The Lorenz System : Hidden Boundary of Practical Stability and the Lyapunov Dimension

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Author(s): Kuznetsov, N. V.; Mokaev, T. N.; Kuznetsova, O. A.; Kudryashova, E. V.

Title: The Lorenz system : hidden boundary of practical stability and the Lyapunov dimension

Year: 2020

Version: Published version

Rights: CC BY 4.0

Rights url: https://creativecommons.org/licenses/by/4.0/

Please cite the original version: Kuznetsov, N. V., Mokaev, T. N., Kuznetsova, O. A., & Kudryashova, E. V. (2020). The Lorenz system : hidden boundary of practical stability and the Lyapunov dimension. Nonlinear Dynamics, 102(2), 713-732. https://doi.org/10.1007/s11071-020-05856-4 Nonlinear Dyn https://doi.org/10.1007/s11071-020-05856-4

ORIGINAL PAPER

The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

N. V. Kuznetsov · T. N. Mokaev · O. A. Kuznetsova · E. V. Kudryashova

Received: 16 March 2020 / Accepted: 29 July 2020 © The Author(s) 2020

Abstract On the example of the famous Lorenz Keywords Global stability · Chaos · Hidden attractor · system, the difficulties and opportunities of reliable Transient set · Lyapunov exponents · Lyapunov numerical analysis of chaotic dynamical systems are dimension · Unstable periodic orbit · Time-delayed discussed in this article. For the Lorenz system, the feedback control boundaries of global stability are estimated and the difficulties of numerically studying the birth of self- excited and hidden attractors, caused by the loss of 1 Introduction global stability, are discussed. The problem of reliable numerical computation of the finite-time Lyapunov In 1963, meteorologist Edward Lorenz suggested an dimension along the trajectories over large time inter- approximate mathematical model (the Lorenz system) vals is discussed. Estimating the Lyapunov dimension for the Rayleigh–Bénard convection and discovered of attractors via the Pyragas time-delayed feedback numerically a chaotic attractor in this model [76]. This control technique and the Leonov method is demon- discovery stimulated rapid development of the chaos strated. Taking into account the problems of reliable theory, numerical methods for attractor investigation, numerical experiments in the context of the shadowing and still attracts much attention of scientists from dif- and hyperbolicity theories, experiments are carried out ferent fields1 (see, e.g., [26,89,96,102,104]). The aim on small time intervals and for trajectories on a grid of of this work is to discuss the estimates of the global initial points in the attractor’s basin of attraction. stability and the difficulties of numerically studying the birth of self-excited and hidden attractors, caused Dedicated to the memory of Gennady A. Leonov. by the loss of global stability. Consider the classical Lorenz system N. V. Kuznetsov (B)· ⎧ T. N. Mokaev · O.A. Kuznetsova · E. V. Kudryashova ⎨⎪x˙ =−σ(x − y), Mathematics and Mechanics Faculty, Saint Petersburg y˙ = rx − y − xz, (1) State University, Saint Petersburg, Russia ⎩⎪ e-mail: [email protected] z˙ =−bz + xy, N. V. Kuznetsov σ, > ∈ ( , ] Faculty of Information Technology, University of with physically sound parameters r 0, b 0 4 Jyväskylä, Jyväskylä, Finland [76]. N. V. Kuznetsov Institute for Problems in Mechanical Engineering, RAS, Saint- 1 The original celebrated work by Lorenz [76] has gained more Petersburg, Russia than 21,500 citations according to Google Scholar. 123 N. V. Kuznetsov et al.

An important task in both qualitative and quanti- the system is called globally stable.2 In this deﬁnition, tative investigation of dynamical systems is the study the stationary set can contain both stable (trivial oscil- of limiting behavior of a system after a transient pro- lations) and unstable equilibrium states, i.e., the local cess, i.e., the problem of revealing and analysis of all stability of all equilibria is not required3. Various ana- possible limiting oscillations. In a theoretical perspec- lytical methods, based on the Lyapunov ideas, have tive, this problem relates to a generalization [44,63] been developed (see survey, e.g., [47]) for the study of of the second part of the celebrated Hilbert’s 16th global stability and estimates of its boundaries in the problem [34] on the number and mutual disposition space of parameters. of attractors and repellers in the chaotic multidimen- Farther, the birth of nontrivial oscillations in a sys- sional dynamical systems, and, in particular, their tem can be observed when crossing the global stability dependence on the degree of polynomials in the model boundary in the space of parameters. In the simplest (see also corresponding discussion, e.g., in [99,113]). case, this is due to the loss of stability of the station- Remark that 16th Hilbert problem is far from com- ary set. Within this framework, it is naturally to clas- plete solution even for two-dimensional polynomial sify oscillations in systems as self-excited or hidden quadratic systems admitting only periodic attractors [39,62,67,68]. Basin of attraction of a hidden oscil- (see, e.g., [42,62]). lation in the phase space does not intersect with small A limiting oscillation (or a set of oscillations) is neighborhoods of any equilibria, whereas a self-excited called an attractor, if the initial data from its open oscillation is excited from an unstable equilibrium. A neighborhood in the phase space (with the exception self-excited oscillation is a nontrivial one if it does not of a minor set of points) lead to a long-term behavior approach the stationary states (i.e., ω-limit set of cor- that approaches it, and its attracting set is called the responding trajectory does not consist of equilibria). basin of attraction (i.e., a set of initial data for which In the general case, when the boundaries of global the trajectories tend to the attractor). stability are violated, the birth of oscillations can occur The examination of all limiting oscillations could be due to local bifurcations in the vicinity of the stationary facilitated if for a system it is possible to prove the dis- set (trivial boundary of the global stability)4 or nonlocal sipativeness in the sense of Levinson (or D-property) bifurcations (hidden boundary of the global stability). ([74], see also [67])—the existence of a bounded con- If an attractor is born via such a nonlocal bifurcation of vex globally absorbing set containing all such oscilla- the loss of global stability, then the attractor is a hidden tions. The Lorenz system (1) is dissipative in the sense one since the basin of its attraction is separated from of Levinson, and one can consider, e.g., the following the locally attractive stationary set. absorbing set [60,61]: In practice, only problem of the birth of new nontrivial attractors is of interest because there are round- 3 2 2 2 B = (x, y, z) ∈ R (x + y + (z − r − σ) ) off and truncation errors in numerical experiments and ≤ b(σ+r)2 , = (σ, , b ). there is noise in physical experiments. That is, if the 2c c min 1 2 (2) 2 ∈ We use the term “global stability” for simplicity of further This implies that all solutions of (1)existfort discussion, while in the literature there are used different terms [0, +∞) and, thus, system (1) generates a dynamical like “globally asymptotically stable” [108, p. 137], [31, p. 144], system. For considered assumptions on parameters, if “gradient-like” [72,p.2],[111, p. 56], “quasi-gradient-like” [72, r < 1, then system (1) has a unique equilibrium p. 2], [111, p. 56] and others, reﬂecting the features of the stationary set and the convergence of trajectories to it. S0 = (0, 0, 0), and if r > 1, then system (1) has three 3 This can be demonstrated on the example suggested by Vino- equilibria: S0 = (0, 0, 0) and two symmetric equilib- 2( − )+ 5 grad [109](seealso[32, p. 191]): x˙ = x y x y , ria: (x2+y2)(1+(x2+y2)2) 2( − ) y˙ = y y 2x . Here, a planar system with a unique (x2+y2)(1+(x2+y2)2) S± = (± (r − 1)b, ± (r − 1)b, r − 1). globally attractive equilibrium of the saddle type has a family of homoclinic orbits. Also, global stability in the presence of locally unstable equilibria is a typical case for systems describing pen- For a system that is dissipative in the sense of Levin- dulums, PLLs [11,21,46,69], and electric machines [64]. son, further study of existence of oscillations is usually 4 See, e.g., Andronov–Hopf bifurcation [2,5] and Bautin’s connected with the stability of a stationary set. If any “safe” and “dangerous” boundaries of stability [9], and corre- trajectory of a system tends to the stationary set, then sponding birth of hidden Chua attractors [43,101]. 123 The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension model has a global bounded convex absorbing set, then the Barbashin–Krasovskii theorem is not applica- over time, the state of the system, observed experimen- ble. In this case, system (1) may possess nontriv- tally, will be attracted to the local attractor contained in ial self-excited oscillations with respect to unstable the absorbing set. This problem can be considered as a equilibria and also hidden oscillations. practical interpretation of the problem of determining (2) For r > 1 and 2σ

