FRACTAL PROPERTIES OF CHAOTIC DYNAMICAL SYSTEMS IN REVERSE TIME AND ITS APPLICATIONS

A. I. Tomashevsky, M.V. Kapranov

Moscow Power Engineering Institute (Technical University), Radio Engineering Faculty Department of Generation of Oscillations and Signals

14 Krasnokazarmennaya str., Moscow 111250, Russia; tel. +7(095)362-7795; fax +7(095)273-3522 e-mail: TomashevskyAI@mpei.ru

Abstract—The dynamical process properties of

chaotic discrete map in reverse time is considered.

The fractal structure of reverse iteration points is

investigated. Several types of ordering of the reverse

points manifold are proposed.

I. INTRODUCTION

Dynamical chaos and its applications to secure communication and other fields have been investigating in literature for a long time. The majority of approaches proposed are in hiding of the fact of information transmission by means of its masking in random-like chaotic signal or other methods. The novel and deeply studied enough theoretical and experimental results have described in a lot of papers and books, for instance [1].

Another progressive example of dynamical chaos application is the research of information encryption algorithm based on chaotic properties of nonlinear dynamical systems [2-4]. By now, there are several of inconsistent opinions about perspectives of this approach. But due to a small number of publications in this direction, it’s necessary to conduct further more detailed investigations with the help of new ideas and approaches to application of dynamical chaos for the final solution of problem.

As such new idea in given paper we propose to use the chaotic processes of the dynamical systems in “reverse” time. This approach is connected with inverse problem of nonlinear dynamics [5], having polymorphism (ambiguous solution) and demonstrating in general case, exceeding difficulty for research.

This idea is also concerned the several problems of various approaches to solution of inverse problem for dynamical systems and its practical applications, that are investigated in literature. They are separation of chaotic signals into components from its sum with the noise [6], decoding the information stored in chaotic sequence by symbolic dynamics

[7], restoration of initial conditions, parameters and nonlinearly of the model from observed chaotic waveform in the presence of noise and other disturbing factors, by means of so called reverseiteration [8], data encryption [3,4] and etc.

In these works, as a rule, only partial questions of reverse time system applications for radiophysicaland telecommunication problems are considered. At the same time, reverse dynamics itself is greatly interesting field for research and has varied properties, which are not enough investigated in literature.

For example, in [3,4] for encryption the noninvertible maps with many-valuedness of 2nd

order in reverse time, where one of two branches is chosen at any step in arbitrary way [4] or corresponding to chaotic dynamics of another map with random initial point [5] have been proposed to use. The decryption is provided by direct dynamics. But authors of these papers, evidently, have not remarked an interesting phenomenon of fractaldistribution of ciphertexts in their algorithm, i.e. reverse iteration points of noninvertible maps.

In our work [9] we have shown briefly that structure of many-valuedness of noninvertible maps in reverse time has clear fractal characteristics. In present paper we propose more detailed investigation of theoretical characteristics of chaotic processes in reverse time. We also consider here how those properties can be used in application to information technologies.

The structure of the paper is as following. In Chapter II, a selection of dynamical system as the discrete map is explained and their basic dynamical and statistical characteristics in direct time are discussed. In Chapter III, properties of these systems for “motion in reverse time” are investigated and compared with characteristics obtained in previous Chapter II. There is a small discussion about example application of reverse map complex structure order for information encryption in Chapter IV.

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II. BASIC PROPERTIES OF THE MAP IN DIRECT TIME

Let’s take as our object of investigation the class of dynamical systems with discrete time tk. In given paper we consider the simplest example of 1-dimension single-parameter map in the general form

xk+1=F(xk; a), (1)

where xk – state of the system in the moment of time k= tk/T=0,1,2…, T – sample period, F – map functions, a – constant parameter.

One of the simplest and widely-used types of maps from this class, that providing the chaotic behavior, is piecewise-linear map with single maximum (or minimum), that is will-known also as tent map (2).

.1,)1()1(,0,

);(1kk

kkkk xaax

axaxaxFx (2)

In our case we present a modification of asymmetrical tent map with fixed maximum, i.e the tent with top, that changing along the line of F(xk;a)=1, when varying the parameter, but the base of tent is fixed in the definitional domain borders of xk,i.e. F(0;a)=F(1;a)=0 (Fig. 1,a). In this instance we deal with asymmetrical linear maximum, that formed by two closing straight line with different tangents, depending on the value of parameter a. Map function becomes symmetrical only in the single situation: a=0.5. Exactly the same map has been investigated theoretically in [10] and called there as skew tent map.

