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Hidden Attractors on One Path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich Systems

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Hidden Attractors on One Path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich Systems Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems G. Chen,1 N.V. Kuznetsov,2, 3 G.A. Leonov,2, 4 and T.N. Mokaev2 1City University of Hong Kong, Hong Kong SAR, China 2Faculty of Mathematics and Mechanics, St. Petersburg State University, Peterhof, St. Petersburg, Russia 3Department of Mathematical Information Technology, University of Jyv¨askyl¨a,Jyv¨askyl¨a,Finland 4Institute of Problems of Mechanical Engineering RAS, Russia (Dated: October 2, 2018) In this report, by the numerical continuation method we visualize and connect hidden chaotic sets in the Glukhovsky-Dolzhansky, Lorenz and Rabinovich systems using a certain path in the parameter space of a Lorenz-like system. I. INTRODUCTION The Glukhovsky-Dolzhansky system describes the con- vective fluid motion inside a rotating ellipsoidal cavity. In 1963, meteorologist Edward Lorenz suggested an ap- If we set proximate mathematical model (the Lorenz system) for a < 0; σ = ar; (7) the Rayleigh-B´enardconvection and discovered numeri- − cally a chaotic attractor in this model [1]. This discovery then after the linear transformation (see, e.g., [8]): stimulated rapid development of the chaos theory, nu- 1 1 1 1 1 merical methods for attractor investigation, and till now (x; y; z) ν1− y; ν1− ν2− h x; ν1− ν2− h z) ; t ν1 t has received a great deal of attention from different fields ! ! [2{7]. The Lorenz system gave rise to various generaliza- with positive ν1; ν2; h, we obtain the Rabinovich system tions, e.g. Lorenz-like systems, some of which are also [10, 11], describing the interaction of three resonantly simplified mathematical models of physical phenomena. coupled waves, two of which being parametrically ex- In this paper, we consider the following Lorenz-like sys- cited: tem 8 x_ = hy ν x yz; 8 <> − 1 − >x_ = σ(x y) ayz y_ = hx ν2y + xz; (8) < − − − − y_ = rx y xz (1) :>z_ = z + xy; − − − :>z_ = bz + xy; − where where parameters r, σ, b are positive and a is real. Sys- 1 1 2 2 1 1 2 σ = ν− ν2; b = ν− ; a = ν h− ; r = ν− ν− h : (9) tem (1) with 1 1 − 2 1 2 Hereinafter, the Lorenz, Glukhovsky-Dolzhansky, and a = 0 (2) Rabinovich systems are studied in the framework of sys- tem (1) under the corresponding assumptions on parame- coincides with the classical Lorenz system. ters ((2), (3), or (7)), respectively. For the considered as- Consider sumptions on parameters, if r < 1, then (1) has a unique1 b = 1; a > 0; σ > ar: (3) equilibrium S0 = (0; 0; 0), which is globally asymptoti- cally Lyapunov stable [8, 12]. If r > 1, then system (1) Then by the following linear transformation (see, e.g., has three equilibria: S0 = (0; 0; 0) and [8]): S = ( x1; y1; z1): (10) arXiv:1705.06183v1 [nlin.CD] 17 May 2017 ± ± ± ζ ζ (x; y; z) x; ; r y ; (4) Here, ! σ ar − σ ar − − σbpξ p σξ x = ; y = ξ; z = ; system (1) is transformed to the Glukhovsky-Dolzhansky 1 σb + aξ 1 1 σb + aξ system [9]: and 8 x_ = σx + ζz + αyz σb h p i <> − ξ = a(r 2) σ + (σ ar)2 + 4aσ : y_ = ρ y xz (5) 2a2 − − − − − :>z_ = z + xy; The stability of equilibria S of system (1) depends on − the parameters r, σ, a and b±. where r(σ ar) ζ2a ζ > 0; ρ = − > 0; α = > 0: (6) 1 ζ (σ ar)2 In general, system (1) can possess up to five equilibria [8]. − 2 Lemma 1 (see, e.g. [13]). For a certain σ > 2, the equilib- [13, 15], which contains the set of equilibria and can be ria S of system (1) with (3) (and, thus, of Glukhovsky- constructed as 't ( ). ± τ>0 t τ Dolzhansky system (5)) are stable if and only if the fol- Computational\ errors[ ≥ (causedB by finite precision arith- lowing condition holds: metic and numerical integration of differential equations) and sensitivity to initial conditions allow one to get a a2σ2(σ 2)r3 a 2σ4 4σ3 3aσ2 + 4aσ + 4a r2+ − − − − reliable visualization of a chaotic attractor by only one + σ2 σ3 + 2(3a 1)σ2 8aσ + 8a r pseudo-trajectory computed on a sufficiently large time − − − interval. For that, one needs to choose an initial point in σ3 σ3 + 4σ2 16a < 0: attractor's basin of attraction and observe how the tra- − − (11) jectory starting from this initial point after a transient Lemma 2 (see, e.g. [14]). The equilibria S of system process visualizes the attractor. Thus, from a computa- ± (1) with (7) (and, thus, of the Rabinovich system (8)) tional point of view, it is natural to suggest the follow- are stable if and only if one of the following conditions