2 2 2

2 1.5 1.5 1.5 2 2 101 1 101 10 1 101 1 1 1 1 1 1 0 101 1 100.1 100.1 99.9 99.9 1 100.1 0 99.9 0.5 99.9 0.5 0 0.5 00.1 0.1 100.1 10 2

9 99.6 -0 99 99 -0.4 9. .4 .6 6 . 6

0 ϑ 0 ϑ 0 -0 ϑ 0 -0 .3 .3 3 1 0

.3 0 0 -

-0. 1 1 101 1 0 2 1 0 .9 0 9 0 0 1 1 99.9 1 .1 99. 9.9 9 -0.5 101 .1 99.9 99.9 -0.5 1 -0.5 9 99. 100. 99. 0 9 100. 9 9 0 1 100.1 100.1 -1 -1 -1 101 1 101 101 2 1 101

2 -1.5 -1.5 2 -1.5

-2 -2 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 χ χ χ (a) υ = −1 (b) υ =0 (c) υ =1

Fig. 1 Level lines of Lyapunov function (5) surrounding the stationary set (red, green) of system (4) with parameters r = 24, σ = 10, b = 2σ + 0.1 in three different projections: υ =−1, υ = 0, υ = 1. (Color ﬁgure online)

special analytical method based on [40,41,56,95]. The −σ + μ 3 3 main idea behind this method is choosing a speciﬁc = − + μ > , 3 3 1 0 0 (8) nonsingular (3 × 3) matrix S and differentiable scalar − + μ 3 0 b function V : U ⊆ R3 → R1, to satisfy the following √ condition: = σ − z σ = y σ where 3 r 2 r , 3 2 r . Condition (8) ˙ λ1(u0, S) + λ2(u0, S) + V (u0)<0, u0 ∈ B, (6) can be rewritten as follows: 2 λ ( , )>λ( , )>λ( , ) r z where 1 u0 S 2 u0 S 3 u0 S are the ((μ − σ )(μ − ) − rσ ) − σ 1 eigenvalues of the following symmetrized matrix: 4 y2 (μ − 1) 1 −1 −1 ∗ − + rz > 0. (9) Q = SJ(u0)S + (SJ(u0)S ) 4 (μ − b) 2 and For the solutions of system (1), the following relation ⎛ ⎞ is known [60,61]: −σσ0 ⎝ ⎠ 2 2 2 2 J(u0) = J(x0, y0, z0) = r − z0 −1 −x0 lim sup[y (t) + (z(t) − r) ]≤l r , t→+∞ y0 x0 −b × 1, if b < 2, is the 3 3 Jacobian matrix of system (1). l = (10) √b , ≥ . Condition (6) yields the inequality 1 < r ≤ − if b 2 (σ+ )( + ) 2 b 1 b b 1 in (3). σ Using relations (9) and (10), we conclude that inequal- (3.1) For b ∈[0, 2] or b ∈[2, 4⎛] and 1 < r ⎞< r ity (6) is true, when σ 00 (σ+ )( + ) ⎜ ⎟ (μ − σ)(μ − ) = 2 b b 1 = ⎝ ⎠ 2 1 r0 σ( + 2) , we consider S 010 σ + 2 − b ≥ 0, r < , 1 l σ(1 + l2) 001 and V ≡ 0. Then, condition (6) is equivalent to the which matches the initial conditions on parameter b following relation: and r. (σ+ )( + ) (3.2) For b ∈[2, 4] and r ≤ r ≤ b b 1 , Q + μI > 0, (7) ⎛ 0 ⎞ σ −A−1 00 μ =− ( ) = σ + + > = ⎝ (b−1) ⎠ ( , , ) = where Tr J u0 b 1 0. Condition we consider S − σ 10 and V x y z (7) means that all leading principal minors 1,2,3 of 001 ( , , ) the corresponding matrix are positive. For the chosen V1 x y z = √ σ 2 1/2 , where A and matrix S,wehave =−σ +μ>0, = 3 + [(σ−1) +4σr] σr+(σ−b)(b−1) 1 2 (−b+μ) 2 2 2 2 V1(x, y, z) = γ1(y + z ) + γ2x 3 and relation (7) can be expressed in the following 4 2 σ way: +γ ( x − x z − y2 − 2xy) − z. 3 4σ 2 σ b 123 The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

σ ≥ If 2 b, then it is known [60,61] the following (0, 0,z0 exp(−bt)) relation for the solutions of system (1): S+ S− x2(t) lim inf z(t) − ≥ 0. (11) 8 t→+∞ 2σ 7 Using this relation, one can obtain the following esti- 6 2 5 mation for all z ≥ x : 2σ z 4

˙ 2 2 2 3 V1 ≤[2rxy − 2dy − 2bz ]γ1 +[2σ xy − 2σ x ]γ2 + 2 5 +[(4σ + 2b)z2 + 2(σ − r + d)xy − 2(σ − d)y2 − 2rx2]γ 3 1 +σ − σ . S 0 z b xy (12) 0 0 y -8 -6 -4 -5 -2 0 2 4 The matrix Q has the following eigenvalues: x 6 8