In this map the band of changing the parameter, corresponding to global stable dynamics, is

)1,0(a ; initial condition definitional domain also is the unit interval: )1,0(kx .

The reason, why we have selected this type of map function is as followings. The map (2) is non-compressing and non-expanding, i.e. it converts the unit interval (0,1) into itself. Thereby reverse map also would be non-compressing and non-expanding. This provides the existence of iteration point of reverse map function for all points from definitional domain of the direct map function, and signifies the mutual correspondence of any reverse dynamical process to direct one for all parameter values and initial conditions. While there is no such a correspondence, in general, in compressing or expanding maps.

Let’s consider a little bit more in detail some properties and characteristics of our piecewise-linear map. This map generates a dynamical chaos for all allowed values of parameter, that is confirmed by solid form of bifurcation diagram and positive Lyapunov exponent, when a is changed (Fig. 1,c-d).An example of the chaotic behavior for the fixed parameter is shown in phase plane (Lamerey-Königs diagram) in Fig. 1,a.

a) Lamerey-Königs diagram for a=0.25, x0=0.2

b) Density of distribution (it is uniform for any

(0,1)a )

c) Bifurcation diagram d) Lyapunov exponent

Fig. 1. Characteristics of piecewise-linear map (2)

Note that density of distribution P(x) the chaotic process when increasing of duration of the time sample tends to uniform law for nearly any )1,0(a(Fig. 1,b). This is also could be marked in bifurcation diagram in Fig. 1,c, where almost the whole volume of phase space is filled uniformly, except of narrow areas, which are visited by trajectory comparatively less, than all rest space – these areas are concentrated near the fixed points of the system (2). One of them is always in origin x1*=0, but coordinate of the other point is defined by expression: x2*=1/(2–a).

The dependency of Lyapunov exponent on the system parameter, calculated numerically and plotted in Fig. 1,d, can be also derived analytically (refer to [10]): )1ln()1(ln)( aaaaa ; as you see, 0)(a for any )1,0(a .

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The map presented here is very suitable from the standpoint of chaos control in secure communication system, since if the parameter is varied slowly the oscillation characteristics practically are not changed, but density of distribution function (consequently mean value, dispersion and etc) is not changed at all.

III. FEATURES OF THE SYSTEM IN REVERSE TIME

For the skew tent map (2) the reverse map can be written as following

11 1 1

1 12 1

( ) , ( ; )

( ) 1 (1 ) .k

k kk

F axx F x a

F a x (3)

It has ambiguous functional dependence or in another words many-valuedness of the 2nd order.

Thereby, for every value xk+1 we have two preceding values xk (figuratively expressing, for each “effects” we have two possible “reasons”). Hence, at every n-th step of the motion backwards, where

],0[ Nn , ambiguity increases as 2n. Graphically this phenomenon can be shown in the manner of treeof reverse iteration, Fig. 2, where all possible trajectories in the time interval (k-N, k), where k>N,getting through given value xk=0.2 (the time axis positive direction – on the left)*. All trajectories, starting from manifold {xk-N}, – values at the right border of the “tree” in Fig. 2 (top of tree) – for direct iteration in n steps will reach the value xk (base of tree). This process (branches of tree) tries to fill the whole volume of attractor, corresponding to direct dynamics, when n increases. However the question appears: what is the structure of top tree manifold?

The density of distribution function Pr(x) of the top tree manifold is presented in Fig. 3. This is quite different distribution as of direct process (compare with Fig. 1,b) and, hence, not the same as of single branch of the tree in Fig. 2 (it seems to be also uniform). Only for a=0.5 (symmetric map) Pr(x) is the same as P(x). This means that backward process is not ergodic, in general case.

Another difference from the direct process is in the dependence of Pr(x) on the parameter a. So, we

* In the paper [6] a calculation of such multiple backward trajectories by means of reverse iteration was used for increasing the separation quality of chaotic process from mixture of a number of signals with noise in communication channel by statistical analysis.

see, reverse dynamics generates statistically different process, then direct one.

Fig. 2.Tree of reverse iteration; N=10, a=0.25, xk=0.2

a) a=0.25 b) a=0.4

c) a=0.5 d) a=0.75

Fig. 3.Density of distribution of top tree manifold; N=20

The characteristics of Fig. 3 already demonstrate self-similar (fractal) structure, especially Fig. 3,d.It’s interesting to present the top tree as some time waveform, so we could consider the tree of reverse iteration as the generator of fractal consequences. But there is another problem: how to order this manifold or, in other words, what parameter of ordering we have to choose?

