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Homoclinic Orbit and Hidden Attractor in the Lorenz-Like System Describing

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Homoclinic Orbit and Hidden Attractor in the Lorenz-Like System Describing Homoclinic orbit and hidden attractor in the Lorenz-like system describing the fluid convection motion in the rotating cavity G. A. Leonov,1 N. V. Kuznetsov,1,2, ∗ and T. N. Mokaev1, 2 1Faculty of Mathematics and Mechanics, St. Petersburg State University, Peterhof, St. Petersburg, Russia 2Department of Mathematical Information Technology, University of Jyv¨askyl¨a, Jyv¨askyl¨a, Finland (Dated: March 20, 2018) In this paper a Lorenz-like system, describing the process of rotating fluid convection, is consid- ered. The present work demonstrates numerically that this system, also like the classical Lorenz system, possesses a homoclinic trajectory and a chaotic self-excited attractor. However, for consid- ered system, unlike the classical Lorenz one, along with self-excited attractor a hidden attractor can be localized. Analytical-numerical localization of hidden attractor is presented. I. INTRODUCTION and λ,µ,β are the coefficients of viscosity, heat conduc- tion, and volume expansion, respectively; q1(t), q2(t), Consider the following physical problem: the convec- and q3(t) are the projections of temperature gradients tion of viscous incompressible fluid flow inside the ellip- on the axes x1, x2 and x3, respectively, in which case q3(t) 0; ω1(t), ω2(t), and ω3(t) are the projections of soid ≡ the vectors of fluid angular velocities on the axis x1, x2, 2 2 2 x1 x2 x3 and x3, in which case + + =1, a1 >a2 >a3 > 0, a1 a2 a3 gβa3 gβa3 ω1 = cos αq1, ω2 = cos αq2. under the condition of stationary inhomogeneous exter- −2a1a2Ω0 −2a1a2Ω0 nal heating. It is assumed that the ellipsoid together The parameters σ, Ta, and Ra are the Prandtl, Taylor, with heat sources rotates with the constant velocity Ω0 and Rayleigh numbers, respectively. around its axis. The axis has a constant angle α with the gravity vector g. This vector is stationary with re- x3 spect to the ellipsoid motion. The value Ω0 is assumed to be such that the centrifugal forces can be neglected Ω0 in comparison with the influence of gravitational field. Consider the case when the ellipsoid rotates around the x2 g axis x3 and the vector is placed in the plane x1x3. The α temperature difference is generated along the axis a1 and ∇Tˆ a constant q0 is a gradient of this temperature. (Fig. 1). Corresponding mathematical model (three-mode g model of convection) was obtained by Glukhovsky and Dolzhansky [1] in the following form x1 x˙ = Ayz + Cz σx, Figure 1: Illustration of the problem setting − y˙ = xz + Ra y, (1) − − After two sequential transformations z˙ = xy z. arXiv:1412.7667v1 [nlin.CD] 24 Dec 2014 − −1 −1 x x, y C y, z C z, Here → → σ → σ x x, y R z, z y, 2 a0R +1 a0R +1 λ Ω0 gβa3qˆ1 → → − → σ = , Ta = 2 , Ra = , µ λ 2a1a2λµ one obtains the following system 2 2 2 a1 a2 2 −1 2a1a2 A = 2 − 2 cos αTa , C = 2 2 σ sin α, x˙ = σ(x y) ayz a + a a3(a + a ) − − − 1 2 1 2 y˙ = rx y xz (2) −1 gβa3 cos α gβa3 − − x = µ ω3 + q3 , y = q1, z˙ = z + xy, 2a1a2Ω0 2a1a2λµ − 2 gβa3 where a0 = A/C , R = Ra C, z = q2, 2a1a2λµ 2 a0σ R a = 2 , r = (a0R + 1). (3) (a0R + 1) σ In the case a =0 system (2) coincides with the classi- ∗ Corresponding author: [email protected] cal Lorenz system [2]. For the first time, system (2) with 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. ∈ xxxxxxxxxxxxxxxx r xxxxxxxxxxxxxxxx III. CHAOTIC ATTRACTOR IN THE xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx 1000 xxxxxxxxxxxxxxxx LORENZ-LIKE SYSTEM xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx r xxxxxxxxxxxxxxxx 2 xxxxxxxxxxxxxxxx A. Local stability analysis and computation of xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxr xxxxxxxxxxxxxxxx1 attractors xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx 500 xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx Let us study the stability of equilibria S1, S2 of system xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx rxxxxxxxxxxxxxxxx (2). By the Routh-Hurwitz criterion, one can get the xxxxxxxxxxxxxxxx3 xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx following xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Proposition 1 .
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