- Email: [email protected]
Experimental identification of chaotic fibers
Chaos, Solitons and Fractals 39 (2009) 9–16 www.elsevier.com/locate/chaos
Experimental identification of chaotic fibers D.M. Maranha˜o a, J.C. Sartorelli a
a,*
, M.S. Baptista
b
Instituto de Fı´sica, Universidade de Sa˜o Paulo, Caixa Postal 66318, 05315-970 Sa˜o Paulo, SP, Brazil b Universita¨t Potsdam, Institut fu¨r Physik Am Neuen Palais 10, D-14469 Potsdam, Germany Accepted 2 January 2007
Communicated by Prof. R. Seydel
Abstract In a 2D parameter space, by using nine experimental time series of a Chua’s circuit, we characterized three codimension-1 chaotic fibers parallel to a period-3 window. To show the local preservation of the properties of the chaotic attractors in each fiber, we applied the closed return technique and two distinct topological methods. With the first topological method we calculated the linking numbers in the sets of unstable periodic orbits, and with the second one we obtained the symbolic planes and the topological entropies by applying symbolic dynamic analysis. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The interplay between chaotic and periodic behaviors has been studied in the parameter space of some dynamical systems. Barreto et al. [1] and Baptista et al. [2] showed that in the 2D parameter space of a fragile chaotic system [3], one should expect to find periodic windows situated along structures called spines. Once high-period stable periodic orbits, located at the border of these periodic windows, are similar, one can expect to find similar chaotic attractors at the chaotic side of this border. Alike the periodic regions where one can have a path in the 2D parameter space to change two parameters to follow the same high-period orbit, we should expect to find selective paths to vary two parameters in such way that we would produce always chaotic attractors with similar characteristics in each path. In fact, Monti et al. [4] showed numerically the existence of such paths in a parameter space of the Lorenz system. In addition, Baptista et al. [5] suggested that the two parameters variation should be a linear function in the Chua’s circuit [6]. This line was called a chaotic fiber, which arguably would have the same codimension [7] of the neighboring periodic windows. The way one has to change two parameters, preserving the attractors characteristics, is provided by the windows conjecture [1], which states that the periodic windows have codimension k in the parameter space, where k is the number of positive Lyapunov exponents of the neighboring chaos in the vicinity of the periodic windows. So, the chaotic regions that preserve their characteristics by a two parameter variation appear aligned to the periodic windows. If the periodic windows have codimension k, it is reasonable to assume that these especial chaotic regions also have codimension k. *
Corresponding author. E-mail address: [email protected] (J.C. Sartorelli).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.131
10
D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
The Chua’s circuit presents the three main characteristics to observe fibers: (i) minimal amount of internal noise, such that periodic windows can be clearly recognized; (ii) a minimal number of positive Lyapunov exponents, so the chaotic fibers can be found by varying a minimal number of accessible parameters; (iii) a chaotic attractor with a simple topology and a small embedded dimension, such that the UPOs can be easily identified by using symbolic dynamic on reconstructed attractors. The chaotic attractors of the Chua’s circuit have just one positive Lyapunov exponent, so we should expect to find codimension-1 periodic structures in the 2D parameter space, which means that the periodic windows have a line structure. If one wants to change the control parameters preserving the topological and metric properties of the chaotic attractors, we should vary simultaneously two control parameters. By the Lyapunov exponents calculation the fibers could not be discerned, but we are providing strong experimental evidences of the existence of them by applying the closed returns technique [8,9], and by two independent topological methods. With one, we analyzed the structure of the unstable periodic orbits (UPOs) and obtained their linking numbers [9–12]. With the other one, by applying a symbolic dynamics, we constructed the symbolic plane and estimated the topological entropy of the attractors [13–15].
2. Experimental setup In Fig. 1, it is shown the diagram of the Chua’s circuit. The RNL is the non-linear resistance, L is the inductor, C1 and C2 are two capacitors. The dynamical variables are iL, the current across the inductor, V C1 and V C2 , the voltages across
VC1
VC2 L
iL R10 ΔR1
ΔR2 C2
C1
RNL
R 20
Fig. 1. Chua’s circuit. Component values are: R10 1:4 kX; R20 37 X; C 1 4:7 nF; C 2 56 nF; L 9:2 mH.
a
b
c
Fig. 2. Bifurcation diagrams by varying DR2 : (a) Z series with DR1 12:5 X, (b) Y series with DR1 7:5 X and (c) X series with DR1 3:0 X. The boxes represent parameter regions where we analyzed the time series.
