A lower-bound estimate of the Lyapunov dimension for the global attractor of the Lorenz system
N.V. Kuznetsov, T.N. Mokaev, R.N. Mokaev, O.A. Kuznetsova, E.V. Kudryashova
AA lower-bound estimate of the Lyapunov dimensionfor the global attractor of the Lorenz system
N. V. Kuznetsov,
1, 2, 3, ∗ T. N. Mokaev, R. N. Mokaev,
1, 2
O. A. Kuznetsova, and E. V. Kudryashova Faculty of Mathematics and Mechanics, St. Petersburg State University, Peterhof, St. Petersburg, Russia Faculty of Information Technology, University of Jyv¨askyl¨a, Jyv¨askyl¨a, Finland Institute for Problems in Mechanical Engineering RAS, Russia (Dated: October 29, 2019)In this short report, for the classical Lorenz attractor we demonstrate the applications of thePyragas time-delayed feedback control technique and Leonov analytical method for the Lyapunovdimension estimation and verification of the Eden’s conjecture. The problem of reliable numericalcomputation of the finite-time Lyapunov dimension along the trajectories over large time intervalsis discussed.
I. LORENZ ATTRACTOR AND PYRAGASSTABILIZATION OF EMBEDDED UNSTABLEPERIODIC ORBITS
Consider the classical Lorenz system [1] ˙ x = − σ ( x − y ) , ˙ y = rx − y − xz, ˙ z = − bz + xy, (1)with physically sound parameters σ, r >
0, and b ∈ [0 , r < S = (cid:0) , , (cid:1) , and for r > S turnsinto a saddle, while two new symmetric equilibria appear: S ± = (cid:0) ± (cid:112) b ( r − , ± (cid:112) b ( r − , r − (cid:1) , (2)which stability depends on the values of parameters.System (1) is dissipative in the sense of Levinson (seee.g. [2]), i.e. there exist a global bounded absorbing setcontaining global attractor A glob , and in some cases thisattractor exhibits chaotic behavior. For some values ofparameters, it is possible to observe a case of multista-bility, when the global attractor consists of several localattractors. To get a visualization of such attractors oneneeds to choose an initial point in the basin of attrac-tion of a particular attractor and observe how the tra-jectory, starting from this initial point, after a transientprocess visualizes the attractor: an attractor is called a self-excited attractor if its basin of attraction intersectswith any open neighborhood of an equilibrium, other-wise, it is called a hidden attractor [2–5]. It was discov-ered numerically by E. Lorenz that in the phase spaceof system (1) with parameters r = 28, σ = 10, b = 8 / chaotic attractor A , which is self-excitedwith respect to all equilibria S , S ± .The ”skeleton” of a chaotic attractor comprises embed-ded unstable periodic orbits (UPOs) (see e.g. [6–8]), andone of the effective methods among others for the com-putation of UPOs is the delay feedback control (DFC) ∗ Corresponding author: [email protected] approach, suggested by K. Pyragas [9] (see also discus-sions in [10–12]). This approach allows Pyragas and hisprogeny to stabilize and study UPOs in various chaoticdynamical systems. Nevertheless, some general analyt-ical results have been obtained [13], showing that DFChas a certain limitation, called the odd number limitation(ONL), which is connected with an odd number of realFloquet multipliers larger than unity. In order to over-come ONL, later Pyragas suggested a modification of theclassical DFC technique, which was called the unstabledelayed feedback control (UDFC) [14].Rewrite system (1) in a general form˙ u = f ( u ) . (3)Let u upo ( t, u upo ) be its UPO with period τ > u upo ( t − τ, u upo ) = u upo ( t, u upo ), and initial condition u upo = u upo (0 , u upo ). To compute the UPO and overcome ONL,we add the UDFC in the following form:˙ u ( t ) = f ( u ( t )) + KB (cid:2) F N ( t ) + w ( t ) (cid:3) , ˙ w ( t ) = λ c w ( t ) + ( λ c − λ ∞ c ) F N ( t ) ,F N ( t ) = C ∗ u ( t ) − (1 − R ) N (cid:88) k =1 R k − C ∗ u ( t − kT ) , (4)where 0 ≤ R < N =1 , , . . . , ∞ defines the number of previous states involvedin delayed feedback function F N ( t ), λ c >
0, and λ ∞ c < B, C are vectors and
K > u upo and T = τ we have F N ( t ) ≡ , w ( t ) ≡ , and, thus, the solution of system (4) coincides with theperiodic solution of initial system (3).For the Lorenz system (1) with parameters r = 28, σ = 10, b = 8 / B ∗ = (0 , , C ∗ =(0 , , R = 0 . N = 100, K = 3 . λ c = 0 . λ ∞ c = −