λ ( , ) =− , 2 u S b Fig. 2 The absence of self-excited and hidden attractors and { , } (σ+1) 1 2σ 2 b−1 2 the global stability of the stationary set S0 S± in the Lorenz λ , (u, S) =− ± (σ+1−2b)2+ Az− +A2 y+ x , 1 3 2 2 A σ system (1) with parameters σ = 16, b = 4, and r = 6.24 < (σ+b)(b+1) (13) σ . Trajectories (blue, purple) in small neighborhoods of the unstable equilibrium S0 attractors to stable equilibria S± for which relation λ1(u, S)>λ2(u, S)>λ3(u, S) (trivial attractors). (Color ﬁgure online) holds. Then, condition (6) takes the following form: then conditions (14) are equivalent to the following ˙ 2(λ1(u, S) + λ2(u, S) + V ) conditions: ≤−(σ + + ) +[(σ − )2 + σ ]1/2 + 1 2b 1 4 r b ≥ 1, rσ + (σ − b)(b − 1)>0, 2 2 ( − )σ( σ+ σ+ 2− ) + 2 −σ z + A z 2r 1 8 b b b > −3σ 2 + 6σ(b − 1) + (b − 1)2, [(σ−1)2+4σr]1/2 4 b(b+1) A2 b−1 2 ˙ which are satisﬁed for b ∈[2, 4] and r ≥ r . + y + σ x + V1 . 0 4 Thus, for these values condition (6) takes the fol- γ ≥ A2 + γ ( 2σ + ) lowing form: If 1 8b 3 b 1 , then using (12), one gets the following estimation: ˙ 2(λ1(u, S) + λ2(u, S) + V ) −σ + A2z2 + A2 ( + b−1 2) ≤−(σ + + b) z 4 4 y σ x 1 2 / + ˙ ≤ 2 + + 2, ∀ , , ≥ x2 , +[(σ − 1)2 + 4σr]1 2 < 0, V1 B1x B2xy B3 y x y z 2σ < (σ+b)(b+1) where which yields estimation r σ in (3).

2( − )2 This completes the proof. B =−2γ σ + A b 1 − 2γ r, 1 2 4σ 2 3 The case corresponding to conditions (3)ofThe- =− γ + A2 − (σ − )γ , orem 1, when all trajectories tend to the stationary B3 2 1 4 2 1 3 2( − ) σ set, however, not all equilibria of the stationary set are B = 2 rγ + A b 1 − + σγ + (σ + 1 − r)γ . 2 1 4σ 2b 2 3 locally stable, is illustrated in Fig. 2. Beyond the estimate (3) in Theorem 1, the analysis 2 + + 2 ≤ The later inequality B1x B2xy B3 y 0 holds of global stability and the birth of nontrivial attractors for any x, y,if can be performed numerically. It is further known that ≤ , ≤ , − 2 ≥ . B1 0 B3 0 4B1 B3 B2 0 (14) the separatrix of saddle S0 can form a homoclinic loop from which unstable cycles can arise and violate the If one considers the following coefﬁcients (see, e.g., global stability (however, a set of measure zero does [58]): not affect the global attraction on a stationary set from γ = 2 3σ+b−1 , 1 A 8σ(b+2) a practical point of view). Using the Fishing principle − [57,67,71] for the Lorenz system (1), it is possible to γ = A2 b 1 , 3 8σ+2 prove the following: ( +σ− ) γ = r−2 (γ − γ ) − A2 b 1 + 1 , 2 σ 3 1 4σ 2 2b 123 N. V. Kuznetsov et al.

equilibria S0, S± (Fig. 4). This gives an outer estimation of the practical global stability. S− S In this manner, Lorenz had discovered numerically 25 + in the phase space of system (1) the existence of the 20 celebrated chaotic Lorenz attractor from a vicinity of unstable zero equilibrium. z 15 10 20 5 4 The boundary of practical stability and absence S 0 of nontrivial attractors 0 0 y −10 −5 0 −20 x 5 10 The presence of an absorbing set (Fig. 5)impliesthe existence of a globally attractor Aglob, which contains Fig. 3 Two symmetric homoclinic orbits (homoclinic butterfly) all local self-excited and hidden attractors, and a sta- σ = = / in the Lorenz system (1) with parameters 10, b 8 3, and tionary set. r = r ≈ 13.926 h Thus, inside the set B it is possible to study numerically the presence of nontrivial self-excited and hidden Theorem 2 For σ and b fixed, there exists r = rh ∈ attractors for parameters r, σ , b not satisfying con- (1, +∞) corresponding to the homoclinic orbit of the ditions (3) of global stability, i.e., by fixing σ and b saddle equilibrium S0 if and only if 3σ>2b + 1. and by decreasing r from rcr.Forσ = 10, b = 8/3, this gives us the following region r ∈ (rgs, rcr), where For instance, for the classical values of parameters rgs ≈ 4.64, rcr ≈ 24.74. σ = 10, b = 8/3ofsystem(1), it is possible to A nontrivial self-exited attractor can be observed find numerically the approximate value of such homo- numerically for 24.06 r < rcr ≈ 24.74 (see, e.g., clinic bifurcation rh ≈ 13.926, when two symmetric [96]). In this case of nontrivial multistability, system (1) homoclinic orbits appear forming a homoclinic butter- possesses a local chaotic attractor A which is self- fly (Fig. 3). A further increase in the parameter r leads excited with respect to equilibrium S0 and coexists with to the birth of two saddle periodic orbits from each the trivial attractors S± (Fig. 6). homoclinic orbit [93].

4.1 Hidden attractor or hidden transient chaotic sets? 3 Outer estimation: the absence of trivial attractors For the Lorenz system (1), it is still an open question [39, p. 14], whether for some parameters there exists a For systems with a global absorbing set and an unstable hidden chaotic attractor, i.e., whether it is possible by stationary set, the existence of self-excited attractors changing parameters to disconnect the basin of attrac- is obvious. From a computational point of view, this tion from equilibria S0, S± (e.g., for the parameters σ = = 8 = allows one to use a standard computational procedure, 10, b 3 :ifr 28, then attractor is connected in which after a transient process, a trajectory, starting with S0, S±;ifr = 24.5, then attractor is connected from a point of unstable manifold in a neighborhood with only S0). The current results on the existence of the of equilibrium, reaches a state of oscillation; therefore, hidden attractors in the Lorenz system are the follow- one can easily identify it. ing. Recently reported hidden attractors in the Lorenz System (1) possesses the absorbing set B (defined system with r < rcr and locally stable equilibria S± σ> + > = σ(σ+b+3 ) by Eq. (2)) and for b 1, r rcr σ−b−1 all turn out to be a transient chaotic set (a set in the phase equilibria are unstable. Thus, in this case, system (1) space, which can persist for a long time, but after all has a nontrivial self-excited attractor. If we consider collapses), but not a sustained hidden chaotic attractor classical values of parameters σ = 10, b = 8/3, then [79,112]. for r > rcr, e.g., for r = 28, it is possible to observe Since in a numerical computation of a trajectory the self-excited chaotic attractor with respect to all three over a finite-time interval it is difficult to distinguish 123 The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

S− S+ S− S+ S− S+

50 50 50

40 40 40 z 30 z30 z 30 20 20 20 50 50 50 10 10 10 S 0 S 0 S 0 0 0 y 0 0 y 0 0 y −20 −20 −20 −10 0 −10 −10 10 −50 0 −50 0 −50 x 20 x 10 20 x 10 20

(a) Initial data near the equilib- (b) Initial data near the equilib- (c) Initial data near the equilibrium S0 rium S+ rium S−

= σ = = 8 Fig. 4 Numerical visualization of the classical self-excited local chaotic attractor in the Lorenz system (1)withr 28, 10, b 3 by integrating the trajectories with initial data from small neighborhoods of the unstable equilibria S0, S±