D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
11
the capacitors C1 and C2 , respectively. We have two control parameters given by two resistances R1 ¼ R01 DR1 and R2 ¼ R02 DR2 , where R01 and R02 are fixed, and DR1 and DR2 are precision potentiometers driven by step motors. These two potentiometers are allowed to vary within the intervals DR1 2 ½0:0; 15:0X and DR2 2 ½0:0; 5:5X, with a step of Dpr ¼ 50 mX. The time series, 100,000 points long, were obtained recording the V C1 voltage with a 12 bits ADC at 400 ksamples/s. In Fig. 2 are shown the bifurcation diagrams V C1 vs. R2 obtained for three values of R1, and a large period three window can be seen in the three cases. The data sets X, Y and Z were named according to the value of DR1 = 3.0, 7.5 and 12.5 X, respectively. We selected three times series of each set X,Y and Z in each DR2 interval defined by the boxes drawn in Fig. 2. We named the time series as XJ, YJ and ZJ, where J = 1, 6, and 11 measured from the border of the large periodthree window (J = 0),as shown by the parameter space ðDR1 DR2 Þ [5] schematically drawn in Fig. 3. Once the period-three window border appears linear in parameter space [5], we expect that the parameter values chosen belongs to three different chaotic linear fibers. Thus, the time series Z11, Y11 and X11 belong to the fiber named (I), Z6, Y6 and X6 to the fiber named (II), and Z1, Y1 and X1 to the fiber named (III). The chaotic attractors were reconstructed in an embedding dimension de = 3 with a time delay s = 45.0 ls. Two-dimensional projections are shown in Fig. 4.
Fig. 3. Illustration of the parameter space showing the positions of the times series X, Y, and Z as well as the positions of the fibers.
Fig. 4. Two-dimensional projection of reconstructed attractors in the fibers (I), (II) and (III) with time delay s ¼ 45:0 ls. In Z11 it is also shown a Poincare´ section V C1 ðtÞ ¼ 2:25 V.
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3. Results and discussion 3.1. Metric characterization The Lyapunov exponents were obtained using the TISEAN package [16]. As shown in Table 1, the Lyapunov exponents and dimensions values for all the nine attractors, are not suitable to state about the existence of different fibers. Since the Lyapunov dimension for all the attractors is smaller than 3 and there is only one positive Lyapunov exponent the topological methods are applicable to find their dynamical characteristics. 3.2. Topological approach 3.2.1. Periodic orbits (UPOs) and linking numbers The topological analysis employed is based on the fact that the measure and average quantities of a chaotic attractor can be obtained from the unstable periodic orbits (UPO) embedded in it [17,18].A typical chaotic trajectory evolving in an attractor can be thought as being a random walk along the infinite number of UPOs. The trajectory approaches an unstable orbit along its stable direction, and it stays in the neighborhood trajectory during an interval of time and after it departs from the UPO along the local unstable direction. In this time interval, a short segment of the trajectory shadows the periodic orbit embedded within the chaotic attractor, allowing to extract approximations to the periodic orbit and obtaining approximations to a finite set of periodic orbits with not too long period, what is appropriate for the finite experimental data. We extracted the periodic orbits by searching in the time series V C1 ðtÞ [8,9] the time intervals ½t; t þ tp for which the difference jV C1 ðtÞ V C1 ðt þ tp Þj remains smaller than a given e (0.05% of the maximum value of V C1 ), and after we also checked the UPOs in the 3D reconstructed attractors. By scanning the data series for the existence of UPOs for many different time intervals tp we constructed the histograms shown in Fig. 5. The peaks position corresponds to the period of an UPO, and all the histograms display the same cycle time T = 86 Dt ðDt ¼ 2:5 lsÞ, which is approximately the characteristic mean period of our Chua’s circuit. The relevant information in these histograms is the height of the peaks, indicating the contribution of the corresponding UPO to the measure of the attractor. For each column the histograms present the same characteristic pattern, showing that we have three linear fibers as we claimed before. Another way of revealing special characteristics of the existing UPOs is by symbolic dynamics techniques. To construct the symbolic sequence of a trajectory or periodic orbit, we first construct a first return map of the reconstructed ðnÞ attractors, by using the Poincare´ section shown in Fig. 4 for the data series Z11. The new variable is represented by V C1 , where n ¼ 1; 2; . . . representing the nth crossing of the reconstructed trajectory with this section. ðnþ1Þ ðnÞ The first return mappings V C1 vs. V C1 can be seen in Fig. 6. They are approximately unimodal maps of an interval onto itself, and the position of the maximum is the partition point as shown in Fig. 6a. The partition point divides the map into two regions: one with a positive slope, labeled 0, and other with a negative slope, labeled 1. As the trajectory or periodic orbit visits these regions, we write down the corresponding symbol to obtain the symbolic sequence S ¼ 010 . . .. An UPO extracted by the closed return method, has period p if its symbolic sequence contains p symbols, represented by the sequence s0 s1 . . . sp1 , and is regarded to be the name of the UPO, for example a period-2 orbit is codified as (10).
