Finite-Time Lyapunov Dimension and Hidden Attractor of the Rabinovich System

Nonlinear Dyn (2018) 92:267–285 https://doi.org/10.1007/s11071-018-4054-z

ORIGINAL PAPER

Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

N. V. Kuznetsov · G. A. Leonov · T. N. Mokaev · A. Prasad · M. D. Shrimali

Received: 18 November 2017 / Accepted: 1 January 2018 / Published online: 31 January 2018 © The Author(s) 2018. This article is an open access publication

Abstract The Rabinovich system, describing the pro- punov dimension for the hidden attractor and hidden cess of interaction between waves in plasma, is con- transient chaotic set in the case of multistability are sidered. It is shown that the Rabinovich system can given. exhibit a hidden attractor in the case of multistability as well as a classical self-excited attractor. The hid- Keywords Hidden attractors · Perpetual points · den attractor in this system can be localized by analyt- Finite-time Lyapunov exponents · Adaptive algorithm ical/numerical methods based on the continuation and for the computation of finite-time Lyapunov dimension perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of attractors. A conjecture on the Lyapunov 1 Introduction dimension of self-excited attractors and the notion of exact Lyapunov dimension are discussed. A com- One of the main tasks of the investigation of dynamical parative survey on the computation of the finite-time systems is the study of established (limiting) behaviour Lyapunov exponents and dimension by different algo- of the system after transient processes, i.e. the problem rithms is presented. An adaptive algorithm for studying of localization and analysis of attractors (limited sets of the dynamics of the finite-time Lyapunov dimension the system states, which are reached by the system from is suggested. Various estimates of the finite-time Lya- close initial data after transient processes) [1–3]. While trivial attractors (stable equilibrium points) can be easily found analytically or numerically, the search of periodic and chaotic attractors can turn out to be a challenging problem (see, e.g. famous 16th Hilbert problem [4] N. V. Kuznetsov (B) · G. A. Leonov · T. N. Mokaev on the number of coexisting periodic attractors in two- Saint-Petersburg State University, St. Petersburg, Russia dimensional polynomial systems, which was formu- e-mail: [email protected] lated in 1900 and is still unsolved, and its generaliza- N. V. Kuznetsov tion for multidimensional systems with chaotic attrac- Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland tors [5]). For numerical localization of an attractor, one needs to choose an initial point in the basin of attraction A. Prasad Department of Physics and Astrophysics, Delhi University, and observe how the trajectory, starting from this ini- Delhi, India tial point, after a transient process visualizes the attrac- M. D. Shrimali tor. Self-excited attractors, even coexisting in the case Central University of Rajasthan, Ajmer, India of multistability [6], can be revealed numerically by 123 268 N. V. Kuznetsov et al. the integration of trajectories, started in small neigh- and time rescaling: bourhoods of unstable equilibria, while hidden attrac- → ν−1 , tors have the basins of attraction, which are not con- t 1 t (3) nected with equilibria and are “hidden somewhere” in the phase space [7–10]. Remark that in numerical com- we obtain a generalized Lorenz system: putation of trajectory over a finite-time interval, it is difficult to distinguish a sustained chaos from a tran- x˙ =−σ(x − y) − ayz, sient chaos (a transient chaotic set in the phase space, y˙ = rx − y − xz, (4) which can persist for a long time) [11]. The search and z˙ =−bz + xy, visualization of hidden attractors and transient sets in the phase space are challenging tasks [12]. where In this paper, we study hidden attractors and transient chaotic sets in the Rabinovich system. It is shown σ = ν−1ν , = ν−1, =−ν2 −2, = ν−1ν−1 2. that the methods of numerical continuation and perpet- 1 2 b 1 a 2 h r 1 2 h ual point are helpful for localization and understanding (5) of hidden attractor in the Rabinovich system. For the study of chaotic dynamics and dimension System (4) with a = 0 coincides with the classical of attractors, the concept of the Lyapunov dimension Lorenz system [23]. As it is discussed in [22], system [13] was found useful and became widely spread [14– (4) can also be used to describe the following physical 18]. Since only a finite time can be considered in the processes: the convective fluid motion inside rotating numerical analysis of dynamical system, in this paper ellipsoid, the rotation of rigid body in viscous fluid, the we develop the concept of the finite-time Lyapunov gyrostat dynamics, the convection of horizontal layer dimension [19] and approaches to its reliable numerical of fluid making harmonic oscillations and the model of computation. A new adaptive algorithm for the compu- Kolmogorov’s flow. tation of finite-time Lyapunov dimension and exponents Note that since parameters ν1, ν2, h are positive, is used for studying the dynamics of the dimension. the parameters σ , b, r are positive and parameter a is Various estimates of the finite-time Lyapunov dimen- negative. From relation (5), we have: sion for the Rabinovich hidden attractor in the case of multistability are given. σ =−ar. (6)

Further, we study system (4) under the assumption 2 The Rabinovich system: interaction between (6). If r < 1, then system (4) has a unique equilib- waves in plasma rium S0 = (0, 0, 0), which is globally asymptotically Lyapunov stable (global attractor) [18,22]. If r > 1, Consider a system, studied in 1978 by Rabinovich then system (4) has three equilibria: S = (0, 0, 0) and [20] and Pikovski et al. [21], 0 S± = (±x1, ±y1, z1), where √ x˙ = hy − ν x − yz, σ b ξ σξ 1 x = , y = ξ, z = 1 σ + ξ 1 1 σ + ξ y˙ = hx − ν2 y + xz, (1) b a b a z˙ =−z + xy, and σ b ξ = a(r − 2) − σ + (σ − ar)2 + 4aσ . describing the interaction of three resonantly coupled 2a2 waves, two of which are parametrically excited. Here, The stability of equilibria S± of system (4) depends on the parameter h is proportional to the pumping ampli- the parameters r, a and b. Using the Routh–Hurwitz tude and the parameters ν , are normalized dumping 1 2 criterion, we obtain the following decrements. After the linear transformation (see, e.g. [22]): Lemma 1 The equilibria S± of system (4) with parameters (5) are stable if and only if one of the following −1 −1 χ : (x, y, z) → (ν1ν2h y,ν1x,ν1ν2h z), (2) conditions holds: 123 Finite-time Lyapunov dimension and hidden attractor 269

(i) 0 ≤ ar + 1 < √2r , Note that system (4)[or(7)] is dissipative in the r− r(r−1) √ ( − )( + ) ( − )+( − )3 sense that it possesses a bounded convex absorbing set (ii) ar+1 < 0,b>b = 4a r 1 ar 1 r r 1 ar 1 . cr (ar+1)2−4ar2 [9,22]: Proof The coefficients of the characteristic polynomial 2 χ( , , ) = λ3 + ( , , )λ2 + ( , , )λ + b(σ + δr) x y z p1 x y z p2 x y z B(r, a, b) = u ∈ R3 | V (u) ≤ , ( + δ) (8) p3(x, y, z) of the Jacobian matrix of system (4)atthe 2c a point (x, y, z) are the following: ( ) = ( , , ) = 2 + δ 2 + ( + where V u V x y z x y a ( , , ) = − + , 2 p1 x y z b ar 1 δ) − σ+δr δ z a+δ , is an arbitrary positive number such 2 2 2 p2(x, y, z) = x + ay − az − ar(b − r + 1) + b, b that a+δ>0 and c = min(σ, 1, ). Thus, the solutions 2 2 2 2 p3(x, y, z) =−a 2xyz + rx − y + bz − br(r − 1) . of (4) exist for t ∈[0, +∞) and system (4) possesses a global attractor [9,24], which contains the set of all One can check that inequalities p1(x1, y1, z1)>0 and equilibria. p3(x1, y1, z1)>0 are always valid. If ar+1 ≥ 0, then Computational errors (caused by a finite precision p1(x1, y1, z1)p2(x1, y1, z1) − p3(x1, y1, z1)>0, arithmetic and numerical integration of differential and if condition (i) also holds, then p2(x1, y1, z1)>0. equations) and sensitivity to initial data allow one to If ar +1 < 0, then p2(x1, y1, z1)>0, and if condi- get a visualization of chaotic attractor by one pseudo- tion (ii) also holds, then p1(x1, y1, z1)p2(x1, y1, z1)− trajectory computed for a sufficiently large time inter- p3(x1, y1, z1)>0. val. One needs to choose an initial point in the basin of attraction of the attractor and observe how the trajectory, starting from this initial point, after a transient 3 Attractors and transient chaos process visualizes the attractor. Thus, from a computational point of view, it is natural to suggest the following Consider system (4) as an autonomous differential classification of attractors, based on the simplicity of equation of general form: finding the basins of attraction in the phase space. Definition 1 [7,9,10,25] An attractor is called a self- u˙ = f (u), (7) excited attractor if its basin of attraction intersects with any open neighbourhood of an equilibrium; otherwise = ( , , ) ∈ R3 where u x y z , and the continuously dif- it is called a hidden attractor. ferentiable vector function f : R3 → R3 represents ( , ) the right-hand side of system (4). Define by u t u0 a For a self-excited attractor, its basin of attraction ( , ) = solution of (7) such that u 0 u0 u0. For system (7), is connected with an unstable equilibrium and, there- a bounded closed invariant set K is fore, self-excited attractors can be found numerically by the standard computational procedure in which after (i) a (local) attractor if it is a locally attractive set a transient process, a trajectory, starting in a neighbour- (i.e. limt→+∞ dist(K, u(t, u0)) = 0 ∀u0 ∈ K (ε), hood of an unstable equilibrium, is attracted to the state where K (ε) is a certain ε-neighbourhood of set of oscillation and then traces it; then, the computations K ), are being performed for a grid of points in vicinity of (ii) a global attractor if it is a globally attractive set 3 the state of oscillation to explore the basin of attraction (i.e. limt→+∞ dist(K, u(t, u0)) = 0 ∀u0 ∈ R ), and improve the visualization of the attractor. Thus, where dist(K, u) = infv∈K ||v − u|| is the distance self-excited attractors can be easily visualized (e.g. the from the point u ∈ R3 to the set K ⊂ R3 (see, e.g. [9]). classical Lorenz and Hénon attractors are self-excited Since the whole phase space is a global attractor and with respect to all existing equilibria and can be easily any finite union of attractors is again an attractor, it is visualized by a trajectory from their vicinities). reasonable to consider only minimal global and local For a hidden attractor, its basin of attraction is not attractors, i.e. the smallest bounded closed invariant set connected with equilibria and, thus, the search and possessing the property (ii) or (i). visualization of hidden attractors in the phase space 123 270 N. V. Kuznetsov et al. may be a challenging task. Hidden attractors are attractors in the systems without equilibria (see, e.g. rotating electromechanical systems with Sommerfeld effect (1902) [26,27]), and in the systems with only one stable equilibrium (see, e.g. counterexamples [7,28] to Aizer- man’s (1949) and Kalman’s (1957) conjectures on the S monostability of nonlinear control systems [29,30]). − S+ One of the first related problems is the second part of 16th Hilbert problem [4] on the number and mutual disposition of limit cycles in two-dimensional polynomial systems, where nested limit cycles (a special case S0 of multistability and coexistence of periodic attractors) exhibit hidden periodic attractors (see, e.g. [7,31,32]). The classification of attractors as being hidden or self- Fig. 1 Multistability in the Rabinovich system (4) with the clas- excited was introduced by Leonov and Kuznetsov in sical values of parameters ν1 = 1, ν2 = 4, h = 4.92 from [21]: connection with the discovery of the first hidden Chua coexistence of three local attractors—two stable equilibria S± attractor [25,33–38] and has captured much attention and a chaotic self-excited attractor (self-excited with respect to of scientists from around the world (see, e.g. [39–68]). the unstable zero equilibrium S0) Since in numerical computation of trajectory over a finite-time interval, it is difficult to distinguish a sus- method starting from a self-excited chaotic attractor. tained chaos from a transient chaos (a transient chaotic We change parameters, considered in [72], in such a set in the phase space, which can nevertheless persist way that the chaotic set is located not too close to the for a long time) [11,69], a similar classification can be unstable zero equilibrium to avoid a situation, when introduced for the transient chaotic sets. numerically integrated trajectory oscillates for a long Definition 2 [70,71]Atransient chaotic set is called time and then falls on the unstable manifold of unsta- a hidden transient chaotic set if it does not involve ble zero equilibrium, leaves the chaotic set and tends and attract trajectories from a small neighbourhood of to one of the stable equilibria. equilibria; otherwise, it is called self-excited. In order to distinguish an attracting chaotic set 3.1 Localization via numerical continuation method (attractor) from a transient chaotic set in numerical experiments, one can consider a grid of points in a One of the effective methods for numerical localiza- small neighbourhood of the set and check the attraction of hidden attractors in multidimensional dynamical tion of corresponding trajectories towards the set. systems is based on the homotopy and numerical con- For system (1) with parameters ν1 = 1, ν2 = 4, and tinuation method (NCM). It is based on the assumption increasing h, it is possible to observe [21] the classi- that the position of the attractor changes continuously cal scenario of transition to chaos (via homoclinic and with the parameters changing. The idea is to construct subcritical Andronov-Hopf bifurcations) similar to the a sequence of similar systems such that for the first scenario in the Lorenz system. For 4.84 h 13.4 (starting) system, the initial point for numerical com- in system (1), there is a self-excited chaotic attractor putation of oscillating solution (starting attractor) can (see e.g. Fig. 1), which coexists with two stable equi- be obtained analytically, for example, it is often possi- libria. The same scenario can be obtained for system ble to consider the starting system with a self-excited (4) when parameters b > 0 and a < 0 are fixed and r starting attractor; then, the transformation of this start- is increasing. Besides self-excited chaotic attractors, a ing attractor in the phase space is tracked numerically hidden attractor was found [71,72] in the system. Note while passing from one system to another; the last sys- that in [9,73], system (4) with a > 0 was studied and tem corresponds to the system in which a hidden attrac- a hidden attractor was also found numerically. tor is searched. Further, we localize a hidden chaotic attractor in For the study of the scenario of transition to chaos, system (4) with a < 0 by the numerical continuation we consider system (7) with f (u) = f (u,λ), where 123 Finite-time Lyapunov dimension and hidden attractor 271

