V. N. 2 ) a > 3 λµ sassumed is σ x Dtd ac 0 2018) 20, March (Dated: 1 sin q x 3 1 α 3 a > , The . α, 1 with and 0 Ω (1) 1 , ). 0 nteaxes the on and q and h etr ffli nua eoiiso h axis the on velocities angular fluid of vectors the n alihnmes respectively. numbers, Rayleigh and a oezsse 2.Frtefis ie ytm( system time, first the For [2]. system Lorenz cal and in n oueepnin respectively; expansion, volume and tion, where n ban h olwn system following the obtains one h parameters The 3 ,2, 1, ( ntecase the In fe w euniltransformations sequential two After ω t trco spresented. is attractor n ) 1 self-excited x ,µ β µ, λ, q x x ∗ ≡ 3 3 = -xie trco a attractor f-excited iue1 lutaino h rbe setting problem the of Illustration 1: Figure nvriy eehf t eesug Russia Petersburg, St. Peterhof, University, a → → nwihcase which in , ( n .N Mokaev N. T. and t 0 0 − ) st fJyv¨askyl¨a, of Jyv¨askyl¨a,rsity Finland ; = oaigfli ovcin sconsid- is convection, fluid rotating f ,y x, ,y x, se,as ietecasclLorenz classical the like also ystem, 2 a r h rjcin ftmeauegradients temperature of projections the are ω a r h offiinso icst,ha conduc- heat viscosity, of coefficients the are A/C gβa = 1 1 x ( a a x t 1 2 ( 2 trco.Hwvr o consid- for However, attractor. ) , a 3 0 = , Ω → → 2 σ 0 x x z y ω 0 , ˙ ˙ ˙ a R , 2 2 R C R 0 = cos = = a T ( σ ytm( system 1) + and t − − = 2 ) − rx − 1 and , and , αq ,z y, z σ aC Ra a α 2 − hidden ( 0 x 1 + r , x g R ω , 3 ,2 1, y σ epciey nwihcase which in respectively, , xy, − 1 + Ra ω 2 − , → onie ihteclassi- the with coincides ) 3 y 2 = xz ( ) trco can attractor t ,z z, r h rnt,Taylor, Prandtl, the are C = − ) R σ ∇ Ω − r h rjcin of projections the are ayz − 0 stem T ( ˆ 1 a z, 2 y x 0 3 a → R gβa 1 a x 1) + a 1 2 3 Ω 0 R 0 q σ . 1 cos 1 + ( t ) , 2 αq x y, with ) 1 q , 2 2 . ( x (3) (2) t 2 ) , , 2 parameters r,σ > 0 was considered in [3, 1978]. After II. HOMOCLINIC ORBIT IN THE the linear change of variables [4] this system can be re- LORENZ-LIKE SYSTEM duced to the Rabinovich system, describing the waves interaction in plasma [5–8]. As is shown in [4] system Denote by x(t) a trajectory of system (2) starting at a (2) describes the following physical processes: the flow certain initial point. In order to compute numerically a of fluid convection inside the rotating ellipsoid [1], the homoclinic trajectory (limt→+∞ x(t) = limt→−∞ x(t) = rotation of rigid body in viscous fluid [9], the gyrostat S0), one integrates system (2) with the initial data x0 dynamics [10, 11], the convection of a horizontal layer from a δ-vicinity of the saddle point S0 and its one- u of fluid making the harmonic oscillations [12], and the dimensional unstable manifold W (S0) that corresponds model of Kolmogorov’s flow [13]. In [14] for system (2) to a positive eigenvalue of the Jacobian matrix J at in the case σ = ar a detailed analysis of the equilibria ± the saddle point S0. For some parameters of system stability and asymptotic behavior of trajectories is given (2) this trajectory after a certain time intersects a two- and the values of parameters are obtained for which sys- dimensional plane M spanned on the eigenvectors that tem (2) is integrable. Remark also the works [15, 16], in correspond to negative eigenvalues of J. The parameters which the analytical and numerical study of some gener- are chosen in such a way that the point of intersection alizations of system (2) and similar systems is presented. belongs to δ-vicinity of S0. In addition, in [17] system (2) was used to describe a specific scenario of transition to chaos in low-dimensional dynamical systems gluing bifurcations. Note that the Glukhovsky-Dolzhansky system is suffi- ciently different from the classical Lorenz system. In the 12
Lorenz system, the flow of the two-dimensional convec- 10 tion is considered only. In the Glukhovsky-Dolzhansky S S 1 system, the flow of the three-dimensional convection is 8 2 considered which can be interpreted as one of the models Z 6 of ocean flows [1]. 4 10
In what follows system (2) will be considered under the S 5 2 0 condition that the parameter a is positive. In this case 0 0 -5 if r< 1, then (2) has a unique equilibrium S0 = (0, 0, 0), -6 -4 -2 0 2 -10 which is globally asymptotically Lyapunov stable [4, 18]. 