Table 1 Estimated Lyapunov exponents of time series in fibers (I), (II) and (III) Fiber
Time series
k1
k2
k3
d KY
(I)
X 11 Y 11 Z 11
0.01257(7) 0.01298(5) 0.01234(5)
0.00519(9) 0.00496(8) 0.00488(10)
0.0831(2) 0.0864(2) 0.0861(2)
2.086 2.095 2.089
(II)
X6 Y6 Z6
0.01342(2) 0.01367(4) 0.01305(9)
0.00503(5) 0.00440(5) 0.00541(3)
0.0870(2) 0.0869(2) 0.0901(2)
2.095 2.107 2.087
(III)
X1 Y1 Z1
0.01362(8) 0.01371(8) 0.01416(2)
0.00463(7) 0.00504(9) 0.00444(2)
0.0896(2) 0.0897(4) 0.0895(6)
2.101 2.098 2.103
The dimension d L is the Lyapunov dimension estimated as d L ¼ 2 þ jk1 =k3 j (Kaplan–Yorke conjecture).
D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
a
b
c
d
e
f
g
h
i
13
Fig. 5. Closed returns histograms for the V C1 ðtÞ time series. The corresponding chaotic fiber of each data set is indicated in the top. Observe the similarity of the histograms in each fiber.
a
b
c
d
e
f
g
h
i
Fig. 6. Poincare´ section V C1 ¼ 2:25 V in the reconstructed attractors. All the maps closely resemble the unimodal maps, that present only one point of maximum, which is used as the partition point resulting in two regions labeled ‘‘0’’ (the left) and ‘‘ 1’’ (the right), as shown in (a).
The invariant linking numbers give the topological properties of the orbits assigning them a signature due to these properties [9,12]. These numbers give an idea of how a periodic orbit in the 3D phase space crosses itself and is called the self-linking number; or how it crosses another orbit, is called the linking number, such that one can determine at each crossing which segment of the orbit passes over the other. The self-linking (linking) number is determined by the sum (half sum) of oriented crossings of the orbit with itself (the orbit pair) [9–11]. The Fig. 7 shows the UPOs spectra of the lowest-period orbits.
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D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
orbits
A
1T
1
0
2T
10
1
1
3T
101
1
2
2
4T
1011
2
3
4
5 8
5Ta
10110
2
4
5
8
5Tb
10111
2
4
5
8
10
8
6Ta
101110
8
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9
14
6Tb
101111
8
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12
10
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17
12
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17
11
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17
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18
7Ta 7Tb
B
7Tc 7Td 8Ta 8Tb 8Tc
C
1T 2T 3T 4T 5Ta 5Tb 6Ta 6Tb 7Ta 7Tb 7Tc 7Td 8Ta 8Tb 8Tc 8Td 8Te
8Td 8Te
1011011 1011111 1011010 1011110 10111110 10111010 10111111 10110111 10110110
3 3 2 3 3 5 3 3 4 3 3
4 5 6 6 5
6 6 7 7 7
5
7
10
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6
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20
Fig. 7. Spectra of UPOs extracted from reconstructed attractors obtained from the time series at parameters along three fibers. We show their symbolic sequence, self-linking (diagonal) and linking numbers. In the fiber (III) all the 17 orbits shown in the table are allowed; along the fiber (II) the orbits in box A and C are forbidden and along the fiber (I) the orbits in boxes A, B and C are forbidden. The dots represent measurements whose value could not be determined.