For the Lorenz system (1), suppose that σ = 10, = 8 ≈ . b 3 are ﬁxed and r varies. Near the point r 24 06, it is possible to observe a long living transient chaotic 100 B Ω set, which is hidden since its basin of attraction does 80 not intersect with the small vicinities of equilibrium S0. 60 z For example, for r = 24 a hidden transient chaotic set 40 can be visualized [112] from the initial point (2, 2, 2) 20 (Fig. 7). In [79], hidden transient chaotic set was 0

-50 = σ = = -20 obtained in system (1) with r 29, 4, b 2. 50 In our experiments, consider system (1) withr = 24. 0 x 0 y For a trajectory with a certain initial point, which is -50 50 computed by a certain solver with speciﬁc parameters, we estimate the moment of the end of transient behavior B Fig. 5 Global absorbing set (see Eq. (2)) and positive invariant as the moment of time when the trajectory falls into a set = (x, y, z) ∈ R3 [y2(t) + (z(t) − r)2]≤r 2 (see Eq. (10)) containing self-excited chaotic attractor in the Lorenz small vicinity of one of the stable equilibria S±.Using = σ = = 8 MATLAB’s standard procedure ode45 with default system (1)withr 28, 10, b 3 , when the stationary set −3 {S0, S±} is unstable parameters (relative tolerance 10 , absolute tolerance 10−6) for system (1) with parameters r = 24, σ = 10, b = 8/3 and for initial point u0 = (20, 20, 20), a tran- a sustained chaos from a transient chaos [30,52], it is sient chaotic behavior is observed on the time interval 4 reasonable to give a similar classiﬁcation for transient [0, 1.8 × 10 ], for initial point u0 = (−7, 8, 22)— chaotic sets [14,18]. A transient chaotic set is called a on the time interval [0, 7.2 × 104], for initial point 5 hidden transient chaotic set if it does not involve and u0 = (2, 2, 2)—on the time interval [0, 2.26 × 10 ], attract trajectories from a small neighborhood of equi- and for initial point u0 = (0, −0.5, 0.5) a transient libria; otherwise, it is called self-excited. chaotic behavior continues over a time interval of more For an arbitrary system possessing a transient than [0, 107]. Remark that, if we consider the same ini- chaotic set, the time of transient process depends tial points, but use MATLAB’s procedure ode45 with − strongly on the choice of initial data in the phase space relative tolerance 10 6, for all these initial points the and also on the parameters of numerical solvers to inte- chaotic transient behavior will last over a time inter- grate a trajectory (e.g., order of the method, step of inte- val of more than [0, 106], and corresponding transient gration, relative and absolute tolerances). This compli- chaotic sets will not collapse. As a conclusion from cates the task of distinguishing a transient chaotic set this numerical study of transient chaotic behavior, we from a sustained chaotic set (attractor) in numerical may suggest to specify precisely numerical solver, its experiments. 123 N. V. Kuznetsov et al.

S− S+

50 50 50

40 40 40 S− S+ S− S+ z 30 z 30 z 30 20 20 20 50 50 50 10 10 10 0 S 0 S 0 S 0 0 0 y 0 0 y 0 0 y -20 −20 −20 −10 -10 −10 0 −50 Y 0 10 -50 0 10 −50 Y x 10 20 x 20 Y x 20 (a) Initial data near the equilib- (b) Initial data near the equilib- (c) Initial data near the equilibrium S0 rium S+ rium S−

= . σ = = 8 Fig. 6 Numerical visualization of the self-excited local chaotic attractor in the Lorenz system (1) r 24 5, 10, b 3 by a trajectory that starts in the vicinity of the unstable equilibrium S0. This attractor coexists with stable equilibria S± (trivial attractors)

z z z 25

S S 10 S − S+ − S+ − S+ 5

0 S S -20 S 30 0 0 -10 0 20 10 0 0 y 10 -10 x y x x -20 y 20 -30 5 (a) u0 =(2, 2, 2), t ∈ [0, 1000] (b) u0 =(2, 2, 2), t∈[0, 2.3 · 10 ] (c) u0 in vicinity of S0

= σ = = 8 Fig. 7 Visualization of the hidden transient chaotic set in the Lorenz system (1)withr 24, 10, b 3 parameters, initial data and time interval, along which with parameters σ,r, b > 0 and parameter a = 0. the transient behavior continues. Consider b = 1, a > 0,σ>ar. (16) For the following linear transformation (see, e.g., [60]): 4.2 The study of existence of the hidden transient chaotic sets and hidden attractors via the ζ ζ (x, y, z) → x, , r − y , (17) continuation method σ − ar σ − ar system (1) is transformed to the Glukhovsky–Dolzhansky In [79,112], the study of existence of the hidden tran- system [27]: sient chaotic sets and hidden attractors in the Lorenz ⎧ ⎪ ˙ −σ + ζ + α , system was done by the numerical study of basins of ⎨x = x z yz y˙ = ρ − y − xz, (18) attraction on some 2d cross sections. Next, we demon- ⎩⎪ strate an approach based on the extension of the param- z˙ = −z + xy, eter space and the numerical continuation method (see where “Appendix A”). r(σ − ar) ζ 2a Consider the following generalization of the classi- ζ>0,ρ= >0,α= > 0. ζ (σ − )2 cal Lorenz system [76]: ar ⎧ (19) ⎪ ˙ =−σ( − ) − , ⎨x x y ayz The Glukhovsky–Dolzhansky system describes the y˙ = rx − y − xz, (15) ⎩⎪ convective ﬂuid motion inside a rotating ellipsoidal z˙ =−bz + xy, cavity. 123 The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