λ ∈ Λ ⊂ Rd is a vector of parameters, whose variation System (9) shows the variation of acceleration in the in the parameter space Λ determines the scenario. Let phase space. λend ∈ Λ define a point corresponding to the system, Similar to the equilibrium points estimation, where where a hidden attractor is searched. Choose a point we set the velocity vector to zero, we can also get a set λbegin ∈ Λ such that we can analytically or numeri- of points, where u¨ = g(upp) = 0in(9), i.e. the points cally localize a certain nontrivial (oscillating) attrac- corresponding to the zero acceleration. At these points, 1 tor A in system (7) with λ = λbegin (e.g. one can the velocity u˙ may be either zero or nonzero. This set consider a self-excited attractor, defined by a trajec- includes the equilibrium points uep with zero veloc- tory u1(t) numerically integrated on a sufficiently large ity as well as a subset of points with nonzero velocity. 1 time interval t ∈[0, T ] with the initial point u (0) These nonzero velocity points upp are termed as per- in the vicinity of an unstable equilibrium). Consider petual points [12,74–76]. a path1 in the parameter space Λ , i.e. a continuous Lemma 2 Perpetual points S = (x , y , z ) of function γ :[0, 1]→Λ for which γ(0) = λbegin pp pp pp pp γ( ) = λ {λ j }k system (4) can be derived from the following system and 1 end, and a sequence of points j=1 1 k on the path, where λ = λbegin, λ = λend, such ⎧ + ⎪ 2 2 2 2 2 2 that the distance between λ j and λ j 1 is sufficiently ⎪ (x − az + ar − 1) −y + z + r (a − 1) ⎪ pp pp pp pp small. On each next step of the procedure, the initial ⎪ ⎨⎪ = r 2(ar − 1)2 − z2 (ar − b − 1)2, point for a trajectory to be integrated is chosen as the pp + 2 − 2 + 2 − ( − ) = , last point of the trajectory integrated on the previous ⎪ 2xpp yppzpp rxpp ypp bzpp br r 1 0 j+1 j ⎪ step: u (0) = u (T ). Following this procedure and ⎪ 2 + ( − − ) − 2 ⎪ rxpp ar b 1 xpp ypp arypp sequentially increasing j, two alternatives are possible: ⎩⎪ z = . pp x2 + ay2 − b2 the points of A j are in the basin of attraction of attractor pp pp A j+1, or while passing from system (7) with λ = λ j to (10) system (7) with λ = λ j+1, a loss of stability bifurcation is observed and attractor A j vanishes. If under chang- The reason why perpetual points may lead to hidden states [perpetual point method (PPM)] is still discussed ing λ from λbegin to λend, there is no loss of stability bifurcation of the considered attractors, then a hidden (see, e.g. [77,78]). k attractor for λ = λend (at the end of the procedure) is localized. 4 Hidden attractor in the Rabinovich system

Next, we apply the NCM for localization of a hid- 3.2 Localization using perpetual points den attractor in the Rabinovich system (4) and check whether the attractor can be also localized using PPM. The equilibrium points of a dynamical system are the In this experiment, we fix parameter r and, using ones at which the velocity and acceleration of the sys- condition (ii) of Lemma 1, define parameters a = tem simultaneously become zero. In this section, we − 1 − ε and b = b − ε .Forr = 100, ε = 10−3, show that there are points, termed as perpetual points r 1 cr 2 1 ε = 10−2, we obtain a = a ≡−1.1 × 10−2, [74], which may help to visualize hidden attractors. 2 0 b = b ≡ 6.7454 × 10−2 and take P (a , b ) as For system (7), the equilibrium points u are 0 0 0 0 ep the initial point of line segment on the plane (a, b). defined by the equation u˙ = f (u ) = 0. Consider ep The eigenvalues of the Jacobian matrix at the equilib- a derivative of system (7) with respect to time: ria S0, S± of system (4) for these parameters are the following: u¨ = J(u) f (u) = g(u), (9)

S0 : 9.4382, −0.0675, − 11.5382, ∂ ( ) 3 ( ) = fi u where J u ∂ is the Jacobian matrix. ± : . ± . , − . . u j i, j=1 S 0 0037 3 6756i 2 1749 Here, g(u) may be termed as an acceleration vector. Consider on the plane (a, b) a line segment, inter- 1 In the simplest case, when d = 1, the path is a line segment. secting a boundary of stability domain of the equilib- 123 272 N. V. Kuznetsov et al.

b perpetual point Spp = (− 0.2385, 49.1403, 0.1 − 101.4613), which allows one to localize a hidden P (a ,b) 0.09 1 1 1 attractor (see Fig. 4). Hence, here both NCM and PPM 0.08 allow one to ﬁnd this hidden attractor. 0.07 Around equilibrium S0, we choose a small spheri- 0.06 P0(a0,b0) P2(a2,b2) cal vicinity of radius δ (in our experiments, we check 0.05 δ ∈[0.1, 0.5]) and take N random initial points on it

0.04 (in our experiment, N = 4000). Using MATLAB, we

0.03 integrate system (4) with these initial points in order to explore the obtained trajectories. We repeat this pro- 0.02

b = bcr(r, a) cedure several times in order to get different initial 0.01 ar +1=0 points for trajectories on the sphere. We get the fol- 0 -0.012 -0.0115 -0.011 -0.0105-0.01 -0.0095 -0.009 a lowing results: all the obtained trajectories attract to either the stable equilibrium S+ or the equilibrium S− ( , ) → ( , ) → ( , ) Fig. 2 Path P0 a0 b0 P1 a1 b1 P2 a2 b2 in param- and do not tend to the attractor. This gives us a reason eters plane (a, b) for the localization of hidden attractor in −2 to classify the chaotic attractor, obtained in system (4), system (4) with r = 100. Here, a0 =−1.1 × 10 , b0 = −2 −2 −2 6.7454 × 10 , a1 =−1.049 × 10 , b1 = 7.2454 × 10 , as the hidden one. −3 −3 a2 =−9.965 × 10 , b2 = 7.7454 × 10 ; (•) P0(a0, b0): Remark that there exist hidden chaotic sets in the (•) ( , ) self-excited attractor with respect to S0, S±; P1 a1 b1 :self- Rabinovich system, which cannot be localized by PPM. (•) ( , ) excited attractor with respect to S0; P2 a2 b2 : hidden attrac- = . =− . ∈ tor. Stability domain is defined using Lemma 1 For example, for parameters r 6 8, a 0 5, b [0.99, 1] [72], the hidden attractor obtained by NCM is not localizable via PPM (see Fig. 5). ria S± with the final point P2(a2, b2), where a2 = −3 −3 a0 +1.035 × 10 =−9.965 × 10 , b2 = b0 +ε2 = −2 7.7454 × 10 , i.e. the equilibrium S0 remains saddle 5 Computation of the Lyapunov dimension and the equilibria S± become stable focus-nodes 5.1 Finite-time and limit values of the Lyapunov dimension and Lyapunov exponents S0 : 8.9842, − 0.0775, − 10.9807, :−. ± . , − . . S± 0 0401 3 9152i 1 9937 The Lyapunov exponents [2] and Lyapunov dimension [13] are widely used for the study of attractors (see, e.g. The initial point P0(a0, b0) corresponds to the [14–18]). Nowadays, various approaches to the defini- parameters for which in system (4), there exists a self- tion of Lyapunov dimension are used. Since in numer- excited attractor. Then for the considered line segment, ical experiments we can consider only finite time, in a sufficiently small partition step is chosen, and at each this paper we develop the concept of finite-time Lya- iteration step of the procedure, an attractor in the phase punov dimension [19] and an approach to its reliable space of system (4) is computed. The last computed numerical computation. This approach is inspirited by point at each step is used as the initial point for the the works of Douady and Oesterlé [79], Hunt [80], and computation at the next step. In this experiment, we Rabinovich et al. [81]. t 3 use NCM with 3 steps on the path P0(a0, b0) → For a fixed t ≥ 0 let us consider the map ϕ : R → ( , ) → ( , ) = 1 ( + ) R3 P1 a1 b1 P2 a2 b2 , with a1 2 a0 a2 , defined by the shift operator along the solutions = 1 ( + ) ϕt ( ) = ( , ) ∈ R3 b1 2 b0 b2 (see Fig. 2). At the first step, we have of system (7): u0 u t u0 , u0 . Since self-excited attractor with respect to unstable equilibria system (7) possesses an absorbing set [see (8)], the S0 and S±; at the second step, the equilibria S± become existence and uniqueness of solutions of system (7) stable but the attractor remains self-excited with respect for t ∈[0, +∞) take place and, therefore, the system t to equilibrium S0; at the third step, it is possible to visu- generates a dynamical system {ϕ }t≥0. Let a nonempty alize a hidden attractor of system (4)(seeFig.3). closed bounded set K ⊂ R3 be invariant with respect t t Using Lemma 2 for parameters r = 100, a = to the dynamical system {ϕ }t≥0, i.e. ϕ (K ) = K for − 9.965 × 10−3, b = 7.7454 × 10−2, we obtain one all t ≥ 0 (e.g. K is an attractor). Further, we use 123 Finite-time Lyapunov dimension and hidden attractor 273