In the first column in Fig. 7, we have the labels of 17 UPOs, whose symbolic sequences are shown in the second column, and the self-linking numbers in the other columns. For the attractors in fiber (III), we find all the 17 periodic orbits shown in Fig. 7. In fiber (II) three of these orbits were not found, i. e. they were pruned by the dynamics, the ones in boxes A and C. In fiber (I) five orbits were also pruned, the ones in boxes A, B and C. Therefore, we can conclude that the topological properties of UPOs are the same in each fiber. 3.2.2. Symbolic planes and topological entropy The fibers were also characterized by the construction of symbolic planes, which is a way to summarize the symbolic sequences in a two-dimensional map, and has been employed to compare similar attractors of different systems [13–15]. Each point in the symbolic plane (a, b) is obtained in an unique way from a symbolic sequence: . . . sm . . . s2 s1 :s0 s1 s2 . . . sm . . .
si 2 0; 1;
ð1Þ
where s0 s1 s2 . . . sm . . . is the forward symbolic sequence (a), and . . . sm . . . s2 s1 is the backward one (b). The decimal real coordinates (a, b) are given by m X ai ð2Þ a ¼ 0:a1 a2 a3 . . . ¼ i; i¼1 2 ai ¼
i1 X j¼0
and
sj ðmod 2Þ;
ð3Þ
D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
b ¼ 0:b1 b2 b3 . . . ¼ bi ¼
i X
m X bi i; i¼1 2
15
ð4Þ
sj ðmod 2Þ:
ð5Þ
j¼1
In Fig. 8, we can see the symbolic planes of the nine attractors. By comparing the patterns in each fiber in Fig. 8, we can see the same symbolic planes. From fiber I to fiber III the number of allowed sequences (dots) increase and in the fiber III the attractors present the most allowed symbolic sequences, which is consistent with the increasing of the number of UPOs. Therefore, the symbolic planes can also be thought as the fingerprint of the fibers. With the symbolic sequences of the nine reconstructed attractors, we estimated the topological entropy
a
b
c
d
e
f
g
h
i
Fig. 8. Symbolic planes in fiber (I) (on the left), in fiber (II) (on the center), and in fiber (III) (on the right). Observe the similarity of the symbolic planes in each fiber.
Fig. 9. Topological entropy of the experimental reconstructed attractors in fibers I, II and III, estimated from their symbolic sequences as a function of the length p of the sequences.
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D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
hT ðSÞ ¼ lim
p!1
ln N p ðSÞ ; p
ð6Þ
where N p ðSÞ is the number of sequences S ¼ s0 s1 . . . sp1 of length p, whose results are shown in Fig. 9. We can see that for every fiber the topological entropies follow the same curve, also indicating the preservation of the topological information.
4. Conclusion We presented experimental evidences about the existence of linear chaotic fibers parallel to a period-three window in the ðDR1 ; DR2 Þ parameter space of a Chua’s circuit. These fibers are parameter lines where the topology of the chaotic attractors are preserved. By topology, we mean the closed return technique, a set of short period unstable period orbits, the orbits characteristics as their linking numbers, the symbolic space and the topological entropy. These involved topological methods were employed because metric methods such as the Lyapunov exponents did not produce convincing evidences in order to state about the existence of such fibers.
Acknowledgements We thank Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo and CNPq for the financial support. M.S.B. also thanks the Alexander von Humboldt Foundation.
