science some new and simple concepts are still needed. For example, in order to analyze the motion of the planetary systems,e in al of ccan

in theIt is s

Let us consider a dynamical system. Its evolution, described by n time-series xjt; j 1;2; . . . ;n, corresponds to atory in the appropriate n-dimensional space. Let us assume that the time-series is subject of experimental measurema set of L values of time: s1; s2; . . . ; sL. The aim is to construct an algorithm which allows to recognize whether the meevolution of the system is periodic, chaotic or hyperchaotic. The periodic motion is by denition self-similar over sufcientlylarge intervals of time. This means overlapping of the trajectories, and as a consequence a nal shape of the trajectory (i.e.

0096-3003/$ - see front matter 2013 Elsevier Inc. All rights reserved.

E-mail address: phwaz@gumed.edu.pl

Applied Mathematics and Computation 227 (2014) 449455

Contents lists available at ScienceDirect

Applied Mathematics and Computationhttp://dx.doi.org/10.1016/j.amc.2013.08.0952. Methodtrajec-ents onasuredperformed for four systems. In particular, an analysis of the properties of the recently introduced hyperchaotic Qi system[20] are considered. Studies on hyperchaos, which may appear in at least four-dimensional autonomous systems is a subjectof many investigations in science and engineering [2128]. Hyperchaotic systems are characterized by more than one posi-tive Lyapunov exponent. Generally, hyperchaos is more disordered than the ordinary chaos and the new indicator is ex-pected to reveal also this property (see subsequent sections).indicators of chaos which could utiliztance [14,15]. In this area the controfamily of the indicators of chaos onecently formulated by us [18,19].

This kind of studies is continuedminimum length orbit is introduced.simple way the data derived from astronomical observations are of a great impor-haotic systems by time-series analysis [16,17] may be particularly useful. To thisassign a simple method based on the statistical properties of the time-series re-

present work. A simple algorithm for the calculation of a value referred to as thehown that it behaves differently for different kinds of solutions. The studies areNew indicators of chaos

Piotr Wa _zDepartment of Nuclear Medicine, Medical University of Gdansk, Tuwima 15, 80-210 Gdansk, Poland

a r t i c l e i n f o

Keywords:HyperchaosChaosIndicators of chaosNonlinear differential equations

a b s t r a c t

An algorithm which allows to recognize whether the evolution of the system is periodic,chaotic or hyperchaotic is proposed. Minimum length orbit is introduced. Its behaviordepends on the kind of solution. The calculations are presented for the Hnon map, forthe damped driven pendulum, for the Rssler system, and for the hyperchaotic Qi system.

2013 Elsevier Inc. All rights reserved.

1. Introduction

A deterministic system can evolve in a way which, in the long term, is unpredictable. The analysis of this kind of evolutionis the objective of the theory of chaos. Applications of this theory include physics, bioinformatics, biomedicine, meteorology,chemistry, sociology, astrophysics, engineering, economy. An excellent review of the history of the concepts underlying the-ory of chaos, from the 17th century to the last decade, has been given by Christophe Letellier in his book Chaos in nature[1].

Though a large number of different kinds of indicators of chaos may be found in the literature [213] in some areas of

journal homepage: www.elsevier .com/ locate/amc

orbit) is a closed curve. Then, the sum of the intervals between the points in the n-dimensional space (the length of the orbit)never exceeds a certain value. In the case of chaos and hyperchaos such a limitation does not exist. The algorithm con-structed is based on this observation.

Let Xsi be a point in the n-dimensional space:Xsi x1si; x2si; . . . ; xnsif g: 1

Let us split the set of points

SL Xsif gLi1; 2to K subsets

SN1 SN2 SNK SL; 3with

SNk Xsif gNki1: 4Now let us construct the minimum length orbit Dk in SNk . Let us dene the interval between a and b in SNk

Da;bk Xnj1

xjsa xjsb 2" #1=2

: 5

Let us start to calculate the length of the orbit at Xsy0 Xs1. Then we nd the closest point Xsy1 to the previous one(Xsy0 ) and calculate

Dy0 ;y1k miniy0 Dy0 ;ik

n o: 6

The consecutive point, Xsy2 , is obtained in a similar way

Dy1 ;sy2k miniy ;y D

y1 ;ik

n o: 7

The m

450 P. Wa _z / Applied Mathematics and Computation 227 (2014) 449455X(5)

X(6) X(10)X(11)

Fig. 1. A model example of the periodic orbit for n 2, K 2. Panel (a): Minimum length orbit D1. Panel (b): Minimum length orbit D2.X(4)X(12) X(13)X(1) X(3) X(9)

X(2)X(7) X(14)

X(8)

(b) orbit5X(4)X( )

X(6)X(1) X(2) X(3)X(7)

(a) orbiti1

Dk

XD

yi1 ;yik : 80 1

inimum length orbit is dened as

Nk1

AsLet

Model

Final intervals Search for the intervals (Eqs. (5)(7))

2 2 2

D2;32D3;4 D < D , i = 5,6,7,10,11,12,13,14

TableParameters in the linear approximations to D for periodic (P), and chaotic (C) solutions for the Hnon map. The notation cd stands for c 10 .