4 6 Y If r> 1, then (2) posesses three equilibria: S0 = (0, 0, 0) X and Figure 2: Approximation of the homoclinic butterfly for system (2). S1,2 = ( x1, y1, z1). (4) ± ± Let us fix the parameters: σ =4 and a =0.0052 (such Here values were considered in [1]). For r = 7.44 there is no intersection of the trajectory x(t) with the plane M σ√ξ σξ (see Fig. 3a) and for r = 7.445 the intersection occurs x1 = , y1 = ξ, z1 = , σ + aξ σ + aξ (see Fig. 3c). So, there exists an intermediate value p r∗ [7.44, 7.445] for which one can get the approxima- ∈ ∗ and the number ξ is defined as tion of homoclinic orbit r =7.4430045820796753... (see Fig. 3b). Note that the approximation for the symmet- ric homoclinic orbit can be obtained by the choose in the σ ξ = a(r 2) σ + (σ ar)2 +4aσ . computational procedure the symmetric (with respect to 2a2 − − − h p i S0) initial data (see Fig. 2). From an analytical point of view, the existence of homoclinic trajectory can be justi- The stability of equilibria S1,2 depends on the parameters fied by Fishing principle [20–22]. The Fishing principle σ, r, a (see. Sec. IIIA). is based on the construction of a special two-dimensional For system (2) with the fixed σ, a (or with the fixed manifold such that the separatrix of the saddle point in- σ only) it is possible to observe a classical scenario of tersects or does not intersect the manifold for two differ- transition to the chaos similar to scenario in the Lorenz ent values of a system parameter. The continuity implies system [19]. To demonstrate this, for system (2) with the existence of some intermediate value of the parameter the fixed parameters σ and a and increasing parameter for which the separatrix touches the manifold. Accord- r > 1, a homoclinic trajectory and a self-excited chaotic ing to construction the only possibility for separatrix is to attractor are obtained numerically. Unlike the Lorenz touch the saddle and thus, one can numerically localize system, for system (2) it is also possible to localize a the birth of the homolcinic orbit by changing the variable hidden chaotic attractor. parameter. 3
12 M 12 M 12 M S S 10 1 10 1 10 S S S 2 2 S 1 8 8 8 2
6 6 6 Z Z Z 4 4 S S 4 0 0 S 2 5 2 5 2 0 10
5 0 0 0 0 0 0 -2 -5 -2 -5 -2 -5 -6 -5 -6 -5 -6 -4 -3 -4 -3 -4 -2 -1 -2 -1 -2 0 0 1 -10 0 1 -10 2 -10 2 3 Y 2 3 Y 4 6 Y
X X X (a) r = 7.44 (b) r = 7.4430045820796753... (c) r = 7.445
Figure 3: The birth of a homoclinic orbit in system (2) with σ =4, a =0.0052, r [7.44, 7.445], δ =0.01. ∈
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xxxxxxxxxxxxxxxx r xxxxxxxxxxxxxxxx 2 xxxxxxxxxxxxxxxx A. Local stability analysis and computation of xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxr xxxxxxxxxxxxxxxx1 attractors xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx
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xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Proposition 1 . If σ > 2 and the parameters r and a 0 0.05 0.1 A satisfy the inequality Figure 4: Domain of stability of the equilibria S1,2 of 3 2 p3(σ, A) r + p2(σ, A) r + p1(σ, A) r + p0(σ, A) < 0, (5) system (2) for σ =4. where
2 2 neighborhood lead to the long-time behavior that ap- p3(σ, A)= a σ (σ 2), proaches the oscillation. Thus, from a computational 4− 3 2 p2(σ, A)= a 2σ 4σ 3aσ +4aσ +4a , point of view it is natural to suggest the following classi- −2 3 − − 2 fication of attractors, based on the simplicity of finding p1(σ, A)= σ σ + 2(3a 1)σ 8aσ +8a , the basin of attraction in the phase space: 3 3 2− − p0(σ, A)= σ σ +4σ 16a , − − Definition [23–26] An attractor is called a hidden at- then the equilibria S1,2 are stable. tractor if its basin of attraction does not intersect with small neighborhoods of equilibria, otherwise it is called a Let us choose the parameter σ =4 and, as in [1], con- self-excited attractor. struct the domains of stability of the equilibria S1,2 of system (2) in dependence on the values of parameters a and r. Then inequality (5) takes the form B. Self-excited attractor in the Lorenz-like system 8a2r3+a(7a 64)r2+(288a+128)r+256a 2048 < 0. (6) − − For a self-excited attractor its basin of attraction is For 0
(16.4961242, 690.6735024). For system (2) the param- C. Hidden attractor in the Lorenz-like system eter r = 687.5 is chosen. For the above parameters the eigenvalues of equilibria For a hidden attractor its basin of attractor is not re- of system (2) are the following lated with unstable equilibria. The hidden attractors, for example, are the attractors in the systems with no S0 : 49.9619, 1, 54.9619 equilibria or with only one stable equilibriumr1 (a special − − case of multistable systems and coexistence of attrac- S1,2 : 0.0295 10.729 i, 6.0591 ± − tors). Recent examples of hidden attractors can be found in [27–38]. Multistability is often an undesired situation Thus, the equilibrium S0 is a saddle and S1,2 are saddle- in many applications but the coexisting self-excited at- tractors can be found by the standard computational pro- cedure. In contrast, there is no regular way to predict the 1200 existence or coexistence of hidden attractors in system. Note that one cannot guarantee the localization of attrac- 1000 tor by the integration of trajectories with random initial 800 data (especially for multidimensional systems) since its S 2 S basin of attractor may be very small. Z 600 1
r 400
725 200
(0.0052, 702.5) 0 1000 -60 S P -40 0 500 2 -20 0 0 0053 687 5 0 700 ( . , . ) 20 -500 Y X P0 P1 (a) Attractor for initial conditions in the neighborhood of S0 675 (0.0052, 687.5)
r2
800 650 700
600 S S a 2 1 0.00515 0.00525 0.00535 500
Z 400 Figure 6: P0 : self-excited attractor, P1 : hidden
300 attractor, P2 : no chaotic attracxtors.