References [1] Barreto E, Hunt BR, Grebogi C, Yorke JA. Phys Rev Lett 1997;78:4561–4. [2] Baptista MS, Barreto E, Grebogi C. Int J Bifurcat Chaos 2003;13(9):2681–8. [3] A fragile chaotic system is defined to be that one whose chaotic behavior can be replace by a periodic one by an arbitrary parameter alteration. [4] Monti M, Pardo WB, Walkenstein JA, Rosa E, Grebogi C. Int J Bifurcat Chaos 1999;9:1459–63. [5] Baptista MS, Reyes MB, Sartorelli JC, Rosa Jr E, Grebogi C. Int J Bifurcat Chaos 2003;13(9):2551–60. [6] Chua LO, Matsumoto T, Komuro M. IEEE Trans Circ Syst 1986;CAS-32:1072–97. [7] Codimension is the number of dimensions of the parameter space minus the dimension of the studied object. [8] Lathrop DP, Kostelich EJ. Phys Rev A 1989;40(7):4028–31. [9] Mindlin GB, Gilmore R. Physica D 1992;58:229–42. [10] Boulant G, Lefranc M, Bielawski S, Derozier D. Phys Rev E 1997;55(5):5082–91. [11] Gilmore R. Rev Mod Phys 1998;70(4):1455–529. [12] Mindlin GB, Hou X-J, Solari HG, Gilmore R, Tufillaro NB. Phys Rev Lett 1990;64(20):2350–3. [13] Cvitanovic´ P, Gunaratne GH, Procaccia I. Phys Rev A 1988;38(3):1503–20. [14] Tufaile A, Sartorelli JC. Phys Lett A 2000;275:211–7. [15] Letellier C, Gouesbet G, Rulkov NF. Int J Bifurcat Chaos 1996;6(12B):2531–55. [16] Hegger R, Kantz H, Schreiber T. Chaos 1999;9:413–39. [17] Auerbach D, Cvitanovic´ P, Eckmann JP, Gunaratne G, Procaccia I. Phys Rev Lett 1987;58:2387–9. [18] Jensen MH, Kadanoff LP, Procaccia I. Phys Rev A 1987;36:1409–20.
Experimental identification of chaotic fibers D.M. Maranha˜o a, J.C. Sartorelli a
a,*
, M.S. Baptista
b
Instituto de Fı´sica, Universidade de Sa˜o Paulo, Caixa Postal 66318, 05315-970 Sa˜o Paulo, SP, Brazil b Universita¨t Potsdam, Institut fu¨r Physik Am Neuen Palais 10, D-14469 Potsdam, Germany Accepted 2 January 2007
Communicated by Prof. R. Seydel
Abstract In a 2D parameter space, by using nine experimental time series of a Chua’s circuit, we characterized three codimension-1 chaotic fibers parallel to a period-3 window. To show the local preservation of the properties of the chaotic attractors in each fiber, we applied the closed return technique and two distinct topological methods. With the first topological method we calculated the linking numbers in the sets of unstable periodic orbits, and with the second one we obtained the symbolic planes and the topological entropies by applying symbolic dynamic analysis. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The interplay between chaotic and periodic behaviors has been studied in the parameter space of some dynamical systems. Barreto et al. [1] and Baptista et al. [2] showed that in the 2D parameter space of a fragile chaotic system [3], one should expect to find periodic windows situated along structures called spines. Once high-period stable periodic orbits, located at the border of these periodic windows, are similar, one can expect to find similar chaotic attractors at the chaotic side of this border. Alike the periodic regions where one can have a path in the 2D parameter space to change two parameters to follow the same high-period orbit, we should expect to find selective paths to vary two parameters in such way that we would produce always chaotic attractors with similar characteristics in each path. In fact, Monti et al. [4] showed numerically the existence of such paths in a parameter space of the Lorenz system. In addition, Baptista et al. [5] suggested that the two parameters variation should be a linear function in the Chua’s circuit [6]. This line was called a chaotic fiber, which arguably would have the same codimension [7] of the neighboring periodic windows. The way one has to change two parameters, preserving the attractors characteristics, is provided by the windows conjecture [1], which states that the periodic windows have codimension k in the parameter space, where k is the number of positive Lyapunov exponents of the neighboring chaos in the vicinity of the periodic windows. So, the chaotic regions that preserve their characteristics by a two parameter variation appear aligned to the periodic windows. If the periodic windows have codimension k, it is reasonable to assume that these especial chaotic regions also have codimension k. *
Corresponding author. E-mail address: [email protected] (J.C. Sartorelli).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.131
10
D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
The Chua’s circuit presents the three main characteristics to observe fibers: (i) minimal amount of internal noise, such that periodic windows can be clearly recognized; (ii) a minimal number of positive Lyapunov exponents, so the chaotic fibers can be found by varying a minimal number of accessible parameters; (iii) a chaotic attractor with a simple topology and a small embedded dimension, such that the UPOs can be easily identified by using symbolic dynamic on reconstructed attractors. The chaotic attractors of the Chua’s circuit have just one positive Lyapunov exponent, so we should expect to find codimension-1 periodic structures in the 2D parameter space, which means that the periodic windows have a line structure. If one wants to change the control parameters preserving the topological and metric properties of the chaotic attractors, we should vary simultaneously two control parameters. By the Lyapunov exponents calculation the fibers could not be discerned, but we are providing strong experimental evidences of the existence of them by applying the closed returns technique [8,9], and by two independent topological methods. With one, we analyzed the structure of the unstable periodic orbits (UPOs) and obtained their linking numbers [9–12]. With the other one, by applying a symbolic dynamics, we constructed the symbolic plane and estimated the topological entropy of the attractors [13–15].