The convenient algorithm, that we propose here, is the ordering in correspondence with branch calculation sequence as the following. If at each n-threverse time step of tree when calculating xk-n due to (3) we’ll give to every value the symbolic digit “0”

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for calculation via 11F , and “1” – for 1

2F , each branch will get the symbolic binary sequence of the length equaled to N (for instance, for most bottom branch of Fig. 2, that was calculated only via 1

1F ,we’ve got the sequence of “00000…” (N zeros)).

In general case, this sequence can be interpreted as some binary number, where each bit corresponds to any reverse iteration n. Then we could link, for example, the upper bit with n=N (accordingly the lower bit with n=1) and present it in decimal form. In such a manner we obtain the unique number of each branch – “branch index” [0, 2 )Nj m , that we’ll use as a parameter of ordering. The dependency of top tree values on the branch index is shown in Fig. 4. This is the fractal structure – each fragment is the similar structure as the whole image. The form of fractal depends on the map parameter a,but not on the xk and N.

Fig. 4. Fractal top tree structure; N=16, a=0.25

The ordering principle play the key part in formation of fractal manifold – any other type of ordering leads to different top tree structure.

IV. DISCUSSION ABOUT APPLICATIONS

Fractal properties of ordering of iteration points in reverse time could find a lot of its applications. It’s possible to propose a simplest application – this is the data encryption algorithm based on reverse maps. This approach with the use of the same skew tent map has been proposed in [3].

As some kind of extension of this approach we can propose the generation of multiple secret keys by using of single initial point. To understand this

mechanism we could imagine that the base of the tree is the initial (confidential) seed, but tops of the tree are the number of generated keys (see Fig. 2). Each of these keys, taken as initial condition for direct dynamics leads to the seed in N steps of iterations. The number of keys is defined by N, the length – by N and the accuracy of calculation required for divergence (for instance, for N=10 we obtain 2N=1024 different keys). This method can be applied not only for encryption but also for control of multiple access to the same SW resources generating of personal ID’s and passwords.

There are also other applications can be proposed, for instance, data compression, identification of parameters, fractal signal generation and etc.

REFERENCES

[1] “Chaotic Electronics in Telecommunications” / Edited by M.P. Kennedy, R. Rovatti, G. Setti. CRC Press LLC,2000, 542 p.

[2] F. Dachselt, W. Schwartz. “Chaos and Cryptography”. IEEE Trans. on Circuits and Systems. I – Fund. Theory and App.l, vol. 48, no. 12, pp. 1498-1509, 2001.

[3] Toshiki Habutsu, Yoshifumi Nishio, Iwao Sasase and Shinsaku Mori. “A Secret Key Cryptosystem Using a Chaotic Map”. Trans. of IEICE, vol. E73, no. 7, pp. 1041-1044, July 1990.

[4] Toru Hiraoka and Yoshifumi Nishio. “Analysis of a Cryptosystem Using a Chaotic Map Extended to Two Dimensions”. Proc. of NCSP'04, Hawaii, USA, Mar. 5-7, 2004.

[5] Anosov, O.L. and Butkovskii, O.Ya. “Predictability of Complex Dynamical Systems”. Berlin: Springer, 1997, 253 p.

[6] Y. V. Andreyev, A. S. Dmitriev, E. V. Efremova and A. N. Anagnostopoulos. “Separation of Chaotic Signals Sum Into Components in the Presence of Noise”. IEEETrans. on Circuits and Systems. I – Fund. Theory and Appl., vol. 50, no. 5, pp. 613-618, 2003.

[7] A.A. Dmitriev. “Perfect Synchronization in Pseudochaotic Systems”. Proc. NDES’03, Scuol, Switzerland, pp. 86-90, 2003.

[8] V. N. Kuleshov, M. V. Larionova. “Solution of Inverse Problem of Chaotic Dynamics for One-Dimensional Logistic Map”. In Proc. 100th Anniversary of A.A. Andronov «Progress in Nonlinear Science». p. 260. Nizhny Novgorod, Russia, 2001.

[9] A.I. Tomashevskiy, M.V. Kapranov, “Generation of Fractal Structure in Reverse Time Chaotic Map”. Proc. ICCSC’2004, Moscow, Russia, elec.vers., 2004

[10] M. Hasler, Y. L. Maistrenko. “An Introduction to the Synchronization of Chaotic Systems: Coupled Skew Tent Map”. IEEE Trans. on Circuits and Systems. I – Fund. Theory and Appl., vol. 44, no. 10, pp. 856-866, 1997.

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