PC

P. Wa _z / Applied Mathematics and Computation 227 (2014) 449455 451culation of the minimum length orbit D1 starts at Xs1. The closest point to Xs1 is Xs7. The appropriate intervals are cal-culated and compared (Fig. 1(a)):

D1;71 < D1;i1 ; i 2;3;4;5;6;

7;6 7;iThe la

SinintervS7 Sand th

Thare shpriateorbit idoes n

Th

is deteTh

as

wherewe shall see, the behavior of the sequence Dk; k 1;2; . . . ;K is different in periodic, chaotic and hyperchaotic cases.us consider a simple model example of a periodic orbit in 2D for Nk 7k (Fig. 1). For k 1 the procedure of the cal-

1.03 2.56(18) 3.15(0)1.4 1.05(3) 22.7(0)k

g q r2d2 2 2

D4;102 D4;102 < D

4;i2 , i = 5,6,7,11,12,13,14

D10;52 D10;52 < D

10;i2 , i = 6,7,11,12,13,14

D5;112 D5;112 < D

5;i2 , i = 6,7,12,13,14

D11;62 D11;62 < D

11;i2 , i = 7,12,13,14

D6;122 D6;122 < D

6;i2 , i = 7,13,14

D12;132 D12;132 < D

12;i2 , i = 7,14

D13;72 D13;72 < D

13;142

D7;142D2;32 < D2;i2 , i = 4,5,6,7,10,11,12,13,14

3;4 3;iD1;82 D1;82 < D

1;i2 , i = 2,3,4,5,6,7,9,10,11,12,13,14

D8;92 D8;92 < D

8;i2 , i = 2,3,4,5,6,7,10,11,12,13,14

D9;2 D9;2 < D9;i , i = 3,4,5,6,7,10,11,12,13,14example of the calculations of D2.

Table 1D1 < D1 ; i 2;3;4;5;D6;51 < D

6;i1 ; i 2;3;4;

D5;21 < D5;i1 ; i 3;4;

D2;31 < D2;41 :

st interval in Eq. (8) is D3;41 . As a consequence, the minimum length orbit is

D1 D1;71 D7;61 D6;51 D5;21 D2;31 D3;41 :ce the number of points is very small, the minimum length orbit is determined with a low precision. In particular theal D5;21 generates an error. For k 2 (Fig. 1(b)) the orbit is covered by larger number of points which belong to S14 and14. The integration of equations of motion is performed for a longer time. New points Xs8;Xs9, . . .Xs14 are addedey are located in between the points belonging to S7.e minimum length orbit D2 starts at Xs1. The closest point to Xs1 is Xs8. The steps of the algorithm (Eqs. (5)(8))own in Table 1. The intervals shown in the left column of the table are the nal components of D2 (Eq. (8)). The appro-intervals add to form the minimum length orbit D2, as it is shown in Fig. 1(b). Let us note that the minimum lengths not closed. For example, in the case presented in Fig. 1(b) the number of intervals is N2 1 13 (cf. Eq. (8)) and D14;12ot belong to the orbit.e density of points increases with increasing k and the minimum length orbit

D2 D1;82 D8;92 D9;22 D2;32 D3;42 D4;102 D10;52 D5;112 D11;62 D6;122 D12;132 D13;72 D7;142rmined more precisely. Relative to D1;D2 is closer to the real length of the orbit (see Fig. 1).e theory can be also applied to discrete systems. In a discrete case a point in the n-dimensional space (Eq. (1)) is dened

Xi x1i; x2i; . . . ; xni

;

i 1;2; . . . L and the algorithm is similar as for the continuous systems.

Parame

452 P. Wa _z / Applied Mathematics and Computation 227 (2014) 449455DDP RO QS

g q r g q r g q r

P 1.07 2.42(6) 1.02(+2) 4 1.28(5) 4.06(+1) 0.1 1.15(4) 1.63(+3)1.47 7.05(4) 1.26(+2) 6 2.33(6) 1.14(+2) 0.45 1.31(4) 3.58(+3)

C 1.15 4.89(2) 2.60(+2) 13 8.59(2) 8.80(+2) 2.15 5.23(0) 2.18(+4)1.50 5.57(2) 2.96(+2) 18 2.29(1) 1.26(+3) 13 2.02(+1) 4.85(+4)

H 23 2.54(+1) 6.58(+4)26 2.67(+1) 7.35(+4)

36Asincreais largSevera

3. Res

Th

Disters in the linear approximations to Dk for periodic (P), chaotic (C), and hyperchaotic (H) solutions. The notation cd stands for c 10d .