2. Experimental setup In Fig. 1, it is shown the diagram of the Chua’s circuit. The RNL is the non-linear resistance, L is the inductor, C1 and C2 are two capacitors. The dynamical variables are iL, the current across the inductor, V C1 and V C2 , the voltages across
VC1
VC2 L
iL R10 ΔR1
ΔR2 C2
C1
RNL
R 20
Fig. 1. Chua’s circuit. Component values are: R10 1:4 kX; R20 37 X; C 1 4:7 nF; C 2 56 nF; L 9:2 mH.
a
b
c
Fig. 2. Bifurcation diagrams by varying DR2 : (a) Z series with DR1 12:5 X, (b) Y series with DR1 7:5 X and (c) X series with DR1 3:0 X. The boxes represent parameter regions where we analyzed the time series.
D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
11
the capacitors C1 and C2 , respectively. We have two control parameters given by two resistances R1 ¼ R01 DR1 and R2 ¼ R02 DR2 , where R01 and R02 are fixed, and DR1 and DR2 are precision potentiometers driven by step motors. These two potentiometers are allowed to vary within the intervals DR1 2 ½0:0; 15:0X and DR2 2 ½0:0; 5:5X, with a step of Dpr ¼ 50 mX. The time series, 100,000 points long, were obtained recording the V C1 voltage with a 12 bits ADC at 400 ksamples/s. In Fig. 2 are shown the bifurcation diagrams V C1 vs. R2 obtained for three values of R1, and a large period three window can be seen in the three cases. The data sets X, Y and Z were named according to the value of DR1 = 3.0, 7.5 and 12.5 X, respectively. We selected three times series of each set X,Y and Z in each DR2 interval defined by the boxes drawn in Fig. 2. We named the time series as XJ, YJ and ZJ, where J = 1, 6, and 11 measured from the border of the large periodthree window (J = 0),as shown by the parameter space ðDR1 DR2 Þ [5] schematically drawn in Fig. 3. Once the period-three window border appears linear in parameter space [5], we expect that the parameter values chosen belongs to three different chaotic linear fibers. Thus, the time series Z11, Y11 and X11 belong to the fiber named (I), Z6, Y6 and X6 to the fiber named (II), and Z1, Y1 and X1 to the fiber named (III). The chaotic attractors were reconstructed in an embedding dimension de = 3 with a time delay s = 45.0 ls. Two-dimensional projections are shown in Fig. 4.
Fig. 3. Illustration of the parameter space showing the positions of the times series X, Y, and Z as well as the positions of the fibers.
Fig. 4. Two-dimensional projection of reconstructed attractors in the fibers (I), (II) and (III) with time delay s ¼ 45:0 ls. In Z11 it is also shown a Poincare´ section V C1 ðtÞ ¼ 2:25 V.