ing classification of attractors, based on the simplicity of holds: finding the basin of attraction in the phase space. (i)0 ar + 1 < 2r , r pr(r 1) ≤ − − Definition. [13, 16{18] An attractor is called a self- 4a(r 1)(ar+1)pr(r 1)+(ar 1)3 − − − excited attractor if its basin of attraction intersects with (ii) ar+1 < 0, b > bcr = (ar+1)2 4ar2 : − any open neighborhood of a stationary state (an equilib- rium); otherwise, it is called a hidden attractor. The particular interest in the considered Lorenz-like systems is due to the existence of chaotic attractors in Remark. Sustained chaos is often (almost) indistinguish- their phase spaces. In the next section, we will present able numerically from transient chaos (transient chaotic the definition of attractor from analytical and numerical set in the phase space), which can nevertheless persist for perspectives. a long time. Similar to the above definition, in general, a chaotic set can be called hidden if it does not involve and attract trajectories from a small vicinities of stationary II. ATTRACTORS OF DYNAMICAL SYSTEMS states; otherwise, it is called self-excited. For a self-excited attractor, its basin of attraction is A. Attractors of dynamical systems connected with an unstable equilibrium and, therefore, self-excited attractors can be localized numerically by the Consider system (1) as an autonomous differential standard computational procedure in which after a tran- equation in a general form: sient process a trajectory, started in a neighborhood of an unstable equilibrium (e.g., from a point of its unsta- u_ = f(u); (12) ble manifold), is attracted to the state of oscillation and then traces it. Thus, self-excited attractors can be eas- where u = (x; y; z) 3, and the continuously differen- R ily visualized (see, e.g. the classical Lorenz, Rossler, and tiable vector-function2 f : 3 3 may represent the R R Hennon attractors can be visualized by a trajectory from right-hand side of system (1).! Define by u(t; u ) a solu- 0 a vicinity of unstable zero equilibrium). tion of (12) such that u(0; u ) = u . For system (12), a 0 0 For a hidden attractor, its basin of attraction is not bounded closed invariant set K is connected with equilibria, and, thus, the search and vi- (i)a (local) attractor if it is a minimal locally attractive sualization of hidden attractors in the phase space may set (i.e., limt + dist(K; u(t; u0)) = 0 for all u0 ! 1 be a challenging task. Hidden attractors are attractors K("), where K(") is a certain "-neighborhood of2 in the systems without equilibria (see, e.g. rotating elec- set K), tromechanical systems with Sommerfeld effect described (ii)a global attractor if it is a minimal globally attrac- in 1902 [19, 20]), and in the systems with only one stable tive set (i.e., limt + dist(K; u(t; u0)) = 0 for all n ! 1 equilibrium (see, e.g. counterexamples [18, 21] to the Aiz- u0 R ), 2 erman's (1949) and Kalman's (1957) conjectures on the where dist(K; u) = infv K v u is the distance from 2 monostability of nonlinear control systems [22, 23]). One the point u 3 to the setjjK − jj3 (see, e.g. [13]). R R of the first related problems is the second part of Hilbert's Note that2 system (1) is dissipative⊂ in the sense that it 16th problem (1900) [24] on the number and mutual dis- possesses a bounded convex absorbing set [8, 13] position of limit cycles in two-dimensional polynomial 2 systems where nested limit cycles (a special case of mul- 3 b(σ + δr) = (x; y; z) R V (x; y; z) ; (13) tistability and coexistence of attractors) exhibit hidden B 2 j ≤ 2c(a + δ) periodic oscillations (see, e.g., [18, 25, 26]). The classi- 2 fication of attractors as being hidden or self-excited was where V (u) = V (x; y; z) = x2 +δy2 +(a+δ) z σ+δr , − a+δ introduced by G. Leonov and N. Kuznetsov in connec- δ is an arbitrary positive number such that a + δ > 0 tion with the discovery of the first hidden Chua attrac- b and c = min(σ; 1; 2 ). Thus, solutions of (12) exist for tor [16, 17, 27{29] and has captured much attention of t [0; + ) and system (1) possesses a global attractor scientists from around the world (see, e.g. [30{48]). 2 1 3 B. Hidden attractor localization via numerical and r = 24 in the Lorenz system, there exists a hid- continuation method den bounded chaotic set (similar to the classical Lorenz attractor), which is numerically indistinguishable from One of the effective methods for numerical localization sustained chaos since it persists for a very long time (see of hidden attractors in multidimensional dynamical sys- corresponding discussions in [50, 51]).
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