If we set Table 1 Initial point (x0, y0, z0) and time interval [0, T ] of numerical integration for each part of the path a < 0,σ=−ar, (20) Path (x0, y0, z0) T then after the linear transformation (see, e.g., [60]): GD → L (10.64, 60.78, 390) 104 ( , , ) → ν−1 ,ν−1ν−1 ,ν−1ν−1 ) , → ν → ( . , . , . ) . × 4 x y z 1 y 1 2 hx 1 2 hz t 1 t L R 0 2 0 2 0 35 1 1 10 with positive ν1,ν2, h, we obtain the Rabinovich system [83,87], describing the interaction of three reso- Consider two line segments, P → P and P → nantly coupled waves, two of which being parametri- GD L L P , defining two parts of the path in the continuation cally excited: R ⎧ procedure. Choose the partition of the line segments ⎨⎪x˙ = hy − ν1x − yz, into Nst = 10 parts and define intermediate values of i = + i ( − ) y˙ = hx − ν y + xz, (21) parameters as follows: PGD→L PGD PL PGD ⎩⎪ 2 Nst ˙ =− + , and Pi = P + i (P −P ), where i = 1,...,N . z z xy L→R L Nst R L st Initial points for trajectories of system (15) that define where hidden chaotic sets are presented in Table 1. At each σ = ν−1ν , = ν−1, =−ν2 −2, = ν−1ν−1 2. 1 2 b 1 a 2 h r 1 2 h iteration of the procedure, a chaotic set defined by the (22) trajectory in the phase space of system (15) is computed using MATLAB’s procedure ode45 with relative and Following [14], let us briefly describe the exper- absolute tolerances equal to 10−8. The last computed iment, which allows to organize the transition from point of the trajectory at the previous step is used as the the hidden attractors in Glukhovsky–Dolzhansky and initial point for computation at the next step. Rabinovich systems to the hidden transient chaotic set Thus, using numerical continuation procedure and in the Lorenz system. starting from the hidden attractor in the Glukhovsky– In this experiment for system (15), we consider three Dolzhansky system (18), it is possible to organize a = , = . ,σ = sets of parameters: PGD r 346 a 0 01 transition and localize numerically the hidden transient , = 4 b 1 (for the Glukhovsky–Dolzhansky system chaotic sets in both Lorenz (1) and Rabinovich (21)sys- = , = ,σ = , = / — GD), PL r 24 a 0 10 b 8 3 tems. One can ensure that in all three cases the trajec- = , = (for the Lorenz system — L), and PR r 24 a tories (including separatrices), starting in small neigh- −1/r − 0.01,σ =−ar, b = bcr + 0.14 (for the borhoods of the unstable equilibrium S0 = (0, 0, 0), Rabinovich system — R). Here, in contrast to the case are not attracted by the computed chaotic set, but tend of classical Lorenz system considered in previous sec- to the symmetric stable equilibria S±. tion, we change the parameters in such a way that hidden Glukhovsky–Dolzhansky and Rabinovich attractors are located not too close to the unstable zero equi- 5 Exact and finite-time Lyapunov dimension & librium so as to avoid a situation that numerically inte- Lyapunov exponents: global attractor, local grated trajectory persists for a long time and then falls attractors, and transient chaotic sets on an unstable manifold of the unstable zero equilibrium, then leaves the transient chaotic set, and finally In this section, we will demonstrate the difficulties in tends to one of the stable equilibria. the reliable numerical computation of the finite-time Hidden chaotic attractors in the Glukhovsky– Lyapunov exponents (FTLE) and finite-time Lyapunov = , = . ,σ= Dolzhansky system with PGD r 346 a 0 01 dimension (FTLD)5, as well as distinguishing with 4, b = 1 [66,67] and in the Rabinovich system with their help an attractor from a transition set. = , =−/ − . ,σ =− , = PR r 24 a 1 r 0 01 ar b Before proceeding to the description and discus- + . bcr 0 14 [44] could be obtained from the correspond- sion of these numerical experiments, let us remark that ing self-excited attractors by the numerical continua- investigations of the numerical reliability of computer- tion method (see “Appendix A”). generated chaotic trajectories have been extensively These sets of parameters define three points, PGD, 5 PL, and PR, in the 4D parameter space (r, a,σ,b). See corresponding definitions in “Appendix B.” 123 N. V. Kuznetsov et al.

S 45 − S + 40 700 S 45 − 35 S S S− + 600 + 30 40

z 35 500 20

15 30

400 10 25

z 5 z 300 S -40 20 0 -20 20 0 15 10 0 5 0 20 -5 -10 -15 -20 40 15 200 x y 10 100 x2 5 S -400 0 S -40 0 -200 0 -20 40 0 15 0 30 20 0 10 0 10 0 200 5 0 20 -10 -20 -5 -30 -40 400 -10 -15 40 x y x y

x2 S S AGD T + − AL S hid hid + S− S+ S−

R S S− A x1 + hid T x3 0 S AL 0 hid S 2 0 xT S0

Step 1 : r = 346,a=0.01,σ =4,b=1, Step 2 : r =24,a=0,σ =10,b=8/3, Step 3 : r =24,a=1/r − 0.01,σ = −ar,

Glukhovsky-Dolzhansky system, b = bcr +0.14,Rabinovich system, Lorenz system,separatrices of S0 → S±; separatrices of S0 → S±; separatrices of S0 → S±;

( + ) i+1 := i Fig. 8 Localization of hidden chaotic sets in Glukhovsky– point on the i 1 th iteration (yellow) is defined as x0 xT Dolzhansky, Lorenz and Rabinovich systems defined by equa- i = i ( ) (light green arrows), where xT x T is the final point (yel- tion (1) using numerical continuation method. Here, trajectories low). Outgoing separatrices of unstable zero equilibrium tend to xi (t) = (xi (t), yi (t), zi (t) (blue) are defined on the time interval two symmetric stable equilibria. (Color figure online) [0, T ],(GD→ L:T = 104;L→ R:T = 1.1×104) and initial

performed in the context of the shadowing theory and existence of fluctuations of the FTLE is a numerical hyperbolicity breakdown (see, e.g., [29,33,81,84,92]). fingerprint of shadowability breakdown via unstable A computer-generated chaotic trajectory shadows a dimension variability (see, e.g., [90,105–107]). “true” chaotic trajectory if the former stays uniformly Since the classical Lorenz system (1) has a property close to the latter and vice versa. Under this assump- of singular hyperbolicity (or quasi-hyperbolicity), only tion, the shadowing of numerical trajectories for a rea- finite shadowability is possible for it (see, e.g., [4,16]). sonable time span would be a minimum requirement Taking into account these problems, the experiments for a meaningful computer simulation of a dynamical are carried out on small time intervals and for trajecto- system. The obstructions to shadowability vary from ries on a grid of initial points in the attractor’s basin of mild to severe. For example, hyperbolic chaotic sys- attraction. tems have been proved by Anosov to be shadowable for an infinite period of time [3]. However, if non- hyperbolic behavior is caused by manifold tangencies, 5.1 The problem of choosing a time interval: attractor it has been proved that computer-generated chaotic tra- vs transient set jectories shadow “true” trajectories for a finite period of time [91]. Finally, strongly non-hyperbolic systems, Consider system (1) with parameters r = 24, σ = like those with unstable dimension variability [38], can 10, b = 8/3 and integrate numerically the trajec- be shown to be practically nonshadowable since the tory with initial data u0 = (20, 20, 20) using MAT- shadowing time may be very small. In particular, the LAB’s standard procedure ode45 with default param- 123 The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

S− S+ S− S+

z z

u 0 u0 S0 S0 x y x y

6 (b) Trajectory u(t, uinit)fort ∈ [0, 10 ]. (a) Trajectory u(t, uinit) for t ∈ [0,T1], T1 ≈ 18082.