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3. Results and discussion 3.1. Metric characterization The Lyapunov exponents were obtained using the TISEAN package [16]. As shown in Table 1, the Lyapunov exponents and dimensions values for all the nine attractors, are not suitable to state about the existence of different fibers. Since the Lyapunov dimension for all the attractors is smaller than 3 and there is only one positive Lyapunov exponent the topological methods are applicable to find their dynamical characteristics. 3.2. Topological approach 3.2.1. Periodic orbits (UPOs) and linking numbers The topological analysis employed is based on the fact that the measure and average quantities of a chaotic attractor can be obtained from the unstable periodic orbits (UPO) embedded in it [17,18].A typical chaotic trajectory evolving in an attractor can be thought as being a random walk along the infinite number of UPOs. The trajectory approaches an unstable orbit along its stable direction, and it stays in the neighborhood trajectory during an interval of time and after it departs from the UPO along the local unstable direction. In this time interval, a short segment of the trajectory shadows the periodic orbit embedded within the chaotic attractor, allowing to extract approximations to the periodic orbit and obtaining approximations to a finite set of periodic orbits with not too long period, what is appropriate for the finite experimental data. We extracted the periodic orbits by searching in the time series V C1 ðtÞ [8,9] the time intervals ½t; t þ tp for which the difference jV C1 ðtÞ V C1 ðt þ tp Þj remains smaller than a given e (0.05% of the maximum value of V C1 ), and after we also checked the UPOs in the 3D reconstructed attractors. By scanning the data series for the existence of UPOs for many different time intervals tp we constructed the histograms shown in Fig. 5. The peaks position corresponds to the period of an UPO, and all the histograms display the same cycle time T = 86 Dt ðDt ¼ 2:5 lsÞ, which is approximately the characteristic mean period of our Chua’s circuit. The relevant information in these histograms is the height of the peaks, indicating the contribution of the corresponding UPO to the measure of the attractor. For each column the histograms present the same characteristic pattern, showing that we have three linear fibers as we claimed before. Another way of revealing special characteristics of the existing UPOs is by symbolic dynamics techniques. To construct the symbolic sequence of a trajectory or periodic orbit, we first construct a first return map of the reconstructed ðnÞ attractors, by using the Poincare´ section shown in Fig. 4 for the data series Z11. The new variable is represented by V C1 , where n ¼ 1; 2; . . . representing the nth crossing of the reconstructed trajectory with this section. ðnþ1Þ ðnÞ The first return mappings V C1 vs. V C1 can be seen in Fig. 6. They are approximately unimodal maps of an interval onto itself, and the position of the maximum is the partition point as shown in Fig. 6a. The partition point divides the map into two regions: one with a positive slope, labeled 0, and other with a negative slope, labeled 1. As the trajectory or periodic orbit visits these regions, we write down the corresponding symbol to obtain the symbolic sequence S ¼ 010 . . .. An UPO extracted by the closed return method, has period p if its symbolic sequence contains p symbols, represented by the sequence s0 s1 . . . sp1 , and is regarded to be the name of the UPO, for example a period-2 orbit is codified as (10).
Table 1 Estimated Lyapunov exponents of time series in fibers (I), (II) and (III) Fiber
Time series
k1
k2
k3
d KY
(I)
X 11 Y 11 Z 11
0.01257(7) 0.01298(5) 0.01234(5)
0.00519(9) 0.00496(8) 0.00488(10)
0.0831(2) 0.0864(2) 0.0861(2)
2.086 2.095 2.089
(II)
X6 Y6 Z6
0.01342(2) 0.01367(4) 0.01305(9)
0.00503(5) 0.00440(5) 0.00541(3)
0.0870(2) 0.0869(2) 0.0901(2)
2.095 2.107 2.087
(III)
X1 Y1 Z1
0.01362(8) 0.01371(8) 0.01416(2)
0.00463(7) 0.00504(9) 0.00444(2)
0.0896(2) 0.0897(4) 0.0895(6)
2.101 2.098 2.103
The dimension d L is the Lyapunov dimension estimated as d L ¼ 2 þ jk1 =k3 j (Kaplan–Yorke conjecture).
D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
a
b
c
d
e
f
g
h
i
13
Fig. 5. Closed returns histograms for the V C1 ðtÞ time series. The corresponding chaotic fiber of each data set is indicated in the top. Observe the similarity of the histograms in each fiber.
a
b
c
d
e
f
g
h
i
Fig. 6. Poincare´ section V C1 ¼ 2:25 V in the reconstructed attractors. All the maps closely resemble the unimodal maps, that present only one point of maximum, which is used as the partition point resulting in two regions labeled ‘‘0’’ (the left) and ‘‘ 1’’ (the right), as shown in (a).
The invariant linking numbers give the topological properties of the orbits assigning them a signature due to these properties [9,12]. These numbers give an idea of how a periodic orbit in the 3D phase space crosses itself and is called the self-linking number; or how it crosses another orbit, is called the linking number, such that one can determine at each crossing which segment of the orbit passes over the other. The self-linking (linking) number is determined by the sum (half sum) of oriented crossings of the orbit with itself (the orbit pair) [9–11]. The Fig. 7 shows the UPOs spectra of the lowest-period orbits.