2, one can stabilize a period-1 UPO u upo ( t, u ) withperiod τ = 1 . u = (1 , , w = 0 (see Fig. 1). Results of this experiment could berepeated using various other numerical approaches (see a r X i v : . [ n li n . C D ] O c t x yz S S + S − u upo u ( t, u upo ) (a) x yz u upo ˜ u ( t, u upo ) u ( t, u upo ) S S + S − (b) Figure 1: Period-1 UPO u upo ( t ) (red, period τ = 1 . u ( t, u upo ) (blue, t ∈ [0 , r = 28, σ = 10, b = 8 / u upo ≈ ( − . , − . , . u upo ( t ) = u ( t, u upo ) we numerically compute the trajectory ofsystem (4) without the stabilization (i.e. with K = 0) onthe time interval [0 , T = 100] (see Fig. 1b). We denoteit by ˜ u ( t, u upo ) to distinguish this pseudo-trajectoryfrom the periodic orbit u ( t, u upo ). One can see that onthe initial small time interval [0 , T ≈ u ( t, u upo ) tracesapproximately the ”true” periodic orbit u ( t, u upo ). Butfor t > T , without a control, the trajectory ˜ u ( t, u upo )diverge from u upo ( t, u upo ) and visualize a local chaoticattractor A .Remark that in numerical computation of trajectoryover a finite-time interval it is also difficult to distinguisha sustained chaos from a transient chaos (a transientchaotic set in the phase space, which can persist for along time) [20]. This challenging task is related to anopen problem about the existence of a hidden chaoticattractor in the Lorenz system (1) (see e.g. discussionsin [2, 21–23]). II. LYAPUNOV DIMENSION ESTIMATIONAND EDEN CONJECTURE
Following [24, 25], let us outline the concept of the finite-time Lyapunov dimension , which is convenient forcarrying out numerical experiments with finite time.For a fixed t ≥ u ( t, · ) : R → R defined by the shift operator along the solu-tions of system (1): u ( t, u ), u ∈ R . Since system (1) possesses an absorbing set, the existence and unique-ness of solutions of system (1) for t ∈ [0 , + ∞ ) take placeand, therefore, the system generates a dynamical system (cid:0) { u ( t, · ) } t ≥ , ( R , | · | ) (cid:1) .Consider linearization of system (1) along the solu-tion u ( t, u ) and its 3 × t, u ): ˙Φ( t, u ) = Df (( u ( t, u ))Φ( t, u ), whereΦ(0 , u ) = I is a unit 3 × σ i ( t, u ) = σ i (Φ( t, u )), i = 1 , ,
3, the singular val-ues of Φ( t, u ) (i.e. the square roots of the eigenval-ues of the symmetric matrix Φ( t, u ) ∗ Φ( t, u ) with re-spect to their algebraic multiplicity) , ordered so that σ ( t, u ) ≥ σ ( t, u ) ≥ σ ( t, u ) > u ∈ R and t > finite-time Lyapunov exponents atthe point u :LE i ( t, u ) = 1 t ln σ i ( t, u ) , t > , i = 1 , , . (5)Here, the set { LE i ( t, u ) } i =1 is ordered by decreasing (i.e.LE ( t, u ) ≥ LE ( t, u ) ≥ LE ( t, u ) for all t > finite-time local Lyapunov dimension [24, 25] can be de-fined via an analog of the Kaplan-Yorke formula withrespect to the set of ordered finite-time Lyapunov expo-nents { LE i ( t, u ) } i =1 :dim L ( t, u ) = j ( t, u ) + LE ( t,u )+ ·· + LE j ( ,u ( t,u ) | LE j ( t,u ( t,u ) | , (6)where j ( t, u ) = max { m : (cid:80) mi =1 LE i ( t, u ) ≥ } . Thenthe finite-time Lyapunov dimension of dynamical systemwith respect to a set A is defined as:dim L ( t, A ) = sup u ∈A dim L ( t, u ) . (7) Symbol ∗ denotes the transposition of matrix. The
Douady–Oesterl´e theorem [26] implies that for anyfixed t > A , defined by (7), is an upper estimateof the Hausdorff dimension: dim H A ≤ dim L ( t, A ). Thebest estimation is called the Lyapunov dimension [24]dim L A = inf t> sup u ∈A dim L ( t, u ) == lim inf t → + ∞ sup u ∈A dim L ( t, u ) . We use the adaptive algorithm for the computation ofthe finite-time Lyapunov dimension and exponents fortrajectories on the local attractor A [25]. In order to dis-tinguish the corresponding values for the stabilized UPO u ( t, u upo ) with a period τ = 1 . u ( t, u upo ) computed without Pyragas stabi-lization in our experiment we use the following notationsfor finite-time Lyapunov dimensions: dim L ( u ( t, · ) , u upo )and dim L (˜ u ( t, · ) , u upo ), respectively.The comparison of the obtained values of finite-timeLyapunov dimensions computed along the stabilizedUPO and the trajectory without stabilization gives usthe following results. On the initial small part of the timeinterval, one can indicate the coincidence of these valueswith a sufficiently high accuracy. For the UPO and forthe unstabilized trajectory the finite-time local Lyapunovdimensions dim L ( u ( t, · ) , u upo ) and dim L (˜ u ( t, · ) , u upo )coincide up to the 4th decimal place inclusive on the in-terval [0 , t m ≈ τ ]. After t > t m the difference in val-ues becomes significant