A note on finite-time Lyapunov dimension of the Rossler attractor
AA note on finite-time Lyapunov dimension of the Rossler attractor
N. V. Kuznetsov
1, 2, ∗ and T. N. Mokaev Faculty of Mathematics and Mechanics, St. Petersburg State University, Peterhof, St. Petersburg, Russia Department of Mathematical Information Technology, University of Jyv¨askyl¨a, Jyv¨askyl¨a, Finland (Dated: July 3, 2018)For the R¨ossler system we verify Eden’s conjecture on the maximum of local Lyapunov dimen-sion. We compute numerically finite-time local Lyapunov dimensions on the R¨ossler attractor andembedded unstable periodic orbits. The UPO computation is done by Pyragas time-delay feedbackcontrol technique.
I. R ¨OSSLER ATTRACTOR AND PYRAGASSTABILIZATION OF EMBEDDED UNSTABLEPERIODIC ORBITS
Consider the following R¨ossler system [1]˙ x = − y − z, ˙ y = x + ay, ˙ z = b − cz + xz, (1)with arbitrary real parameters a, b, c ∈ R . If c ≥ ab ,then system (1) has the following equilibria: O ± = ( ap ± , − p ± , p ± ) , where p ± = c ±√ c − ab a . (2)For some values of parameters system (1) exhibitschaotic behavior. To get a visualization of chaotic at-tractor one needs to choose an initial point in the basinof attraction of the 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 R¨ossler that in the phase space ofsystem (1) with parameters a = 0 . b = 0 . c = 5 . chaotic attractor of spiral shape, which isself-excited with respect to both equilibria O ± .One of the building blocks of chaotic attractor areembedded unstable periodic orbits (UPOs). An effec-tive method for the computation of UPOs is the time-delay feedback control (TDFC) approach, suggested by K.Pyragas [6] (see also discussions in [7–10]). Let u upo ( t )be an UPO with period τ > u upo ( t − τ ) = u upo ( t ),satisfying a differential equation˙ u = f ( u ) . (3)To compute the UPO, we add the TDFC:˙ u = f ( u ) + kBC ∗ (cid:0) u ( t − T ) − u ( t ) (cid:1) , (4)where B, C are vectors and k is a real gain. If T = τ ,then kBC ∗ (cid:0) u ( t − T ) − u ( t ) (cid:1) = 0 along the UPO, and ∗ Corresponding author: [email protected] periodic solution of system (4) coincides with periodicsolution of system (3).For the R¨ossler system (1) we solved numerically sys-tem (4) and stabilized a period-1 UPO u upo ( t, u ) withperiod τ = 5 . u upo0 , chosen on the UPO u upo = (cid:8) u upo ( t ), t ∈ [0 , τ ] (cid:9) , we numerically compute thetrajectory ˜ u ( t, u upo0 ) of system (4) without the stabiliza-tion (i.e. with k = 0) on sufficiently large time interval[0 , T = 500] (see Fig. 1b). One can see that on the ini-tial small time interval [0 , T ≈ u ( t, u upo0 ) traces approx-imately the ”true” periodic orbit u upo ( t, u upo0 ). But for t > T without control the trajectory ˜ u ( t, u upo0 ) divergefrom u upo and wind on the attractor A . II. FINITE-TIME LYAPUNOV DIMENSIONAND EDEN CONJECTURE
For an attractor, an interesting question [11, p.98](known as Eden conjecture) is whether the supremumof the local Lyapunov dimensions is achieved on a sta-tionary point or an unstable periodic orbit embedded inthe strange attractor. In general, a conjecture on theLyapunov dimension of self-excited attractor [12, 13] isthat for a typical system the Lyapunov dimension of aself-excited attractor does not exceed the Lyapunov di-mension of one of unstable equilibria, the unstable man-ifold of which intersects with the basin of attraction andvisualize the attractor.Below we follow the concept of the finite-time Lya-punov dimension [12, 13], which is convenient for car-rying out numerical experiments with finite time. The finite-time local Lyapunov dimension [12, 13] can be de-fined via an analog of the
Kaplan-Yorke formula withrespect to the set of finite-time Lyapunov exponents:dim L ( t, u ) = d KYL ( { LE i ( t, u ) } i =1 ) = j ( t, u ) + LE ( t,u )+ ·· + LE j ( t,u ) ( t,u ) | LE j ( t,u )+1 ( t,u ) | , (5)where j ( t, u ) = max { m : (cid:80) mi =1 LE i ( t, u ) ≥ } . Thenthe finite-time Lyapunov dimension (of dynamical sys-tem generated by (3) on compact invariant set A ) is de- a r X i v : . [ n li n . C D ] J un y xz O − u upo u upo ( t ) (a) x
10 05 5 y
010 -5-5 z
15 -10 -10-152025 u upo0 u upo ( t ) (b) Figure 1: Period-1 (red, period τ = 5 . a = 0 . b = 0 . c = 5 .
7, stabilizedusing TDFC method.fined as dim L ( t, A ) = sup u ∈A dim L ( t, u ) . (6)The Douady–Oesterl´e theorem [14] implies that for anyfixed t > H A ≤ dim L ( t, A ). The best estimation is called the Lyapunov dimension [12]dim L A = inf t> sup u ∈ K dim L ( t, u ) = lim inf t → + ∞ sup u ∈ K dim L ( t, u ) . For the R¨ossler attractor the Lyapunov dimension wasestimated as 2 .
014 [15], 2 .
01 [16], 2 . . A , i.e. max u ∈ C grid dim L ( t, u ); finite-time Lya-punov dimensions dim L (500 , · ) for the stabilized UPOwith periods τ = 5 . ( t, u upo0 )and dim L ( t, u upo0 ) computed along the stabilized UPOand the trajectory without stabilization gives us the fol-lowing results. On the initial part of the time interval,one can indicate the coincidence of these values with asufficiently high accuracy. For the period-1 UPO and forthe unstabilized trajectory the largest Lyapunov expo-nents LE ( t, u upo0 ) coincide up to the 5th decimal placeinclusive on the interval [0 , . t > . O ± has simple eigenvalues and, thus, wehave dim L O + = d KYL ( { Re λ i ( O + ) } i =1 ) = 3 , dim L O − = d KYL ( { Re λ i ( O − ) } i =1 ) = 2 . u upo with period τ = 5 . ρ = − . ρ =1, ρ = − . · − . Thus, for the localLyapunov dimension of the UPO u upo ( t ) we obtaindim L u upo = d KYL ( { τ log ρ j } j =1 ) = 2 . (cid:47) . L (500 , u upo ). III. CONCLUSION
In this note we have confirmed the Eden conjecturefor the R¨ossler system (1) and obtained the followingrelations between the Lyapunov dimensions:3 = dim L O + > . L O − > . L u upo > . u ∈ C grid dim L (500 , u ) ≥ dim L A ≥ dim H A . 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. However, for two dif-ferent long-time pseudo-trajectories ˜ u ( t, u ) and ˜ u ( t, u )visualizing the same attractor, the corresponding finite-time LEs can be, within the considered error, similardue to averaging over time and similar sets of points { ˜ u ( t, u ) } t ≥ and { ˜ u ( t, u ) } t ≥ . At the same time, thecorresponding real trajectories u ( t, u , ) may have differ-ent LEs, e.g. u may correspond to an unstable periodictrajectory u ( t, u ) which is embedded in the attractorand does not allow one to visualize it. x
10 05 5 y
010 -5-5 z
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