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D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
orbits
A
1T
1
0
2T
10
1
1
3T
101
1
2
2
4T
1011
2
3
4
5 8
5Ta
10110
2
4
5
8
5Tb
10111
2
4
5
8
10
8
6Ta
101110
8
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9
14
6Tb
101111
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12
10
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17
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17
11
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17
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18
7Ta 7Tb
B
7Tc 7Td 8Ta 8Tb 8Tc
C
1T 2T 3T 4T 5Ta 5Tb 6Ta 6Tb 7Ta 7Tb 7Tc 7Td 8Ta 8Tb 8Tc 8Td 8Te
8Td 8Te
1011011 1011111 1011010 1011110 10111110 10111010 10111111 10110111 10110110
3 3 2 3 3 5 3 3 4 3 3
4 5 6 6 5
6 6 7 7 7
5
7
10
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24
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22
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6
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Fig. 7. Spectra of UPOs extracted from reconstructed attractors obtained from the time series at parameters along three fibers. We show their symbolic sequence, self-linking (diagonal) and linking numbers. In the fiber (III) all the 17 orbits shown in the table are allowed; along the fiber (II) the orbits in box A and C are forbidden and along the fiber (I) the orbits in boxes A, B and C are forbidden. The dots represent measurements whose value could not be determined.
In the first column in Fig. 7, we have the labels of 17 UPOs, whose symbolic sequences are shown in the second column, and the self-linking numbers in the other columns. For the attractors in fiber (III), we find all the 17 periodic orbits shown in Fig. 7. In fiber (II) three of these orbits were not found, i. e. they were pruned by the dynamics, the ones in boxes A and C. In fiber (I) five orbits were also pruned, the ones in boxes A, B and C. Therefore, we can conclude that the topological properties of UPOs are the same in each fiber. 3.2.2. Symbolic planes and topological entropy The fibers were also characterized by the construction of symbolic planes, which is a way to summarize the symbolic sequences in a two-dimensional map, and has been employed to compare similar attractors of different systems [13–15]. Each point in the symbolic plane (a, b) is obtained in an unique way from a symbolic sequence: . . . sm . . . s2 s1 :s0 s1 s2 . . . sm . . .
si 2 0; 1;
ð1Þ
where s0 s1 s2 . . . sm . . . is the forward symbolic sequence (a), and . . . sm . . . s2 s1 is the backward one (b). The decimal real coordinates (a, b) are given by m X ai ð2Þ a ¼ 0:a1 a2 a3 . . . ¼ i; i¼1 2 ai ¼
i1 X j¼0
and
sj ðmod 2Þ;
ð3Þ
D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
b ¼ 0:b1 b2 b3 . . . ¼ bi ¼
i X
m X bi i; i¼1 2
15
ð4Þ
sj ðmod 2Þ:
ð5Þ
j¼1
In Fig. 8, we can see the symbolic planes of the nine attractors. By comparing the patterns in each fiber in Fig. 8, we can see the same symbolic planes. From fiber I to fiber III the number of allowed sequences (dots) increase and in the fiber III the attractors present the most allowed symbolic sequences, which is consistent with the increasing of the number of UPOs. Therefore, the symbolic planes can also be thought as the fingerprint of the fibers. With the symbolic sequences of the nine reconstructed attractors, we estimated the topological entropy
a
b
c
d
e
f
g
h
i
Fig. 8. Symbolic planes in fiber (I) (on the left), in fiber (II) (on the center), and in fiber (III) (on the right). Observe the similarity of the symbolic planes in each fiber.
Fig. 9. Topological entropy of the experimental reconstructed attractors in fibers I, II and III, estimated from their symbolic sequences as a function of the length p of the sequences.
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D.M. Maranha˜o et al. / Chaos, Solitons and Fractals 39 (2009) 9–16
hT ðSÞ ¼ lim
p!1
ln N p ðSÞ ; p
ð6Þ
where N p ðSÞ is the number of sequences S ¼ s0 s1 . . . sp1 of length p, whose results are shown in Fig. 9. We can see that for every fiber the topological entropies follow the same curve, also indicating the preservation of the topological information.
4. Conclusion We presented experimental evidences about the existence of linear chaotic fibers parallel to a period-three window in the ðDR1 ; DR2 Þ parameter space of a Chua’s circuit. These fibers are parameter lines where the topology of the chaotic attractors are preserved. By topology, we mean the closed return technique, a set of short period unstable period orbits, the orbits characteristics as their linking numbers, the symbolic space and the topological entropy. These involved topological methods were employed because metric methods such as the Lyapunov exponents did not produce convincing evidences in order to state about the existence of such fibers.
Acknowledgements We thank Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo and CNPq for the financial support. M.S.B. also thanks the Alexander von Humboldt Foundation.
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