S S + − S− u1 S+ S− 0 1 uT

S+ A1 se

S0 S0 A 2 u 2 se 0 S + S − u 2 T S+ S− S 0

S− A3 A1 u1 hid se T S + u2 0 u3 S+ S S 0 0 − A2 se 2 uT u1 S0 0 S0

Step 1 : a = −1.1 · 10−2,b=6.7454 · 10−2, Step 2 : a = −1.049 · 10−2,b=7.2454 · 10−2, Step 3 : a = −9.965 · 10−2,b=7.7454 · 10−2, self-exc. attractor A1 w. r. t. equil. S ,S self-exc. attractor A2 w. r. t. equilibrium S hidden attractor A3 , separatrices of S → S se 0 ± se 0 hid 0 ±

( + ) i+1 := i Fig. 3 Localization, by NCM, of a hidden attractor in system i 1 -th iteration is defined as u0 uT (violet arrows), (4)withr = 100, a =−9.965 × 10−3, b = 7.7454 × 10−2. i = i ( ) where uT u T is a final point (yellow). (Color figure online) Trajectories ui (t) = (xi (t), yi (t), zi (t) (blue) are defined on the time interval [0, T = 103], and initial point (yellow) on

S S− S− S+ +

A3 hid

A hid S0 S S 0 pp Spp

Fig. 4 Localization of hidden attractor in system (4)withr = Fig. 5 Hidden attractor in system (4)withr = 6.8, a =−0.5, −3 −2 100, a =−9.965 × 10 , b = 7.7454 × 10 from the per- b = 0.99, which can be localized via NCM (initial point u0 = petual point Spp = (− 0.2385, 49.1403, − 101.4613) (− 0.1629, − 0.2154, 1.9553)), but cannot be localized from the perpetual point Spp = (0.7431, − 12.6109, − 7.4424) (yellow). (Color ﬁgure online)

t compact notations for the ﬁnite-time local Lyapunov sion of dynamical system {ϕ }t≥0 on invariant set K ): t t dimension:dimL(t, u) = dimL(ϕ , u),theﬁnite-time dimL K = dimL({ϕ }t≥0, K ). t Lyapunov dimension:dimL(t, K ) = dimL(ϕ , K ) and Consider linearization of system (7) along the solu- for the Lyapunov dimension (or the Lyapunov dimen- tion ϕt (u): 123 274 N. V. Kuznetsov et al.

v˙ = J(ϕt (u))v, J(u) = Df(u), (11) allows one to introduce the Lyapunov dimension of K as [19] where J(u) is the 3×3 Jacobian matrix, the elements dim K = lim inf sup dim (t, u) (15) of which are continuous functions of u, and suppose L →+∞ L t u∈K that det J(u) = 0 ∀u ∈ R3. Consider a fundamental t matrix of solutions of linear system (11), Dϕ (u), such and get an upper estimation of the Hausdorff dimen- that Dϕ0(u) = I , where I is a unit 3×3 matrix. Denote sion: t by σi (t, u) = σi (Dϕ (u)), i = 1, 2, 3, the singular t 2 dimH K ≤ dimL K. values of Dϕ (u) (i.e. σi (t, u)>0 and σi (t, u) are the t ∗ t eigenvalues of the symmetric matrix Dϕ (u) Dϕ (u) Recall that a set K with noninteger Hausdorff dimen- 2 with respect to their algebraic multiplicity), ordered sion is referred to as a fractal set [15] and, when such so that σ1(t, u) ≥ σ2(t, u) ≥ σ3(t, u)>0 for any set K is an attractor, it is called a strange attractor 3 u ∈ R and t > 0. [82,83]. A singular value function of order d ∈[0, 3] is Consider a set of ﬁnite-time Lyapunov exponents3 at deﬁned as the point u:

t d−d ω (Dϕ (u)) = σ (t, u) ···σ (t, u)σ + (t, u) , d 1 d d 1 ( , ) = 1 σ ( , ), > , = , , . t t LEi t u ln i t u t 0 i 1 2 3 (16) ω0(Dϕ (u)) = 1,ω3(Dϕ (u)) = σ1(t, u)σ2(t, u)σ3(t, u), t

{ ( , )}3 where d is the largest integer less or equal to d.For Here, the set LEi t u i=1 is ordered by decreas- a certain moment of time t ≥ 0, the finite-time local ing (i.e. LE1(t, u) ≥ LE2(t, u) ≥ LE3(t, u) for all Lyapunov dimension at the point u is defined as [19] t > 0). Then for j(t, u) =dimL(t, u) < 3 and ( , ) = ( , ) − ( , ) = s t u dimL t u dimL t u ,wehave0 1 j(t,u) ln(ω ( , )+ ( , )(Dϕt (u))) = LE (t, u) + ( , ) = { ∈[ , ]:ω ( ϕt ( )) ≥ } t j t u s t u i=1 i dimL t u max d 0 3 d D u 1 ( , ) ( , ) ( , ) = s t u LE j(t,u)+1 t u and j t u max (12) { : m ( , ) ≥ } m i=1 LEi t u 0 . Thus, we get an analog of the Kaplan–Yorke formula [13] with respect to the { ( , )}3 and the finite-time Lyapunov dimension of K is defined set of finite-time Lyapunov exponents LEi t u i=1 as [19]:

KY 3 LE1(t,u)+···+LE j(t,u)(t,u) d ({LE (t,u)} = )= j(t, u)+ , L i i 1 | LE j(t,u)+1(t,u)| dimL(t, K ) = sup dimL(t, u). (13) u∈K (17)

which gives the finite-time local Lyapunov dimension: The Douady–Oesterlé theorem [79] implies that for ≥ ( , ) = KY({ ( , )}3 ). any fixed t 0, the Lyapunov dimension of the map dimL t u dL LEi t u i=1 ϕt with respect to a closed bounded invariant set K , Thus, in the above approach, the use of Kaplan–Yorke defined by (13), is an upper estimate of the Hausdorff formula (17) with the finite-time Lyapunov exponents dimension of the set K :dim K ≤ dim (t, K ). H L {LE (t, u)}3 is rigorously justified by the Douady– For the estimation of the Hausdorff dimension of i i=1 Oesterlé theorem. invariant closed bounded set K , one can use the map Note that the Lyapunov dimension is invariant under ϕt with any time t (e.g. t = 0 leads to the trivial esti- ≤ Lipschitz diffeomorphisms [19,86], i.e. if the dynami- mate dimH K 3) and, thus, the best estimation is t cal system {ϕ }t≥0 and closed bounded invariant set dimH K ≤ inft≥0 dimL(t, K ). The following property 3 The Lyapunov exponents, LEs (see, e.g. [84]) characterize the inf sup dim (t, u) = lim inf sup dim (t, u) (14) rates of exponential growth of the singular values of fundamental L →+∞ L t≥0 u∈K t u∈K matrix of the linearized system. The singular values correspond to the semiaxes of n-dimensional ellipsoid, which is the image of the unit sphere with respect to the linearized system. See [85] 2 Symbol ∗ denotes the transposition of matrix. for various related notions. 123 Finite-time Lyapunov dimension and hidden attractor 275