Fig. 9 The trajectory forms a chaotic set, which looks like an “attractor” (navy blue) and then tends to S+ (cyan). (Color figure online) eters (relative tolerance 10−3, absolute tolerance 10−6). a time interval of more than [0, 106], and the transient We numerically approximate the finite-time Lyapunov chaotic set will not collapse. If now we choose a new exponents and finite-time Lyapunov dimension, using initial point u0 = (10, 10, 10), it would be possible to the algorithm from [44]. Integration with t > T1 ≈ observe a chaotic transient process over the time inter- 5 18082 leads to the collapse of the “attractor,” i.e., the val [0, T1 ≈ 6.85 × 10 ], while LE1(t, u0)>0 and “attractor” turns out to be a transient chaotic set. How- dimL(t, u0)>2 would be satisfied on the time inter- 5 6 ever, on the time interval t ∈[0, T3 ≈ 6.9 × 10 ] vals t ∈[0, T2,3], where T2,3 10 . we have LE1(t, u0)>0 and, thus, may conclude that the behavior is chaotic, and for the time interval 5 t ∈[0, T2 ≈ 3.443 × 10 ] we have dimL(t, u0)>2. 5.2 The problem of choosing the time interval and Thus, the estimation of duration of the transient behav- initial data: attractor and embedded unstable ior by analyzing the sign of the largest Lyapunov expo- periodic orbits nent could lead to the wrong conclusion. This numerical phenomenon can be explained due Consider the challenges of the finite-time Lyapunov to the fact that the finite-time Lyapunov exponents and dimension computation along the trajectories over large Lyapunov dimension are averaged during computation time intervals (see, e.g., [48–51]), which is connected over the considered time interval. Since the “lifetime” with the existence of unstable periodic orbits (UPOs) of a transient chaotic process can be extremely long embedded in chaotic attractor. The “skeleton” of a and in view of the limitations of reliable integration chaotic attractor comprises embedded unstable peri- of chaotic ODEs (which we will discuss in the next odic orbits (UPOs) (see, e.g., [1,6,17]), and one of subsection), even longtime computation of the finite- the effective methods among others for the computa- time Lyapunov exponents and the finite-time Lyapunov tionofUPOsisthedelayed feedback control (DFC) dimension does not guarantee a relevant approximation approach, suggested by K. Pyragas [85](seealsodis- of the Lyapunov exponents and the Lyapunov dimen- cussions in [15,45,55]). This approach allows Pyragas sion. and his progeny to stabilize and study UPOs in various Remark that, if we choose the same initial point chaotic dynamical systems. Nevertheless, some gen- u0 = (20, 20, 20), as in the previous experiment, but eral analytical results have been obtained [35], showing use MATLAB’s procedure ode45 with relative toler- that DFC has a certain limitation, called the odd num- ance 10−6, the chaotic transient process will last over ber limitation (ONL), which is connected with an odd number of real Floquet multipliers larger than unity. 123 N. V. Kuznetsov et al.

1 2.5

0.9

0.8 2

0.7 ) 0 ) 2.055 0 0.6 1.5

0.82 t, u 2.054 ( t, u 0.5 L ( 0.81

1 2.053 0.4 0.8 1 dim LE 0.79 2.052 0.3 0.78 2.051 0.2 0.77 0.5 0.76 2.05 0 0.5 1 1.5 2 0 0.5 1 1.5 2 4 0.1 10 4 10

0 t 0 t 0 0.5 1 1.5 2 0 0.5 1 1.5 2 4 104 10

(a) LE 1(t, u0), t ∈ [0,T1], T1 ≈ 18082. (b) dim L(t, u0), t ∈ [0,T1], T1 ≈ 18082.

Fig. 10 Numerical computation of the largest FTLE LE1(t, u0) and FTLD dimL(t, u0) for the time interval [0, T1 ≈ 18082]

2.5 1

0.9 2 0.8 0.01

0.7 ) 0 )

0 1.5 0.6 t, u

0 ( t, u 6 6.5 7 7.5 8 L 0.5 5 ( 10 1 0.4 1 dim LE

0.3 -0.01

0.2 0.5

0.1

0 t 0 t 0246810 0246810 5 10 LE > 0, LD > 2 LE > 0, LD < 2 105 LE1 > 0, LD > 2 LE1 > 0, LD < 2 1 1 T1 T2 T3 T1 T2 T3 6 6 (a) LE 1(t, u0), t ∈ [0,T=10 ]. (b) dim L(t, u0), t∈[0,T =10 ].

6 Fig. 11 Numerical computation of the largest FTLE LE1(t, u0) and FTLD dimL(t, u0) for the time interval [0, 10 ]

In order to overcome ONL, later Pyragas suggested a u˙(t) = f (u(t)) + KB FN (t) + w(t) , 0 0 ∞ modiﬁcation of the classical DFC technique, which was w(˙ t) = λ w(t) + (λ − λ )FN (t), c c c called the unstable delayed feedback control (UDFC) ( ) = ∗ ( ) − ( − ) N k−1 ∗ ( − ), FN t C u t 1 R k=1 R C u t kT (24) [86]. Rewrite system (1) in a general form where 0 ≤ R < 1 is an extended DFC parameter, = , ,...,∞ ˙ = ( ). N 1 2 deﬁnes the number of previous states u f u (23) ( ) λ0 > involved in delayed feedback function FN t , c 0, upo( , upo1 ) τ> λ∞ < Let u t u0 be its UPO with period 0, and c 0 are additional unstable degree of freedom uupo(t − τ,uupo1 ) = uupo(t, uupo1 ), and initial condi- parameters, B, C are vectors and K > 0 is a feedback 0 0 upo upo1 = upo( , upo1 ) gain. For initial condition u 1 and T = τ,wehave tion u0 u 0 u0 . To compute the UPO and 0 overcome ONL, we add the UDFC in the following form: FN (t) ≡ 0,w(t) ≡ 0, 123 The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

upo1 50 S− u(t, u ) S u(t, uupo1 ) 0 + 0 40

30 z z S− S+ 20

upo1 upo1 upo1 0 u u u˜(t, u ) -20 0 0 0 S0 20 S -10 10 0 0 0 x 10 -10 y 20 -20 x y (a) (b)

upo upo1 ( ) τ = . ˜( , 1 ) Fig. 12 Period-1 UPO u t (red, period 1 1 5586) stabilized using UDFC method, and pseudo-trajectory u t u0 (blue, t ∈[0, 100])insystem(1) with parameters r = 28, σ = 10, b = 8/3. (Color figure online) and, thus, the solution of system (24) coincides with We use the adaptive algorithm for the computation the periodic solution of initial system (23). of the finite-time Lyapunov dimension and exponents For the Lorenz system (1) with parameters r = 28, for trajectories on the local attractor A [44]. In order σ = 10, b = 8/3using(24) with B∗ = (0, 1, 0), to distinguish the corresponding values for the stabi- ∗ = ( , , ) = . = = . ( , upo1 ) C 0 1 0 , R 0 7, N 100, K 3 5, lized UPO u t u0 and for the pseudo-trajectory λ0 = . λ∞ =− ˜( , upo1 ) c 0 1, c 2, one can stabilize a period-1 UPO u t u0 computed without Pyragas stabilization in upo u 1 (t, u0) with period τ1 = 1.5586... from the ini- our experiment, we use the following notations for = ( , , ) w = ( ( , ·), upo1 ) tial point u0 1 1 1 , 0 0 (Fig. 12). Results of finite-time Lyapunov dimensions: dimL u t u0 ( ˜( , ·), upo1 ) this experiment could be repeated using various other and dimL u t u0 , respectively. numerical approaches (see, e.g., [13,82,110]) and are The comparison of the obtained values of finite- in agreement with similar results on the existence of time Lyapunov dimensions computed along the sta- UPOs embedded in the Lorenz attractor [8,25]. How- bilized UPO and the trajectory without stabilization ever, the Pyragas procedure, in general, is more con- gives us the following results. On the initial small part venient for UPOs numerical visualization and finds of the time interval, one can observe the coincidence widespread application in a rich variety of chaos con- of these values with a sufficiently high accuracy. For trol problems (see, e.g., [12,23,94]). the UPO and for the unstabilized trajectory, the finite- upo1 ≈ (− . , − . , ( ( , ·), upo1 ) For the initial point u0 6 2262 11 0027 time local Lyapunov dimensions dimL u t u0 upo upo . ) upo1 ( ) = ( , 1 ) ( ˜( , ·), 1 ) 13 0515 on the UPO u t u t u0 ,we and dimL u t u0 coincide up to the 4th deci- [ , 1 ≈ τ ] numerically compute the trajectory of system (24) mal place inclusive on the interval 0 tm 7 1 .After = > 1 without the stabilization (i.e., with K 0) on the t tm, the difference in values becomes significant time interval [0, T = 100] (Fig. 12b). We denote it by and the corresponding graphics diverge in such a way ˜( , upo1 ) u t u0 to distinguish this pseudo-trajectory from that the part of the graph corresponding to the unsta- ( , upo1 ) the periodic orbit u t u0 . One can see that on the bilized trajectory is lower than the part of the graph initial small time interval [0, T1 ≈ 11], even without corresponding to the UPO (see Figs. 13b, 14). ˜( , upo1 ) the control, the obtained trajectory u t u0 traces The Jacobi matrix at the saddle-foci equilibria S± ( , upo1 ) approximately the “true” periodic orbit u t u0 .But has simple eigenvalues, which give the following: upo > ˜( , 1 ) = . upo1 for t T1, without a control, the trajectory u t u0 dimL S± 2 0136. The UPO u with period upo upo1 ( , 1 ) τ = . diverges from u t u0 and visualizes a local 1 1 5586 has the following Floquet multipliers: −10 chaotic attractor A. ρ1 = 4.7127, ρ2 = 1, ρ3 =−1.19 × 10 and corresponding Lyapunov exponents: { 1 log ρ }3 . τ1 i i=1 123 N. V. Kuznetsov et al.