200
100 One of the effective methods for numerical localiza- tion of hidden attractors in multidimensional dynamical 0 1000 -60 800 -50 -40 600 systems is based on a homotopy and numerical continu- -30 400 -20 S -10 0 200 0 0 ation: it is necessary to construct a sequence of similar Y X systems such that for the first (starting) system the ini- (b) Attractor for initial conditions in the tial data for numerical computation of oscillating solution neighborhood of S1 (starting oscillation) can be obtained analytically, e.g, it is often possible to consider the starting system with self- Figure 5: A self-excited attractor of system (2) for excited starting oscillation. Then the transformation of r = 687.5, σ =4, a =0.0052. this starting oscillation is tracked numerically in passing from one system to another. focuses. Having taken an initial point on the unstable Let us construct on the plane (a, r) a line segment, manifold of one of equilibria S0,1,2 (Fig. 5), one can easily intersecting a boundary of the domain of stability of the be vizualized a self-excited chaotic attractor by standard equilibria S1, S2 (see Fig. 6). Let us choose the point computational procedure. P1 : r = 687.5, a =0.0053 as the finite point of the line segment. To these parameters correspond the following For r (690.6735024, 830.4169122) the equilibria S1,2 become∈ stable and the trajectories, starting from the eigenvalues of the equilibria of system (2): neighborhood of equilibrium S0, are attracted to S1 or S0 : 49.9619, 1, 54.9619 S2. The question arises whether there exists a hidden − − S1,2 : 0.0968 10.4269i, 5.8063 chaotic attractor in system (2) for such values of param- − ± − eters? Next for the computation of hidden attractor in It means that the equilibria S1,2 become stable focus- system (2) a special numerical procedure is considered. nodes. 5
Let us choose the point Remark that hidden attractor does not exist for all points of the shaded domain in Fig. 6. E.g., there is no P0 : r = 687.5,a =0.0052 chaotic attractor for the point P2 : r = 702.5, a =0.0052. as the initial point of the line segment. This point corre- Note also that in the work [44] the upper estimation sponds to the parameters for which in system (2) there of the Lyapunov dimension (LD) of attractor of system exists a self-excited attractor, which can be computed (2) is presented: for r = 687.5, a = 0.0053, σ = 4 it by the standard procedure. Then for the considered line was obtained analytically the estimation LD < 2.8909 segment a sufficiently small partition step is chosen and that is in a good agreement with the numerical result a chaotic attractor in the phase of system (2) space at LD=2.1293. each iteration of the procedure is computed. The last IV. CONCLUSIONS computed point at each step is used as the initial point for the computation of the next step. Our experiment has 8 iterations and the partition step equals 1.25 10−5, respectively. At each iteration for the In the present work by numerical methods the scenario current trajectory· that describes the attractor one com- of transition to chaos in physical model (2), describing putes the largest Lyapunov exponent (LLE) [39] and the a flow of rotating fluid convection inside the ellipsoid Lyapunov dimension (LD) [18, 40] 1. under horizontal heating, is demonstrated. Similarly to Thus, for the selected path and selected partition it is scenario in the classical Lorenz system, in system (2) a possible to visualize a hidden attractor of system (2) (see homoclinic trajectory and self-excited chaotic attractor Fig. 7). are constructed. However, unlike the Lorenz system for system (2) one is able to localize numerically a hidden attractor.
s s 1400 W (S ) W (S ) 2 1 1200 A 1000 hidden S S 800 2 1
600 Z
400
200 S 0 0 1000 ACKNOWLEDGMENTS 500 −200 s −60 W (S ) u −40 0 W (S ) 0 −20 0 0 −500 20 40 60 −1000 Y X This work was supported by Russian Scientific Foun- dation (project 14-21-00041) and Saint-Petersburg State Figure 7: Hidden attractor for system (2). University.
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