K under a smooth change of coordinates w = χ(u) Note also that even if a numerical approximation t are transformed into the dynamical system {ϕχ }t≥0 and (visualization) K of the attractor K is obtained, it is closed bounded invariant set χ(K ), respectively, then not straightforward how to get a point on the attrac- t t dimL({ϕ }t≥0, K ) = dimL({ϕχ }t≥0,χ(K )). Also the tor itself: u0 ∈ K . Thus, an easy way to get reliable Lyapunov dimension is invariant under positive time estimation of the Lyapunov dimension of attractor K rescaling t → at, a > 0. is to localize the attractor K ⊂ K ε, to consider a grid ε ε of points Kgrid on K , and to find the maximum of the corresponding finite-time local Lyapunov dimen- T 5.2 Adaptive algorithm for the computation of the sions for a certain time t = T :maxdimL(ϕ , u) = ∈ ε finite-time Lyapunov dimension and exponents u Kgrid LE1(T,u)+···+LE j(T,u)(T,u) max j(T, u) + | ( , )| . u∈K ε LE j(T,u)+1 T u Applying the statistical physics approach and assum- grid ing the ergodicity (see, e.g. [13,87–89]), the Lyapunov Concerning the time T , remark that while the time series obtained from a physical experiment are assumed dimension of attractor dimL K is often estimated by the to be reliable on the whole considered time interval, the local Lyapunov dimension dimL(t, u0), corresponding to a “typical” trajectory, which belongs to the attrac- time series produced by the integration of mathematical dynamical model (7) can be reliable on a limited tor: {u(t, u0), t ≥ 0}, u0 ∈ K , and its limit value time interval only5 due to computational errors (caused limt→+∞ dimL(t, u0). However, from a practical point of view, the rigorous verification of ergodicity is a by finite precision arithmetic and numerical integra- challenging task [84,90] and hardly it can effectively tion of ODE). Thus, in general, the closeness of the ( , ) be done in a general case (see, e.g. discussions in real trajectory u t u0 and the corresponding pseudo- ˜( , ) [91], [92, p. 118], [93], [94,p.9],[95, p. 19], and trajectory u t u0 calculated numerically can be guar- the works [96,97]onthePerron effects of the largest anteed on a limited short time interval only. For the Lyapunov exponent sign reversals). Examples of the numerical visualization of a chaotic attractor, the com- rigorous use of the ergodic theory for the computa- putation of a pseudo-trajectory on a longer time interval tion of the Lyapunov exponents and dimension can be often allows one to obtain a more complete visualiza- found, for example, in [87,98–100]. In the work by tion of the attractor (pseudo-attractor) due to compu- Yorke et al. [88, p. 190], the exact limit values of finite- tational errors and sensitivity to initial data. However, { ( , )}3 = for two different long-time pseudo-trajectories u˜(t, u1) time Lyapunov exponents limt→+∞ LEi t u0 i 0 3 and u˜(t, u2) visualizing the same attractor, the corre- {LEi (u0)}, if they exist and are the same for all u0 ∈ R , 0 i.e. sponding finite-time LEs can be, within the considered error, similar due to averaging over time [see (16)] and lim LE (t, u ) = LE (u ) ≡ LE ∀u ∈ R3, i = 1, 2, 3, →+∞ i 0 i 0 i 0 {˜( , 1)} {˜( , 2)} t similar sets of points u t u0 t≥0 and u t u0 t≥0. are called the absolute ones, and it is noted that the At the same time, the corresponding real trajectories ( , 1,2) absolute Lyapunov exponents rarely exist4 (in this u t u0 mayhavedifferentLEs(e.g.u0 may corre- = KY({ }3) = + spond to an unstable periodic trajectory u(t, u0), which case, one also has dimL K dL LEi 1 j LE1 +···+LE j is embedded in the attractor and does not allow one to | | ). LE j+1 visualize it). Here, one may recall the question [105,p. 98] (known as Eden conjecture) whether the supremum 4 For example, let us consider the Hénon map with parameters = . = . 1 = of the local Lyapunov dimensions is achieved on a sta- a 0 97 and b 0 466. In this case [101] for initial points u0 (− . , − . ) 2 = ( . , . ) tionary point or an unstable periodic orbit embedded in 1 287342 1 287401 and u0 0 840503 0 704249 after 5 a transient process during [0, Ttrans = 10 ], we get points a1 a2 5 u0 and u0 , respectively, and compute finite-time Lyapunov In [102,103] for the Lorenz system the time interval of reliable exponents and finite-time local Lyapunov dimension for the computation with 16 significant digits and error 10−4 is estimated time interval [0, T = 104] by the adaptive algorithm with as [0, 36], with error 10−8 is estimated as [0, 26], and reliable δ = −8 ( , a1)≈ . , ( , a1)≈ . 10 :LE1 T u0 0 01198 dimL T u0 1 01545 and computation for a longer time interval, e.g. [0, 10,000] in [104], ( , a2)≈ . , ( , a2)≈ . , LE1 T u0 0 01405 dimL T u0 1 01807 the Lyapunov is a challenging task. Also if u0 belongs to a transient chaotic dimensions of two stationary points are 1.42787 and 1.56938 and set,thenu(t, u0) may have positive finite-time Lyapunov expo- some trajectories u(t, u0) and corresponding LE1(t, u0) tend to nent on a very large time interval, e.g. [0, 15,000], but finally infinity. In general, the important question [5]iswhetherthe u(t, u0) converges to a stable stationary point as t →∞and has system possesses coexisting attractors and how to find them all? nonpositive limit Lyapunov exponents (see, e.g. [71]). 123 276 N. V. Kuznetsov et al. the strange attractor. Thus, the numerical computation and its singular value decomposition (SVD)6: of trajectory for a longer time may not lead to a more SVD ∗ precise approximation of the Lyapunov exponents and Φ(T, u0) = U(T, u0)(T, u0)V(T, u0) , dimension (see, also example (24) below). Also any whereU(T, u )∗U(T, u ) ≡ I ≡ V(T, u )∗V(T, u ), long-time computation may be insufficient to reveal the 0 0 0 0 (T, u ) = diag{σ (T, u ), σ (T, u ), σ (T, u )} is limit values of LEs if the trajectory belongs to a tran- 0 1 0 2 0 3 0 a diagonal matrix composed by the singular values of sient chaotic set, which can be (almost) indistinguish- Φ(T, u ), and compute the finite-time Lyapunov expo- able numerically from sustained chaos (see Figs. 9a, 0 nents {LE (T, u )}3 from (T, u ) as in (16). Further, 10a). Note that there is no rigorous justification of the i 0 1 0 we also need the QR decomposition7 choice of t and it is known that unexpected jumps of dimL(t, K ) can occur (see, e.g. Fig. 7). Thus, for a QR Φ(T, u0) = Q(T, u0)R(T, u0), finite time interval t ∈[0, T ), it is reasonable to com- ( , ) ( , ) pute inft∈[0,T ) dimL t K instead of dimL T K ,but, where R(T, u0) is an upper triangular matrix with non- T (T, K ) { [ , ]= [ , ]( , )}3 at the same time, for any the value dimL gives negative diagonal elements R i i R i i T u0 1 ∗ an upper estimate of dimH K . and Q(T, u0) Q(T, u0) ≡ I . Finally, in the numerical experiments, based on the To avoid the exponential growth of values in the finite-time Lyapunov dimension definition (15)[19] and computation, the time interval has to be represented the Douady–Oesterlé theorem [79], we have as a union of sufficiently small intervals, for example, (0, T ]=(0,τ]∪(τ, 2τ]···∪((k−1)τ, kτ = T ]. Then, dimH K ≤ dimL K ≈ inf max dimL(t, u) ∈[ , ] ∈ ε using the cocycle property, the fundamental matrix can t 0 T u Kgrid be represented as LE1(t,u)+···+LE j(t,u)(t,u) = inf max j(t, u) + | ( , )| ∈[ , ] ∈ ε LE j(t,u)+1 t u t 0 T u Kgrid Φ( τ, ) = Φ(τ, − ) ... Φ(τ, )Φ(τ, ). ≤ max dimL(T, u) ≈ dimL(T, K ). k u0 uk 1 u1 u0 (20) ∈ ε u Kgrid (18) Here, if Φ(mτ,u0) and um = u(mτ,u0) are known, then Φ((m + 1)τ, u0) = Φ(τ,um)Φ(mτ,u0), where Additionally, we can consider a set of random points in Φ(τ,um) is the solution of initial value problem (19) K ε. If the maximums of the finite-time local Lyapunov with u(0) = um on the time interval [0,τ]. dimensions for random points and grid points differ By sequential QR decomposition of the product of δ more than by , then we decrease the distance between matrices in (20), we get grid points. This may help to improve the reliability of the result and at the same time to ensure its repeatability. Φ(kτ,u0) = Φ(τ,uk−1)...Φ(τ,u1) Φ(τ,u0) Nowadays, there are several widely used approaches = Φ(τ, )... Φ(τ, ) 0 0 to numerical computation of the Lyapunov exponents; uk−1 u1 Q1 R1 thus, it is important to state clearly how the LEs Q R are being computed [92, p. 121]. Next, we demon- =··· QR= 0 0 ... 0 . strate some differences in the approaches. For a certain Qk Rk R1 ∈ R3 u0 , to compute the finite-time Lyapunov expo- Then, the matrix with singular values nents according to (16), one has to find the fundamental T Σ( τ, ) = ∗( τ, )Φ( τ, ) ( τ, ) matrix Φ(T, u0) = Dϕ (u0) of (11) from the follow- k u0 U k u0 k u0 V k u0 ing variational equation: 6 See, for example, implementation in MATLABor GNU Octave ˙( , )= ( ( , )), ( , ) = , (https://octave-online.net): [U,S,V]=svd(A). u t u0 f u t u0 u 0 u0 u0 ∈[ , ], ˙ t 0 T 7 Φ(t, u0)= J(u(t, u0)) Φ(t, u0), Φ(0, u0)= I, For example, it can be done by the Gram–Schmidt orthogo- nalization procedure or the Householder transformation. MAT- (19) LAB and GNU Octave (https://octave-online.net) provide an implementation of the QR decomposition [Q, R] = qr(A). To have matrix R with positive diagonal elements, one can additionally use Q = Q*diag(sign(diag(R))); R = R*diag(sign(diag(R))). See also [106]. 123 Finite-time Lyapunov dimension and hidden attractor 277 in the SVD can be approximated by sequential QR Table 1 Approximation of the finite-time Lyapunov exponents decomposition of the product of matrices: p ( ) p ( ) p ( ) p LE1,2 5 LE1,2 25 LE1,2 100 Σ0 = Φ( τ, )∗ 0 = ( 0)∗ ...( 0)∗ QR= 1 1 ... 1, k u0 Qk R1 Rk Qk Rk R1 ∗ ∗ ∗ QR 00 0 0 Σ1 = (Q0) Φ(kτ,u ) Q1 = (R1) ...(R1) = Q2 R2 ...R2, k 0 k 1 k k k 1 1 ± 0.00797875 ± 0.04394912 ± 0.09360078 ··· 2 ± 0.01585661 ± 0.07379280 ± 0.09986978 where ⎛ ⎞ 3 ± 0.02353772 ± 0.08902280 ± 0.09999751 σ p 1 00 ± . ± . ± . Σ p = ( p)∗..( p)∗ = ⎝ · σ p ⎠ 4 0 03093577 0 09563887 0 09999995 R1 Rk 2 0 ± . ± . ± . ··σ p 5 0 03797757 0 09830568 0 09999999 3 10 ± 0.06638388 ± 0.09998593 ± 0.10000000 is a lower triangular matrix and [107,108] 50 ± 0.09993286 ± 0.09999999 ± 0.10000000 p p p σ = R [i, i] ...R [i, i]−→σi (kτ,u0), i = 1, 2, 3. i 1 k p→∞ 100 ± 0.09999998 ± 0.09999999 ± 0.10000000 Thus, the finite-time Lyapunov exponents can be approximated as k 1 k ( τ, )≈ 0( τ, ) = 0[ , ]. LCEi k u0 LEi k u0 ln Rl i i (T, u )≈ p(kτ,u )= 1 σ p = 1 R p[i, i]. kτ LEi 0 LEi 0 t ln i kτ ln l l=1 = l 1 (23) (21) The LCEs may differ from LEs, thus, the corre- The MATLAB implementation of the above method sponding Kaplan–Yorke formulas with respect to LEs for the computation of finite-time Lyapunov exponents and LCEs: dim (t, u ) = dKY({LE (t, u )}3) and with the fixed number of iterations p can be found, L 0 L i 0 1 dKY({LCE (t, u )}3),9 may not coincide. The follow- for example, in [9]. For large k, the convergence can be L i 0 1 ing artificial analytical example demonstrates the pos- very rapid: for example, for the Lorenz system with the sible difference between LEs and LCEs. The matrix classical parameters (r = 28, σ = 10, b = 8/3, a = [19,86] 0), k = 1000 and τ = 1, the number of approximations = 1 g(t) − g−1(t) p 1 is taken in [108, p. 44]. For a more precise R(t)= , g(t) = exp( t ) approximation of the finite-time Lyapunov exponents, 01 10 = ( ) = ,..., we can adaptively choose p p l , l 1 k so as has the following exact limit values to obtain a uniform estimate: = −1 ( ) = . , = , LCE1 lim t ln g t 0 1 LCE2 0 t→+∞ p(l)−1( τ, ) − p(l)( τ, ) <δ. = ,..., . max LEi l u0 LEi l u0 l 1 k −1 ±1 i=1,2,3 LE1,2 = lim t ln g (t) =±0.1, t→+∞ (22) where LCE2 = LE2. For the finite-time values we have Remark that there is another widely used defini- 1 1 LCE (t)= 1 ln (g(t)− )2 + 1 2 ∈(0, 0.1], tion of the “Lyapunov exponents” via the exponen- 1 t g(t) tial growth rates of norms of the fundamental matrix LCE (t)≡0, columns v (t, u ), v (t, u ), v (t, u ) = Φ(t, u ): 2 1 0 2 0 3 0 0 ( ) ≡ =± . . the finite-time Lyapunov characteristic exponents LE1,2 t LE1,2 0 1 { ( , )}3 { 1 ||vi ( , )||}3 LCEi t u0 1 are the set t ln t u0 1 ordered Approximations by the above algorithm with k = 1are 8 by decreasing. Relying on ergodicity [84], Benet- given in Table 1. tin et al. [109](Benettin’s algorithm; see also Wolf et Remark that here the approximation of LCEs and al. [110]) approximate the LCEs by (21) with p = 0: LEs by Benettin’s algorithm, i.e. by (23), becomes 8 To obtain all possible limit values of the finite-time Lyapunov worse with increasing time: { ( , )} characteristic exponents (LCEs) [2] lim supt→+∞ LCEi t u0 of linear system, one has to consider a normal fundamental 9 Rabinovich et al. [81, p. 203, p. 262] refer this value as local matrix, whose sum of LCEs is less or equal to the sum of LCEs dimension and note that it is a function of time and may be dif- of any other fundamental matrix. ferent in different parts of the attractor. 123 278 N. V. Kuznetsov et al.