and the corresponding graphicsdiverge in such a way that the part of the graph corre-sponding to the unstabilized trajectory is lower than thepart of the graph corresponding to the UPO (see Fig. 2b,Fig. 3).The Jacobi matrix at the saddle-foci equilibria S ± hassimple eigenvalues, which give the following: dim L S ± =2 . u upo with period τ = 1 . ρ = 4 . ρ = 1, ρ = − . · − and corresponding Lyapunov exponents: { τ log ρ i } i =1 . Thus, for the local Lyapunov dimensionof this UPO we obtain: dim L u upo = 2 . (cid:47) . L ( u (100 , · ) , u upo ).Using an effective analytical technique, proposed byLeonov [24, 27], which is based on a combination ofthe Douady-Oesterl´e approach and the direct Lyapunovmethod, it is possible to obtain [28, 29] the exact formulaof the Lyapunov dimension for the global attractor A glob of the Lorenz system (1):dim L A glob = 3 − σ + b +1) σ +1+ √ ( σ − +4 σr (8)for the case, when rσ > ( σ + b )( b + 1). III. CONCLUSION
In this note, for the Lorenz system (1) with classicalvalues of parameters r = 28, σ = 10, b = 8 / L A glob = dim L S = 3 − σ + b +1) σ +1+ √ ( σ − +4 σr = 2 . >> dim L A ≥ dim L u upo = 2 . > dim L (˜ u (100 , · ) , u upo )= 2 . > dim L S ± = 2 . . Here, since the global Lorenz attractor contains aperiod-1 UPO: A glob ⊃ u upo , we have the follow-ing lower-bound estimate for the Lyapunov dimension:dim L A glob ≥ . L u upo . Similar experimentand results for the R¨ossler system [31] are presented in[32, 33].Concerning the time of integration, remark that whilethe time series obtained from a physical experiment areassumed to be reliable on the whole considered time inter-val, the time series produced by the integration of math-ematical dynamical model can be reliable on a limitedtime interval only due to computational errors (causedby finite precision arithmetic and numerical integrationof ODE). Thus, in general, the closeness of the real tra-jectory u ( t, u ) and the corresponding pseudo-trajectory˜ u ( t, u ) calculated numerically can be guaranteed on alimited short time interval only.In our experiment, if we continue computation overa long time interval [0 , L ( u ( t, · ) , u upo ) along the stabilized UPOwill tend to the analytical value dim L u upo = 2 . L (˜ u ( t, · ) , u upo ) along the pseudo-trajectory will con-verge to the value 2 . . These results are in goodagreement with the rigorous analysis of the time inter-val choices for reliable numerical computation of trajec-tories for the Lorenz system: the time interval for re-liable computation with 16 significant digits and error10 − is estimated as [0 , − is estimatedas [0 ,
26] (see [47, 48]), and reliable computation for alonger time interval, e.g. [0 , The following results on the dimension of the Lorenz attrac-tor with parameters r = 28, σ = 10, b = 8 / . ± . .For the correlation dimension the following results are known: . ± . in [34, p. 193] and [36, p. 456]; . ± . in [37,p. 47]; . ± . in [38, p. 1874]; . in [39, p. 80]. Forthe Lyapunov dimension the following values have been com-puted: . in [40, p. 92] and [41, p. 1957]; . in [42, p. 267]; . in [38, p. 1874], [43, p. 115] and [44, p. 53]; . [45,p. 033124-3] and [39, p. 83]. Also, let us mention estimates forthe global attractor: 2 . ≤ dim L A glob ≤ .
409 [46, p. 170]and dim L A glob ≈ . ... in [42, p. 267]. x yz u upo ˜ u ( t, u upo ) u ( t, u upo ) S S + S − (a)
50 60 70 80 90 1002.062.0652.072.075 dim L S ± analytical value dim L A glob = dim L S (b) Figure 2: Evolution of FTLDs dim L ( u ( t, · ) , u upo ) (red) and dim L (˜ u ( t, · ) , u upo ) (blue) computed on the timeinterval t ∈ [0 , u upo ( t ) = u ( t, u upo ) (red) and the trajectory ˜ u ( t, u upo ) (blue) integratedwithout stabilization, respectively. Both trajectories start from the point u upo = ( − . , − . , . t t s . . .. . .. . . Lorenz systemFinite-time Lyapunov dimension along UPO (with Pyragas stabilization, dde23)along pseudo-trajectory (no stabilization, ode45)anlytical value via Floquet multipliers
Figure 3: Evolution of FTLDs dim L ( u ( t, · ) , u upo ) (red) and dim L (˜ u ( t, · ) , u upo ) (blue) computed on the long timeinterval t ∈ [0 , u upo ( t ) = u ( t, u upo ) (red) and the trajectory ˜ u ( t, u upo ) (blue) integratedwithout stabilization, respectively. Both trajectories start from the point u upo = ( − . , − . , . ACKNOWLEDGEMENT
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