SystemTable 3we shall see, in the periodic cases Dk reaches quickly a constant value while in the chaotic and hyperchaotic ones itses with increasing k, where k 1;2; . . . ;K. Since the degree of disorder is the largest in hyperchaotic cases, also Dker for the hyperchaos than for the ordinary chaos. Then, the behavior of Dk may be used as a simple indicator of chaos.l examples are discussed in the next section.

ults and discussion

e behavior of Dk is illustrated on several representative examples of different kinds of dynamical systems:

crete systems2D Hnon map [29]

x1i1 1 gx12i 0:3x2ix2i1 x1i

): 9

0

9

18

27

0 2000 4000 6000 8000 10000

Dk

Nk

Dk=qNk+rg=1.4 chaosg=1.03

Fig. 2. Dk for the Hnon map.

0

300

600

900

0 2000 4000 6000 8000 10000

Dk

Nk

Dk=qNk+rg=1.15 chaosg=1.50 chaosg=1.07g=1.47

Fig. 3. Dk for the damped driven pendulum.

0

1300

2600

3900

0 2000 4000 6000 8000 10000

Dk

Nk

Dk=qNk+rg=18 chaosg=13 chaosg=4g=6

Fig. 4. Dk for the Rssler system.

0

100000

200000

300000

0 2000 4000 6000 8000 10000

Dk

Nk

Dk=qNk+rg=26 hyperchaosg=23 hyperchaosg=2.15 chaosg=13 chaosg=0.1g=0.45

Fig. 5. Dk for the Qi system.

55

85

115

145

0 2000 4000 6000 8000 10000

Dk

Nk

30

60

90

120

0 2000 4000 6000 8000 10000

Dk

Nk

1000

2000

3000

4000

0 2000 4000 6000 8000 10000

Dk

Nk

Fig. 6. Dk for the periodic solutions for the damped driven pendulum (top), for the Rssler system (middle), and for the Qi system (bottom).

P. Wa _z / Applied Mathematics and Computation 227 (2014) 449455 453

structhyper

454 P. Wa _z / Applied Mathematics and Computation 227 (2014) 449455[8] Ch. Skokos, Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits, J. Phys. A 34 (2001) 1002910043.[9] Ch. Skokos, Ch. Antonopoulos, T.C. Bountis, M.N. Vrahatis, Detecting order and chaos in Hamiltonian systems by the SALI method, J. Phys. A 37 (2004)

62696284.ential equations have been solved numerically with L 10000.For the continuous systems, the main part of the numerical integration code constitutes procedure RA15 [32] and the

time sequences sif gLi1 with a step si1 si 0:1, and s1 0:1 have been taken.The number of points in the k-th subset is equal to Nk 100k, where k 1;2; . . .K and K 100. In particular for the larg-

est subset NK L 10000 and for the smallest one N1 100. Figs. 25 show Dk as functions of the number of points Nk forthe Hnon map (Fig. 2), for the damped driven pendulum (Fig. 3), for the Rssler system (Fig. 4), and for the Qi system(Fig. 5). The distinction between periodic, chaotic, and hyperchaotic systems is clearly seen. In periodic cases Dk is smalland approximately constant. For chaos Dk increases and is signicantly larger than in the periodic cases (Figs. 25). Forhyperchaos in Qi system one can observe further increase of Dk its values for hyperchaos are larger than for chaos(Fig. 5). For the periodic cases enlargements of Figs. 35 are displayed in Fig. 6. As one can see, for the Rssler and for theQi systems the constant asymptotic values of Dk are reached after several steps; for the damped driven pendulum Dk stabi-lizes around Nk 3000.

For sufciently large Nk the behavior of Dk is approximately linear. The coefcients of the least square t

Dk qNk r; 13obtained for 3000 6 Nk 6 10000 are collected in Tables 2 and 3. As one should expect, the smallest values of q and r corre-spond to the periodic solutions and the largest to the hyperchaotic ones.

Summarizing, the presented algorithm is simple and it facilitates a convenient classication of the solutions. The newestimators Dk behave in a different way for different kinds of the solutions. In particular, for the periodic cases Dk is constant.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.amc.2013.08.095.

References

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9>=>;; 11

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_x1 50x2 x1 x2x3_x2 gx1 x2 x1x3_x3 13x3 33x4 x1x2_x4 8x4 30x3 x1x2

9>>>=>>>;: 12

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P. Wa _z / Applied Mathematics and Computation 227 (2014) 449455 455

New indicators of chaos1 Introduction2 Method3 Results and discussionAppendix A Supplementary dataReferences