( ) −→ ≡ 0( ) = 1 ≡ , dimension with respect to linear change of variables LCE1 t 0 LE1 t ln 1 0 t→+0 t implies ( ) −→ . ≡ ( ) = 0( ) = 1 ≡ . LCE1 t 0 1 LE1 t LE1 t ln 1 0 = KY({λ ( )}3 ). t→+∞ t dimL ueq dL i ueq i=1 (26) (24) If the maximum of local Lyapunov dimensions on Thus, although the notions of LCEs and LEs often do the global attractors, which involves all equilibria, cr = not differ (see, e.g. Eckmann and Ruelle [15, p. 620, is achieved at an equilibrium point: dimL ueq maxu∈K dimL u, then this allows one to get analytical p. 650], Wolf et al. [110, p. 286, pp. 290–291] and 10 Abarbanel et al. [17, p. 1363, p. 1364]) relying on the formula for the exact Lyapunov dimension. In gen- Oseledec ergodic theorem [84], in general case, the eral, a conjecture on the Lyapunov dimension of self- excited attractor computations of LCEs by (21) and LEs by (23)may [19,114] is that for a typical system, give unreliable results. See also numerical examples the Lyapunov dimension of a self-excited attractor does below and counterexamples in [111, p. 289], [97,p. not exceed the Lyapunov dimension of one of the unsta- 1083]. ble equilibria, the unstable manifold of which intersects with the basin of attraction and visualizes the attractor. To avoid numerical computation of the eigenval- 5.3 Estimation of the Lyapunov dimension without ues, one can use an effective analytical approach integration of the system and the exact Lyapunov [19,22,115], which is based on a combination of the dimension Douady-Oesterlé approach with the direct Lyapunov method: for example, in [22] for system (4) with b = 1, While analytical computation of the Lyapunov expo- it analytically obtained the following estimate nents and Lyapunov dimension is impossible in a (σ + ) general case, they can be estimated by the eigen- 2 2 dimL K ≤ 3 − . values of the symmetrized Jacobian matrix [79,112]. 2 16 r σ + 1 + (σ − 1) + σ {λ ( )}3 3 Let i u i=1 be the eigenvalues of the symmetrized 1 ( ( ) + ( )∗) The proof of the above conjecture and analytical deriva- Jacobian matrix 2 J u J u , ordered so that λ1(u) ≥ λ2(u) ≥ λ3(u).TheKaplan–Yorke formula tion of the exact Lyapunov dimension formula for sys- with respect to the ordered set of eigenvalues of the tem (4) with all possible parameters is an open problem. symmetrized Jacobian matrix [19] gives an upper esti- In [116,117], it is demonstrated how the above tech- mation of the Lyapunov dimension: nique can be effectively used to derive constructive upper bounds of the topological entropy of dynamical ≤ KY {λ ( )}3 . dimL K sup dL i u i=1 u∈K systems. In a general case, one cannot get the same values of {λ ( )}3 i u i=1 at different points u; thus, the maximum 6 The ﬁnite-time Lyapunov dimension in the case KY {λ ( )}3 of dL i u i on K has to be computed. To avoid of hidden attractor and multistability numerical localization of the set K , we can consider an analytical localization, e.g. by the absorbing set B ⊃ t Consider the dynamical system {ϕ }t≥0, generated by K . Then for the corresponding grid of points B ,we grid system (4) with parameters (5), and its attractor K . expect in numerical experiments the following: ϕt ( , , ) Here, x0 y0 z0 is a solution of (4) with the ≤ ≤ KY {λ ( )}3 initial data (x , y , z ). Since the dynamical system dimH K dimL K sup dL i u 1 0 0 0 u∈K {ϕt } R t≥0, generated by the Rabinovich system (1), t ≤ KY {λ ( )}3 ≈ ( ) can be obtained from {ϕ }t≥0 by the smooth trans- supdL i u 1 max j u ∈B u∈B χ −1 u grid formation ,inverseto(2), and inverse rescaling λ ( )+··+λ ( ) t 1 u j(u) u time (3) t → ν1t,wehavedimL({ϕ }t≥0, K ) = + |λ ( )| . (25) j(u)+1 u ({ϕt } ,χ−1( )) dimL R t≥0 K . In our experiments, we con- If the Jacobian matrix J(ueq) at one of the equilibria sider system (4) with parameters r = 100, a = {λ ( )}3 λ ( ) ≥ has simple real eigenvalues: i ueq i=1, i ueq 10 λi+1(ueq), then [19,86] the invariance of the Lyapunov This term was suggested by Doering et al. in [113]. 123 Finite-time Lyapunov dimension and hidden attractor 279

σ + δr 2 b(σ + δr)2 Bh = (x, y, z) ∈ R3 x2 + δy2 +(a + δ) z − ≤ For the considered set of parameters, we compute: a + δ 2c(a + δ) (i) Finite-time local Lyapunov dimensions dimL(100, ·) at the point O1 = (0, 1, 98), which h = belongs to the grid Cgrid, and at the point O2 (0.0099, 0.0995, 0) on the unstable manifold of zero equilibrium S0; (ii) Maximum of the finite-time local Lyapunov dimensions at the points of grid, max ∈ h dimL(t, u), u Cgrid for the time points t = t = 0.1 k (k = Ch ⊂ Ch k grid 1,...,1000); (iii) The corresponding values, given by the Kaplan– Yorke formula with respect to finite-time Lya- punov characteristic exponents. Fig. 6 Localization of the hidden attractor of system (4) withr = The results are given in Table 2. The dynamics of =− . × −3 = . × −2 100, a 9 965 10 , b 7 7454 10 by the absorbing finite-time local Lyapunov dimensions for different set Bh with δ =−a + 0.1 and cuboid Ch =[−11, 11]× [−17, 19]×[80, 117] points and their maximums on the grid of points are shown in Fig. 7. − − −9.965 × 10 3, b = 7.7454 × 10 2 corresponding For the absorbing set Bh and the corresponding grid to the hidden chaotic attractor. of points Bh (the distance between grid points is 5), h grid In Fig. 6 is shown the grid of points Cgrid filling the by estimation (25), we get the following estimate: h = hidden attractor: the grid of points fills cuboid C ≤ ≤ KY {λ ( )}3 [− , ]×[− , ]×[ , ] dimH K dimL K sup dL j u i=1 11 11 17 19 80 117 with the distance u∈Bh between points being equal to 0.5. The time interval is ≈ sup dKY {λ (u)}3 = 2.97001... . (27) [0, T = 100], k = 1000, τ = 0.1, and the integration L j i=1 u∈Bh method is MATLAB ode45 with predefined parame- grid σ + ≥ ters. The infimum on the time interval is computed at Assuming 1 b, the eigenvalues of the unstable the points {t }k with time step τ = t + − t = 0.1. zero equilibrium S0: i 1 i 1 i Note that if for a certain time t = ti the computed tra- 1 λ , (S ) =− (σ + 1)∓ (σ − 1)2 + 4σr , jectory is out of the cuboid, the corresponding value of 1 3 0 2 finite-time local Lyapunov dimension is not taken into λ2(S0) =−b, account in the computation of maximum of the finite- are ordered by decreasing, i.e. λ1(S0)>λ2(S0) ≥ time local Lyapunov dimension (there are trajectories λ3(S0). Therefore, for the considered values of param- with initial data in cuboid, which are attracted to the eters by (26), we get zero equilibria and belong to its stable manifold, e.g. KY 3 λ1(S0)+λ2(S0) dimL S0 = d ({λi (S0)} = ) = 2 + |λ ( )| system (4) with x = y = 0isz˙ =−bz). For the finite- L i 1 3 S0 (σ+ + ) time Lyapunov exponents (FTLEs) computation, we = 3 − 2 √ b 1 = 2.8111... . (σ+ )+ (σ− )2+ σ use MATLAB realization from [9] based on (21) with 1 1 4 r p = 2. For computation of the finite-time Lyapunov (28) characteristic exponents (FTLCEs), we use MATLAB The above numerical experiments lead us to the fol- realization from [118] based on (23). lowing important remarks. While the Lyapunov dimen-

Table 2 Numerical estimation of ﬁnite-time Lyapunov dimension in the case of hidden attractor (see Fig. 3)

t = 100 t = 100 t = 100 inf max t∈[0, 100] ∈ h u = (0, 1, 98) u = (0.0099, 0.0995, 0) max ∈ h u Cgrid u Cgrid

( , ) KY({ ( , )}3 ) . . . . dimL t u = dL LEi t u i=1 2 1474 1 4987 2 2063 2 2050 KY({ ( , )}3 ) . . . . dL LCEi t u i=1 2 1338 1 1213 2 2105 2 2076

123 280 N. V. Kuznetsov et al.