1 dim A =dim S S u(t, uupo ) analytical value L glob L 0 − 0 S+ 2.4

2.35 2.075

2.3 2.07 z 2.25 2.065

2.2 2.06

2.15 50 60 70 80 90 100 uupo1 u˜(t, uupo1 ) 2.1 0 0 S0 2.05 dimL S± x y 2 020406080100

(a) (b)

( ( , ·), upo1 ) ˜( , upo1 ) Fig. 13 Evolution of FTLDs dimL u t u0 (red) and trajectory u t u0 (blue) integrated without stabilization, ( ˜( , ·), upo1 ) ∈ upo1 = dimL u t u0 (blue) computed on the time interval t respectively. Both trajectories start from the point u0 upo [ , ] upo1 ( ) = ( , 1 ) (−6.2262, −11.0027, 13.0515). (Color ﬁgure online) 0 100 along the UPO u t u t u0 (red) and the

Fig. 14 Evolution of 2.095 Lorenz system ( ( , ·), upo1 ) Finite-time Lyapunov dimension FTLDs dimL u t u0 (red) and 2.09 along UPO (with Pyragas stabilization, dde23) dim (u˜(t, ·), uupo1 ) (blue) along pseudo-trajectory (no stabilization, ode45) L 0 2.085 anlytical value via Floquet multipliers computed on the long time interval t ∈[0, 10000] 2.08 along the UPO 2.075 upo upo1 ( ) = ( , 1 ) u t u t u0 (red) and the trajectory 2.07 ... ˜( , upo1 ) u t u0 (blue) integrated 2.065 without stabilization, ... respectively. Both 2.06 trajectories start from the 2.055 upo1 = point u0 (−6.2262, −11.0027, 13.0515). 2.05 (Color ﬁgure online) 2.045 t 1 t ... 0m 20406080100s 9960 9980 10000

Thus, for the local Lyapunov dimension of this UPO, aging of these values across the attractor, but to tending upo we obtain: dimL u 1 = 2.0678 2.0679 = of these values to the finite-time local Lyapunov expo- ( ( , ·), upo1 ) dimL u 100 u0 . nents of S0. Also remark that system (1) has the analytical solution u(t) = (0, 0, z0 exp(−bt)) which tends to the equilibrium S0 = (0, 0, 0) from any initial point 5.3 Rigorous analytical computation for the global 3 (0, 0, z0) ∈ R . In general, the existence of such solu- attractor tions in the phase space complicates the procedure of visualization of a chaotic attractor (pseudo-attractor) by Using an effective analytical technique, proposed by one pseudo-trajectory with arbitrary initial data com- Leonov [40,56], it is possible to obtain [65,70]the puted for a sufficiently large time interval (see, e.g., exact formula of the Lyapunov dimension for the global [79,97]). In particular, the numerical computation of attractor Aglob of the Lorenz system (1): finite-time local Lyapunov exponents along this trajec- 2√(σ+b+1) dimL Aglob = 3 − (25) tory during for any time interval does not lead to aver- σ+1+ (σ−1)2+4σr 123 The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension for the case, when conditions (3) of the Theorem 1 are good agreement with the rigorous analysis of the time not satisfied. interval choices for reliable numerical computation of For the Lorenz system (1) with classical values of trajectories for the Lorenz system. The time interval parameters r = 28, σ = 10, b = 8/3, we have the for reliable computation with 16 significant digits and following relations: error 10−4 is estimated as [0, 36], with error 10−8 is estimated as [0, 26] (see [36,37]), and reliable compu- 2√(σ+b+1) dimL Aglob = dimL S0 = 3 − =2.4013 > σ +1+ (σ−1)2+4σr tation for a longer time interval, e.g., [0, 10000] in [75], > A ≥ upo1 = . > ( ˜( , ·), upo1 ) is a challenging task that requires a significant increase dimL dimL u 2 0678 dimL u 100 u0 in the precision of the floating-point representation and = 2.0621 > dimL S± = 2.0136. the use of supercomputers. Here, since the global Lorenz attractor contains a upo period-1 UPO: Aglob ⊃ u 1 , we have the following lower-bound estimate for the Lyapunov dimension: 6 Conclusion upo dimL Aglob ≥ 2.0678 = dimL u 1 . Similar experiment and results for the Rössler system [88] are pre- In this work, for the classical Lorenz system, the bound- sented in [48,50]. aries of practical global stability are estimated and the Concerning the time of integration, remark that difficulties of numerically studying the birth of self- while the time series obtained from a physical exper- excited and hidden attractors, caused by the loss of iment are assumed to be reliable on the whole con- global stability, are discussed. Difficulties in localiza- sidered time interval, the time series produced by the tion and numerical analysis of hidden transient chaotic integration of mathematical dynamical model can be sets in the Lorenz system are emphasized in the exper- reliable on a limited time interval only due to computa- iments on computation of the largest finite-time Lya- tional errors (caused by finite precision arithmetic and punov exponent and finite-time Lyapunov dimension numerical integration of ODE). Thus, in general, the along trajectories with different initial data and large closeness of the real trajectory u(t, u0) and the corre- time intervals. The classical self-excited Lorenz attrac- sponding pseudo-trajectory u˜(t, u0) calculated numer- tor is considered, and the applications of the Pyragas ically can be guaranteed on a limited short time interval time-delayed feedback control technique and Leonov only. analytical method are demonstrated for the Lyapunov In our experiment, if we continue computation over dimension estimation, as well as for the verification of a long time interval [0, 10000] of FTLD along the sta- the famous Eden’s conjecture on the maximum value of bilized UPO and the pseudo-trajectory obtained with- this characteristic that could be achieved on the attrac- out Pyragas stabilization, as a result, completely dif- tor. The problem of reliable numerical computation of ferent values will be obtained (Fig. 14). Evolution the finite-time Lyapunov dimension along the trajecto- ( ( , ·), upo1 ) of dimL u t u0 along the stabilized UPO will ries over large time intervals is discussed. upo tend to the analytical value dimL u 1 = 2.0678, computed via Floquet multipliers, while evolution of Acknowledgements Open access funding provided by Uni- ( ˜( , ·), upo1 ) versity of Jyväskylä (JYU). The authors declare that they have dimL u t u0 along the pseudo-trajectory will no conflict of interest. We dedicate this paper to the memory 6 converge to the value 2.0622 . These results are in of Gennady A. Leonov (1947–2018), with whom we began this work in 2018. This study was partially funded by the Russian 6 The following results on the dimension of the Lorenz attrac- Science Foundation (Project 19-41-02002). tor with parameters r = 28, σ = 10, b = 8/3 can be found in the literature. In [28, p. 193] and [10, p. 3529], the fractal Open Access This article is licensed under a Creative Com- (box counting, capacity) dimension is estimated as 2.06 ± 0.01. mons Attribution 4.0 International License, which permits use, For the correlation dimension, the following results are known: sharing, adaptation, distribution and reproduction in any medium 2.05 ± 0.01 in [28, p. 193] and [103, p. 456]; 2.06 ± 0.03 in [78, or format, as long as you give appropriate credit to the original p. 47]; 2.049 ± 0.096 in [100, p. 1874]; 2.05 in [24, p. 80]. For author(s) and the source, provide a link to the Creative Com- the Lyapunov dimension, the following values have been commons licence, and indicate if changes were made. The images or puted: 2.063 in [77, p. 92] and [80, p. 1957]; 2.05 in [19, p. 267]; other third party material in this article are included in the article’s 2.062 in [100, p. 1874], [97, p. 115] and [53,p.53];2.06215 [98, Creative Commons licence, unless indicated otherwise in a credit p. 033124-3] and [24, p. 83]. Also, let us mention estimates for line to the material. If material is not included in the article’s Cre- the global attractor: 2.401 ≤ dimL Aglob ≤ 2.409 [22, p. 170] ative Commons licence and your intended use is not permitted by and dimL Aglob ≈ 2.401... in [19, p. 267]. statutory regulation or exceeds the permitted use, you will need 123 N. V. Kuznetsov et al.