max dKY({LCE (t, u)}3 ) max dKY({LE (t, u)}3 ) L i i=1 L i i=1 u∈Ch u∈Ch grid grid dKY({LCE (t, O )}3 ) dKY({LE (t, O )}3 ) L i 1 i=1 L i 1 i=1 dKY({LCE (t, O )}3 ) dKY({LE (t, O )}3 ) L i 2 i=1 L i 2 i=1

3 3 max dKY({LE (100,u)}3 ) = max dim (100,u)=2.2063 L i i=1 L u∈Ch u∈Ch grid grid 2.8 max dKY({LCE (100,u)}3 )=2.2105 2.8 L i i=1 u∈Ch grid 2.6 2.6 dKY({LE (100,O )}3 ) = dim (100,O )=2.1474 L i 1 i=1 L 1

2.4 2.4

2.2 2.2

2 dKY({LCE (100,O )}3 )=2.1338 2 L i 1 i=1 1.8 1.8 KY 3 d ({LCEi(100,O2)}i )=1.1213 1.6 L =1 1.6

1.4 1.4 dKY({LE (100,O )}3 ) = dim (100,O )=1.4987 L i 2 i=1 L 2 1.2 1.2

1 t 1 t 0 102030405060708090100 0 102030405060708090100 tinf =98.5 tinf =98

Fig. 7 Dynamics of the finite-time local Lyapunov dimension (light red), at the point O2 = (0.0099, 0.0995, 0) from the one- estimations on the time interval t ∈[0, 100]: the maximum on dimensional unstable manifold of S0 (blue). (Color figure online) = ( , , ) ∈ h the grid of points (dark red), at the point O1 0 1 98 Cgrid sion, unlike the Hausdorff dimension, is not a dimen- dimension (i.e. numerical approximation of the limit sion in the rigorous sense [119] (e.g. the Lyapunov values of the finite-time Lyapunov dimensions). dimension of the saddle point S0 in (28) is noninteger), Consider system (4) with parameters r = 6.485, it gives an upper estimate of the Hausdorff dimension. a =−0.5, b = 0.85 for which equilibrium S0 is a If the attractor K or the corresponding absorbing set saddle point and equilibria S± are stable focus-nodes B ⊃ K is known [see, e.g. (8)] and our purpose is and integrate numerically11 the trajectory with initial to demonstrate that dimH K ≤ dimL K < 3, then it data u0 = (0.5, 0.5, 0.5) in the vicinity of the S0. can be achieved without integration of the considered We discard the part of the trajectory, corresponding to dynamical system [see, e.g. (27)]. If our purpose is to the initial transition process (for [0, ttp = 25,000]), get a precise estimation of the Hausdorff dimension, and get the point uinit = ( −2.089862710574761, then we can use (18) and compute the finite-time Lya- −2.500837780529156, 2.776106323157132). Further, punov dimension, i.e. to find the maximum of the finite- we numerically approximate the finite-time Lyapunov time local Lyapunov dimensions on a grid of points for exponents and dimension for the time interval [0, T ] a certain time. To be able to repeat a computation of by (23) (see MATLAB code in [120]). finite-time Lyapunov dimension, one need to know the { }k = initial points of considered trajectories u i i=1 Kgrid 11 Our experiment was carried out on the 2.5 GHz Intel Core ( , ]= k−1( , ] on the set K , time interval 0 T i=0 ti ti+1 i7 MacBook Pro laptop. For numerical integration, we use and how the finite-time Lyapunov exponents were MATLAB R2016b. To simplify the repetition of results, we computed (e.g. by (23), (21), or (22) with the parameter use the single-step fifth-order Runge–Kutta method ode5 from δ https://www.mathworks.com/matlabcentral/answers/98293-is- ). there-a-fixed-step-ordinary-differential-equation-ode-solver-in- matlab-8-0-r2012b#answer107643. Corresponding numerical simulation of the considered trajectory can be performed 7 Computation of the Lyapunov dimension and using the following code: phiT_u = feval(’ode5’, transient chaos @(t, u) [-3.2425*u(1) + 3.2425*u(2) + 0.5* u(2)*u(3); 6.485*u(1) - u(2) - u(1)*u(3); u(1)*u(2) - 0.85*u(3)], 0 : 0.01 : 15295, Consider an example, which demonstrates difficulties [-2.089862710574761, -2.500837780529156, in the reliable numerical computation of the Lyapunov 2.776106323157132]); 123 Finite-time Lyapunov dimension and hidden attractor 281

S+ S− S+ S−

uinit uinit

S0 S0

(a) (b)

Fig. 8 The trajectory forms a chaotic set, which looks like an “attractor” (navy blue) and then tends to S+ (cyan). a Trajectory u(t, uinit) for t ∈[0, T1], T1 ≈ 15,295. b Trajectory u(t, uinit) for t ∈[0, T ], T = 500,000. (Color ﬁgure online)

LCE (t, u ) dKY({LCE (t, u )}3 ) 1 init L i init i=1 2.5 0.4 2

0.3 2.1 1.5

0.2 2 1

0.1 1.9 0.5 0 20 40 60 80 100

0 t 0 t 0 5000 10000 15000 0 5000 10000 15000 (a) (b)

( , ) KY({ ( , )}3 ) [ , ≈ , ] ( , ) Fig. 9 Numerical computation of LCE1 t uinit and dL LCEi t uinit i=1 for the time interval 0 T1 15 295 . a LCE1 t uinit , ∈[ , ] ≈ , KY({ ( , )}3 ) ∈[ , ] ≈ , t 0 T1 , T1 15 295. b dL LCEi t uinit i=1 , t 0 T1 , T1 15 295

The trajectory computed on the time interval Since the lifetime of transient chaotic process can be [0, T1 ≈ 15295] traces a chaotic set in the phase space, extremely long and taking into account the limitations which looks like an “attractor” (see Fig. 8a, Fig. 9a). of reliable integration of chaotic ODEs, even long- Further integration with t > T1 leads to the collapse of time numerical computation of the finite-time Lya- the “attractor”, i.e. the “attractor” (see Fig. 8a) turns punov exponents and the finite-time Lyapunov dimen- out to be a transient chaotic set (see Fig. 8b). How- sion does not necessarily lead to a relevant approxi- ever on the time interval t ∈[0, T3 ≈ 431,560], mation of the Lyapunov exponents and the Lyapunov we have LCE1(t, uinit)>0 (see Fig. 10a) and, thus, dimension [see also effects in (24)]. may conclude that the behaviour is chaotic, and for the time interval t ∈[0, T2 ≈ 223,447],wehave KY({ ( , )}3 )> dL LCEi t uinit i=1 2 (see Fig. 10b). This 8 Conclusion effect is due to the fact that the finite-time Lyapunov exponents and finite-time Lyapunov dimension are In this work, the Rabinovich system, describing the the values averaged over the considered time interval. process of interaction between waves in plasma, is 123 282 N. V. Kuznetsov et al.

LCE (t, u ) dKY({LCE (t, u )}3 ) 1 init L i init i=1 2.5 0.4 0.005 2

0.3 1.5

0 0.2 300000 400000 500000 1

0.1 0.5

0 t 0 t 0 100000 200000 300000 400000 500000 0 100000 200000 300000 400000 500000

LCE1 > 0, LD > 2 LCE1 > 0, LD < 2 LCE1 > 0, LD > 2 LCE1 > 0, LD < 2 T1 T2 T3 T1 T2 T3 (a) (b)

( , ) KY({ ( , )}3 ) [ , · 5] ( , ) Fig. 10 Numerical computation of LCE1 t uinit and dL LCEi t uinit i=1 for the time interval 0 5 10 . a LCE1 t uinit , ∈[ , ] = · 5 KY({ ( , )}3 ) ∈[ , ] = · 5 t 0 T , T 5 10 . b dL LCEi t uinit i=1 , t 0 T , T 5 10 considered. It is shown that the methods of numeri- 3. Leonov, G., Reitmann, V.: Attraktoreingrenzung fur Nicht- cal continuation and perpetual point are helpful for the lineare Systeme. Teubner, Leipzig (1987) (in German) localization and understanding of hidden attractor in 4. Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8: 437–479 (1901–1902) the Rabinovich system. For the study of dimension of 5. Leonov, G., Kuznetsov, N.: On differences and similarities the hidden attractor, the concept of the finite-time Lya- in the analysis of Lorenz, Chen, and Lu systems. Appl. punov dimension is developed. An approach to reli- Math. Comput. 256, 334–343 (2015) able numerical estimation of the finite-time Lyapunov 6. Pisarchik, A., Feudel, U.: Control of multistability. Phys. Rep. 540(4), 167–218 (2014) exponents [see (21) and (22)] and finite-time Lyapunov 7. Leonov, G., Kuznetsov, N.: Hidden attractors in dynamical dimension [see (18)] is suggested. Various numerical systems. From hidden oscillations in Hilbert–Kolmogorov, estimates of the finite-time Lyapunov dimension for the Aizerman, and Kalman problems to hidden chaotic attrac- hidden attractor in the case of multistability are given. tors in Chua circuits. Int. J. Bifurc Chaos 23(1), 1330002 (2013) 8. Kuznetsov, N., Leonov, G.: Hidden attractors in dynam- Acknowledgements The work in Sects. 1–4 is done within ical systems: systems with no equilibria, multistability the joint Grant from DST and RFBR (INT/RUS/RFBR/P-230 and coexisting attractors. IFAC Proc. Vol. 47, 5445–5454 and 16-51-45002); Sects. 5–7 are done within Russian Science (2014) Foundation Project (14-21-00041). 9. Leonov, G., Kuznetsov, N., Mokaev, T.: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like sys- Open Access This article is distributed under the terms of the tem describing convective fluid motion. Eur. Phys. J. Spe- Creative Commons Attribution 4.0 International License (http:// cial Top. 224(8), 1421–1458 (2015) creativecommons.org/licenses/by/4.0/), which permits 10. Kuznetsov, N.: Hidden attractors in fundamental problems unrestricted use, distribution, and reproduction in any medium, and engineering models. A short survey. Lecture Notes provided you give appropriate credit to the original author(s) and Electr. Eng. 371, 13–25 (2016) (Plenary lecture at Inter- the source, provide a link to the Creative Commons license, and national Conference on Advanced Engineering Theory indicate if changes were made. and Applications 2015) 11. Grebogi, C., Ott, E., Yorke, J.: Fractal basin boundaries, long-lived chaotic transients, and unstable–unstable pair References bifurcation. Phys. Rev. Lett. 50(13), 935 (1983) 12. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N., Leonov, G., Prasad, A.: Hidden attractors in dynamical sys- 1. Poincare, H.: Les methodes nouvelles de la mecanique tems. Phys. Rep. 637, 1–50 (2016) celeste, vol. 1–3. Gauthiers-Villars, Paris (1892, 1893, 13. Kaplan, J., Yorke, J.: Chaotic behavior of multidimensional 1899) (English transl. edited by D. Goroff: American Insti- difference equations. In: Peitgen, H.-O., Walther, H.-O. tute of Physics, NY, 1993) (eds.) Functional Differential Equations and Approxima- 2. Lyapunov, A.: The General Problem of the Stability of tions of Fixed Points, pp. 204–227. Springer, Berlin (1979) Motion, Kharkov (1892) (English transl.: Academic Press, NY, 1966). (in Russian) 123 Finite-time Lyapunov dimension and hidden attractor 283

14. Grassberger, P., Procaccia, I.: Measuring the strangeness The Aizerman and Kalman conjectures and Chua’s circuits. of strange attractors. Phys. D Nonlinear Phenom. 9(1–2), J. Comput. Syst. Sci. Int. 50(4), 511–543 (2011) 189–208 (1983) 35. Leonov, G., Kuznetsov, N., Vagaitsev, V.: Hidden attrac- 15. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and tor in smooth Chua systems. Phys. D Nonlinear Phenom. strange attractors. Rev. Mod. Phys. 57(3), 617–656 (1985) 241(18), 1482–1486 (2012) 16. Constantin, P., Foias, C., Temam, R.: Attractors represent- 36. Kuznetsov, N., Kuznetsova, O., Leonov, G., Vagaitsev, V.: ing turbulent flows. Mem. Am. Math. Soc. 53(314), 1–67 Analytical-numerical localization of hidden attractor in (1985) electrical Chua’s circuit. Lecture Notes Electr. Eng. 174(4), 17. Abarbanel, H., Brown, R., Sidorowich, J., Tsimring, L.: 149–158 (2013) The analysis of observed chaotic data in physical systems. 37. Kiseleva, M., Kudryashova, E., Kuznetsov, N., Kuznetsova, Rev. Mod. Phys. 65(4), 1331–1392 (1993) O., Leonov, G., Yuldashev, M., Yuldashev, R.: Hidden and 18. Boichenko, V., Leonov, G., Reitmann, V.: Dimension The- self-excited attractors in Chua circuit: synchronization and ory for Ordinary Differential Equations. Teubner, Stuttgart SPICE simulation. Int. J. Parallel Emergent Distrib. Syst. (2005) (2017). https://doi.org/10.1080/17445760.2017.1334776 19. Kuznetsov, N.: The Lyapunov dimension and its estimation 38. Stankevich, N., Kuznetsov, N., Leonov, G., Chua, L.: Sce- via the Leonov method. Phys. Lett. A 380(25–26), 2142– nario of the birth of hidden attractors in the Chua circuit. 2149 (2016) Int. J. Bifurc. Chaos 27(12), 1730038 (2017) 20. Rabinovich, M.I.: Stochastic self-oscillations and turbu- 39. Burkin, I., Khien, N.: Analytical-numerical methods of lence. Soviet Phys Uspekhi 21(5), 443–469 (1978) finding hidden oscillations in multidimensional dynamical 21. Pikovski, A., Rabinovich, M., Trakhtengerts, V.: Onset of systems. Differ. Equ. 50(13), 1695–1717 (2014) stochasticity in decay confinement of parametric instabil- 40. Li, C., Sprott, J.: Coexisting hidden attractors in a 4-D ity. Sov. Phys. JETP 47, 715–719 (1978) simplified Lorenz system. Int. J. Bifurc. Chaos 24(03), 22. Leonov, G., Boichenko, V.: Lyapunov’s direct method in 1450034 (2014) the estimation of the Hausdorff dimension of attractors. 41. Li, Q., Zeng, H., Yang, X.S.: On hidden twin attractors and Acta Appl. Math. 26(1), 1–60 (1992) bifurcation in the Chua’s circuit. Nonlinear Dyn. 77(1–2), 23. Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 255–266 (2014) 20(2), 130–141 (1963) 42. Pham, V.T., Rahma, F., Frasca, M., Fortuna, L.: Dynam- 24. Chueshov, I.: Introduction to the Theory of Infinite- ics and synchronization of a novel hyperchaotic system dimensional Dissipative Systems. Electronic Library of without equilibrium. Int. J. Bifurc. Chaos 24(06), 1450087 Mathematics. ACTA, Kharkiv (2002) (2014) 25. Leonov, G., Kuznetsov, N., Vagaitsev, V.: Localization of 43. Chen, M., Li, M., Yu, Q., Bao, B., Xu, Q., Wang, J.: Dynam- hidden Chua’s attractors. Phys. Lett. A 375(23), 2230–2233 ics of self-excited attractors and hidden attractors in gener- (2011) alized memristor-based Chua’s circuit. Nonlinear Dyn. 81, 26. Sommerfeld, A.: Beitrage zum dynamischen ausbau der 215–226 (2015) festigkeitslehre. Z. Vereins Dtsch. Ing. 46, 391–394 (1902). 44. Kuznetsov, A., Kuznetsov, S., Mosekilde, E., Stankevich, (in German) N.: Co-existing hidden attractors in a radio-physical oscil- 27. Kiseleva, M., Kuznetsov, N., Leonov, G.: Hidden attractors lator system. J. Phys. A Math. Theor. 48, 125101 (2015) in electromechanical systems with and without equilibria. 45. Saha, P., Saha, D., Ray, A., Chowdhury, A.: Memristive IFAC-PapersOnLine 49(14), 51–55 (2016) non-linear system and hidden attractor. Eur. Phys. J. Special 28. Leonov, G., Kuznetsov, N.: Algorithms for searching for Top. 224(8), 1563–1574 (2015) hidden oscillations in the Aizerman and Kalman problems. 46. Semenov, V., Korneev, I., Arinushkin, P., Strelkova, G., Dokl. Math. 84(1), 475–481 (2011) Vadivasova, T., Anishchenko, V.: Numerical and exper- 29. Aizerman, M.A.: On a problem concerning the stability imental studies of attractors in memristor-based Chua’s in the large of dynamical systems. Uspekhi Mat. Nauk 4, oscillator with a line of equilibria. Noise-induced effects. 187–188 (1949). (in Russian) Eur. Phys. J. Special Top. 224(8), 1553–1561 (2015) 30. Kalman, R.E.: Physical and mathematical mechanisms of 47. Sharma, P., Shrimali, M., Prasad, A., Kuznetsov, N., instability in nonlinear automatic control systems. Trans. Leonov, G.: Control of multistability in hidden attractors. ASME 79(3), 553–566 (1957) Eur. Phys. J. Special Top. 224(8), 1485–1491 (2015) 31. Bautin, N.N.: On the number of limit cycles generated on 48. Zhusubaliyev, Z., Mosekilde, E., Churilov, A., Medvedev, varying the coefficients from a focus or centre type equilib- A.: Multistability and hidden attractors in an impulsive rium state. Dokl. Akad. Nauk SSSR 24(7), 668–671 (1939). Goodwin oscillator with time delay. Eur. Phys. J. Special (in Russian) Top. 224(8), 1519–1539 (2015) 32. Kuznetsov, N., Kuznetsova, O., Leonov, G.: Visualization 49. Wei, Z., Yu, P., Zhang, W., Yao, M.: Study of hidden of four normal size limit cycles in two-dimensional poly- attractors, multiple limit cycles from Hopf bifurcation and nomial quadratic system. Differ. Equ. Dyn. Syst. 21(1–2), boundedness of motion in the generalized hyperchaotic 29–34 (2013) Rabinovich system. Nonlinear Dyn. 82(1), 131–141 (2015) 33. Kuznetsov, N., Leonov, G., Vagaitsev, V.: Analytical- 50. Danca, M.F., Kuznetsov, N., Chen, G.: Unusual dynamics numerical method for attractor localization of generalized and hidden attractors of the Rabinovich–Fabrikant system. Chua’s system. IFAC Proc. Vol. 43(11), 29–33 (2010) Nonlinear Dyn. 88, 791–805 (2017) 34. Bragin, V.,Vagaitsev, V.,Kuznetsov, N., Leonov, G.: Algo- 51. Jafari, S., Pham, V.T., Golpayegani, S., Moghtadaei, M., rithms for finding hidden oscillations in nonlinear systems. Kingni, S.: The relationship between chaotic maps and 123 284 N. V. Kuznetsov et al.