+ to obtain permission directly from the copyright holder. To view to system (23) with λ = λ j 1, a loss of stability bifur- a copy of this licence, visit http://creativecommons.org/licenses/ cation is observed and attractor A j vanishes. If, while by/4.0/. changing λ from λbegin to λend, there is no loss of stability bifurcation of the considered attractors, then a k hidden attractor for λ = λend (at the end of the proce- Appendix A: Hidden attractors localization via dure) is localized. numerical continuation method

One of the effective methods for numerical localiza- Appendix B: Finite-time and limit values of Lya- tion of hidden attractors in multidimensional dynam- punov exponents and Lyapunov dimension ical systems is based on the homotopy and numerical continuation method (NCM). The idea is to construct a Following [40,44], let us outline the concept of the sequence of similar systems such that for the first (start- finite-time Lyapunov dimension, which is convenient ing) system the initial point for numerical computa- for carrying out numerical experiments with finite time. tion of oscillating solution (starting oscillation) can be For a fixed t ≥ 0, let us consider the map u(t, ·) : obtained analytically. Thus, it is often possible to con- R3 → R3 defined by the shift operator along the solu- 3 sider the starting system with self-excited starting oscil- tions of system (1): u(t, u0), u0 ∈ R . Since system (1) lation; then, the transformation of this starting oscilla- possesses an absorbing set, the existence and unique- tion in the phase space is tracked numerically while ness of solutions of system (1)fort ∈[0, +∞) take passing from one system to another; the last system place and, therefore, the system generates a dynamical 3 corresponds to the system in which a hidden attractor system {u(t, ·)}t≥0,(R , |·|) . is searched. Consider linearization of system (1) along the solu- For studying the scenario of transition to chaos, we tion u(t, u0) and its 3 × 3 fundamental matrix of ( ) = ( ,λ) ˙ consider system (23) with f u f u , where solutions (t, u0): (t, u0) = Df(u(t, u0))(t, u0), λ ∈ ⊂ Rd is a vector of parameters, whose variation where (0, u0) = I is a unit 3×3 matrix. Denote in the parameter space determines the scenario. Let by σi (t, u0) = σi ((t, u0)), i = 1, 2, 3, the singular λ ∈ end define a point corresponding to the system, values of (t, u0) (i.e., the square roots of the eigen- ∗ where a hidden attractor is searched. Choose a point values of the symmetric matrix (t, u0) (t, u0) with 8 λbegin ∈ such that we can analytically or numeri- respect to their algebraic multiplicity) , ordered so that 3 cally localize a certain nontrivial (oscillating) attractor σ1(t, u0) ≥ σ2(t, u0) ≥ σ3(t, u0)>0 for any u0 ∈ R 1 A in system (23) with λ = λbegin (e.g., one can con- and t > 0. sider an initial self-excited attractor defined by a tra- Consider a set of finite-time Lyapunov exponents at 1( ) jectory u t numerically integrated on a sufficiently the point u0: ∈[, ] 1( ) large time interval t 0 T with initial point u 0 1 in the vicinity of an unstable equilibrium). Consider LEi (t, u0) = ln σi (t, u0), t > 0, i = 1, 2, 3. (26) t a path7 in the parameter space , i.e., a continuous Here, the set {LE (t, u )}3 is ordered by decreasing function γ :[0, 1]→,forwhichγ(0) = λbegin i 0 i=1 γ( ) = λ {λ j }k (i.e., LE1(t, u0) ≥ LE2(t, u0) ≥ LE3(t, u0) for all and 1 end, and a sequence of points j=1 1 k t > 0). The finite-time local Lyapunov dimension [40, on the path, where λ = λbegin, λ = λend, such + 44] can be defined via an analog of the Kaplan–Yorke that the distance between λ j and λ j 1 is sufficiently formula with respect to the set of ordered finite-time small. On each next step of the procedure, the initial Lyapunov exponents {LE (t, u )}3 : point for a trajectory to be integrated is chosen as the i 0 i=1 LE (t,u )+···+LE (, )(t,u ) last point of the trajectory integrated on the previous ( , ) = ( , ) + 1 0 j u0 0 , + dimL t u0 j t u0 | LE ( , )+ (t,u )| step: u j 1(0) = u j (T ). Following this procedure and j t u0 1 0 sequentially increasing j, two alternatives are possible: (27) A j m the point of are in the basin of attraction of attractor where j(t, u0) = max{m : = LEi (t, u0) ≥ 0}. j+1 j i 1 A , or while passing from system (23) with λ = λ Then, the finite-time Lyapunov dimension of dynamical

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