some chaotic systems with hidden attractors. Int. J. Bifur- exciting homopolar disc dynamo. Chaos 27(3), 033101 cat. Chaos 26(13), 1650211 (2016) (2017) 52. Menacer, T., Lozi, R., Chua, L.: Hidden bifurcations in 68. Zhang, G., Wu, F., Wang, C., Ma, J.: Synchronization the multispiral Chua attractor. Int. J. Bifurc. Chaos 26(14), behaviors of coupled systems composed of hidden attrac- 1630039 (2016) tors. Int. J. Mod. Phys. B 31, 1750180 (2017) 53. Ojoniyi, O., Njah, A.: A 5D hyperchaotic Sprott B system 69. Lai, Y., Tel, T.: Transient Chaos: Complex Dynamics on with coexisting hidden attractors. Chaos Solitons Fractals Finite Time Scales. Springer, New York (2011) 87, 172–181 (2016) 70. Danca, M.F., Kuznetsov, N.: Hidden chaotic sets in a Hop- 54. Pham, V.T., Volos, C., Jafari, S., Vaidyanathan, S., Kapita- field neural system. Chaos Solitons Fractals 103, 144–150 niak, T., Wang, X.: A chaotic system with different families (2017) of hidden attractors. Int. J. Bifurc. Chaos 26(08), 1650139 71. Chen, G., Kuznetsov, N., Leonov, G., Mokaev, T.: Hidden (2016) attractors on one path: Glukhovsky–Dolzhansky, Lorenz, 55. Rocha, R., Medrano-T, R.O.: Finding hidden oscillations in and Rabinovich systems. Int. J. Bifurc. Chaos 27(8), the operation of nonlinear electronic circuits. Electr. Lett. 1750115 (2017) 52(12), 1010–1011 (2016) 72. Kuznetsov, N., Leonov, G., Mokaev, T., Seledzhi, S.: Hid- 56. Wei, Z., Pham, V.T., Kapitaniak, T., Wang, Z.: Bifurca- den attractor in the Rabinovich system, Chua circuits and tion analysis and circuit realization for multiple-delayed PLL. AIP Conf. Proc. 1738(1), 210008 (2016) Wang–Chen system with hidden chaotic attractors. Non- 73. Leonov, G., Kuznetsov, N., Mokaev, T.: Hidden attractor linear Dyn. 85(3), 1635–1650 (2016) and homoclinic orbit in Lorenz-like system describing con- 57. Zelinka, I.: Evolutionary identification of hidden chaotic vective fluid motion in rotating cavity. Commun. Nonlinear attractors. Eng. Appl. Artif. Intell. 50, 159–167 (2016) Sci. Numer. Simul. 28, 166–174 (2015) 58. Borah, M., Roy, B.: Hidden attractor dynamics of a novel 74. Prasad, A.: Existence of perpetual points in nonlinear non-equilibrium fractional-order chaotic system and its dynamical systems and its applications. Int. J. Bifurc. synchronisation control. In: 2017 Indian Control Confer- Chaos 25(2), 1530005 (2015) ence (ICC), pp. 450–455 (2017) 75. Dudkowski, D., Prasad, A., Kapitaniak, T.: Perpetual points 59. Brzeski, P., Wojewoda, J., Kapitaniak, T., Kurths, J., Per- and hidden attractors in dynamical systems. Phys. Lett. A likowski, P.: Sample-based approach can outperform the 379(40–41), 2591–2596 (2015) classical dynamical analysis–experimental confirmation of 76. Prasad, A.: A note on topological conjugacy for perpetual the basin stability method. Sci. Rep. 7, 6121 (2017) points. Int. J. Nonlinear Sci. 21(1), 60–64 (2016) 60. Feng, Y., Pan, W.: Hidden attractors without equilib- 77. Nazarimehr, F., Saedi, B., Jafari, S., Sprott, J.: Are perpet- rium and adaptive reduced-order function projective syn- ual points sufficient for locating hidden attractors? Int. J. chronization from hyperchaotic Rikitake system. Pramana Bifurc. Chaos 27(03), 1750037 (2017) 88(4), 62 (2017) 78. Dudkowski, D., Prasad, A., Kapitaniak, T.: Perpetual 61. Jiang, H., Liu, Y.,Wei, Z., Zhang, L.: Hidden chaotic attrac- points: new tool for localization of coexisting attractors in tors in a class of two-dimensional maps. Nonlinear Dyn. dynamical systems. Int. J. Bifurc. Chaos 27(04), 1750063 85(4), 2719–2727 (2016) (2017) 62. Kuznetsov, N., Leonov, G., Yuldashev, M., Yuldashev, R.: 79. Douady, A., Oesterle, J.: Dimension de Hausdorff des Hidden attractors in dynamical models of phase-locked attracteurs. C.R. Acad. Sci. Paris Ser. A (in French) loop circuits: limitations of simulation in MATLAB and 290(24), 1135–1138 (1980) SPICE. Commun. Nonlinear Sci. Numer. Simul. 51, 39– 80. Hunt, B.: Maximum local Lyapunov dimension bounds 49 (2017) the box dimension of chaotic attractors. Nonlinearity 9(4), 63. Ma, J., Wu, F., Jin, W., Zhou, P., Hayat, T.: Calculation 845–852 (1996) of Hamilton energy and control of dynamical systems with 81. Rabinovich, M., Ezersky, A., Weidman, P.: The Dynamics different types of attractors. Chaos Interdiscip. J. Nonlinear of Patterns. World Scientific, Singapore (2000) Sci. 27(5), 053108 (2017) 82. Ruelle, D., Takens, F.: On the nature of turbulence. Com- 64. Messias, M., Reinol, A.: On the formation of hidden chaotic mun. Math. Phys. 20(3), 167–192 (1971) attractors and nested invariant tori in the Sprott A system. 83. Grebogi, C., Ott, E., Yorke, J.: Chaos, strange attractors, Nonlinear Dyn. 88(2), 807–821 (2017) and fractal basin boundaries in nonlinear dynamics. Sci- 65. Singh, J., Roy, B.: Multistability and hidden chaotic attrac- ence 238(4827), 632–638 (1987) tors in a new simple 4-D chaotic system with chaotic 2-torus 84. Oseledets, V.: A multiplicative ergodic theorem. Char- behaviour. Int. J. Dyn. Control (2017). https://doi.org/10. acteristic Lyapunov exponents of dynamical systems. 1007/s40435-017-0332-8 Trudy Mosk. Matematicheskogo Obshchestva 19, 179–210 66. Volos, C., Pham, V.T., Zambrano-Serrano, E., Munoz- (1968). (in Russian) Pacheco, J.M., Vaidyanathan, S., Tlelo-Cuautle, E.: Analy- 85. Vallejo, J., Sanjuan, M.: Predictability of Chaotic sis of a 4-D hyperchaotic fractional-order memristive sys- Dynamics: A Finite-time Lyapunov Exponents Approach. tem with hidden attractors. In: Vaidyanathan, S., Volos, C. Springer, Cham (2017) (eds.) Advances in Memristors, Memristive Devices and 86. Kuznetsov, N., Alexeeva, T., Leonov, G.: Invariance of Systems, pp. 207–235. Springer, Berlin (2017) Lyapunov exponents and Lyapunov dimension for regular 67. Wei, Z., Moroz, I., Sprott, J., Akgul, A., Zhang, W.: Hidden and irregular linearizations. Nonlinear Dyn. 85(1), 195– hyperchaos and electronic circuit application in a 5D self- 201 (2016)

123 Finite-time Lyapunov dimension and hidden attractor 285

87. Ledrappier, F.: Some relations between dimension and Lya- 105. Eden, A.: An abstract theory of L-exponents with applica- punov exponents. Commun. Math. Phy. 81(2), 229–238 tions to dimension analysis. Ph.D. thesis. Indiana Univer- (1981) sity (1989) 88. Frederickson, P., Kaplan, J., Yorke, E., Yorke, J.: The Lya- 106. Ramasubramanian, K., Sriram, M.: A comparative study punov dimension of strange attractors. J. Differ. Equ. 49(2), of computation of Lyapunov spectra with different algo- 185–207 (1983) rithms. Phys. D Nonlinear Phenom. 139(1–2), 72–86 89. Farmer, J., Ott, E., Yorke, J.: The dimension of chaotic (2000) attractors. Phys. D Nonlinear Phenom. 7(1–3), 153–180 107. Rutishauser, H., Schwarz, H.: The LR transformation (1983) method for symmetric matrices. Numer. Math. 5(1), 273– 90. Dellnitz, M., Junge, O.: Set oriented numerical methods for 289 (1963) dynamical systems. In: Handbook of Dynamical Systems, 108. Stewart, D.: A new algorithm for the SVD of a long product vol. 2. Elsevier, Amsterdam, pp. 221–264 (2002) of matrices and the stability of products. Electron. Trans. 91. Barreira, L., Schmeling, J.: Sets of “Non-typical” points Numer. Anal. 5, 29–47 (1997) have full topological entropy and full Hausdorff dimension. 109. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lya- Israel J. Math. 116(1), 29–70 (2000) punov characteristic exponents for smooth dynamical sys- 92. Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, tems and for hamiltonian systems. A method for comput- G.: Chaos: Classical and Quantum. Niels Bohr Institute, ing all of them. Part 2: Numerical application. Meccanica Copenhagen. http://ChaosBook.org (2016) 15(1), 21–30 (1980) 93. Ott, W., Yorke, J.: When Lyapunov exponents fail to exist. 110. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Deter- Phys.Rev.E78, 056203 (2008) mining Lyapunov exponents from a time series. Phys. D 94. Young, L.S.: Mathematical theory of Lyapunov exponents. Nonlinear Phenom. 16(D), 285–317 (1985) J. Phys. A Math. Theor. 46(25), 254001 (2013) 111. Bylov, B.E., Vinograd, R.E., Grobman, D.M., Nemytskii, 95. Pikovsky, A., Politi, A.: Lyapunov Exponents: A Tool to V.V.: Theory of Characteristic Exponents and Its Applica- Explore Complex Dynamics. Cambridge University Press, tions to Problems of Stability. Nauka, Moscow (1966). (in Cambridge (2016) Russian) 96. Kuznetsov, N., Leonov, G.: On stability by the first approx- 112. Smith, R.: Some application of Hausdorff dimension imation for discrete systems. In: 2005 International Con- inequalities for ordinary differential equation. Proc. R. Soc. ference on Physics and Control, PhysCon 2005. Volume Edinb. 104A, 235–259 (1986) Proceedings Volume 2005. IEEE, pp. 596–599 (2005) 113. Doering, C., Gibbon, J., Holm, D., Nicolaenko, B.: Exact 97. Leonov, G., Kuznetsov, N.: Time-varying linearization and Lyapunov dimension of the universal attractor for the the Perron effects. Int. J. Bifurc. Chaos 17(4), 1079–1107 complex Ginzburg–Landau equation. Phys. Rev. Lett. 59, (2007) 2911–2914 (1987) 98. Benedicks, M., Young, L.S.: Sinai-Bowen-Ruelle mea- 114. Kuznetsov, N., Leonov, G.: A short survey on Lyapunov sures for certain Henon maps. Invent. Math. 112(1), 541– dimension for finite dimensional dynamical systems in 576 (1993) Euclidean space (2016). arxiv:1510.03835 99. Tasaki, S., Gilbert, T., Dorfman, J.: An analytical construc- 115. Leonov, G.: On estimations of Hausdorff dimension of tion of the SRB measures for Baker-type maps. Chaos Inter- attractors. Vestnik St. Petersb. Univ. Math. 24(3): 38– discip.J.NonlinearSci.8(2), 424–443 (1998) 41 (1991) [Transl. from Russian: Vestnik Leningradskogo 100. Schmeling, J.: A dimension formula for endomorphisms— Universiteta. Mathematika, 24(3): 41–44 (1991)] the Belykh family. Ergod. Theory Dyn. Syst. 18, 1283– 116. Boichenko, V., Leonov, G.: Lyapunov’s direct method in 1309 (1998) estimates of topological entropy. J. Math. Sci. 91(6), 3370– 101. Kuznetsov, N., Leonov, G., Mokaev, T.: Finite-time and 3379 (1998) exact Lyapunov dimension of the Henon map (2017). 117. Pogromsky, A., Matveev, A.: Estimation of topological arxiv:1712.01270 entropy via the direct Lyapunov method. Nonlinearity 102. Kehlet, B., Logg, A.: Quantifying the computability of the 24(7), 1937–1959 (2011) Lorenz system using a posteriori analysis. In: Proceedings 118. Kuznetsov, N., Mokaev, T., Vasilyev, P.: Numerical justi- of the VI International Conference on Adaptive Modeling fication of Leonov conjecture on Lyapunov dimension of and Simulation (ADMOS 2013) (2013) Rossler attractor. Commun. Nonlinear Sci. Numer. Simul. 103. Kehlet, B., Logg, A.: A posteriori error analysis of round- 19, 1027–1034 (2014) off errors in the numerical solution of ordinary differential 119. Hurewicz, W., Wallman, H.: Dimension Theory. Princeton equations. Numer. Algorithms 76(1), 191–210 (2017) University Press, Princeton (1941) 104. Liao, S., Wang, P.: On the mathematically reliable long- 120. Siu, S. www.mathworks.com/matlabcentral/fileexchange/ term simulation of chaotic solutions of Lorenz equation 233-let (1998) in the interval [0, 10000]. Sci. China Phys. Mech. Astron. 57(2), 330–335 (2014)

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