On a classification of dynamic systems subject to noise

Chaos, Solitons and Fractals 11 (2000) 297±302

www.elsevier.nl/locate/chaos

On a classi®cation of dynamic systems subject to noise John Argyris a,*, Ioannis Andreadis a, Georgios Pavlos b, Michalis Athanasiou b a

Institute for Computer Applications I, Pfa€enwaldring 27, University of Stuttgart, D-70569 Stuttgart, Germany b Department of Electrical Engineering, University of Thrace, GR-67100 Xanthi, Greece

Abstract A classi®cation of alternative mathematical schemes which determine possible impositions of noise on dynamic systems either of a continuous or a discrete formulation in time is presented. We refer to the results obtained concerning the in¯uence of noise on the correlation dimension and on the largest Lyapunov exponent. A method of constructing models for e€ecting predictions of time series with a ®nite correlation dimension is presented. Numerical results concerning the in¯uence of various level of additive noise to the Henon attractor are also provided. Ó 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction In this paper, we recall a classi®cation of a number of mathematical schemes which describe how a noise can in¯uence a dynamic system having a discrete or continuous in time formulation as in [1]. When a noise interferes with the evolution of a dynamic system, it is called a dynamic noise. Such a dynamic noise may take the form of an additive or a multiplicative expression which illustrate the kind of parameters by which noise may be generated in the equations of a dynamic system. We also consider the case of an output noise which does not in¯uence the evolution of the dynamic system. We recall that the evolution of a dynamic system in Rr is de®ned through a functional vector x…t† for a continuous in time formulation or a map xn for a discrete in time formulation [2]. Consequently, in the case of an output noise we study a new evolution X…t† (resp. X n ) by applying noise to x…t† (resp. xn ). The output noise is again divided into additive and multiplicative forms which illustrates how noise may be generated in the de®nition of X…t† (resp. X n ). We also present our past theoretical results with respect to the in¯uence of noise on the correlation dimension and on the largest Lyapunov exponent of chaotic attractors. Numerical results for the case of the Lorentz attractor and for the Henon-map has been provided in [3]. A model for prediction for time series with a ®nite correlation dimension based on the theorem of Takens [4] is also presented. 2. Dynamic systems subject to noise Let us ®rst consider a dynamic system continuous or discrete in time de®ned through the equations x_ ˆ f …x; l; t† or

xn‡1 ˆ F…xn ; k; n†;

where x 2 Rr , xn 2 Rr are vectors, l 2 Rk , k 2 Rm are parameters, t 2 R and n 2 N . *

Corresponding author. Tel.: +49-711-685-3593.

0960-0779/00/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 8 ) 0 0 2 9 7 - 5

…1†

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J. Argyris et al. / Chaos, Solitons and Fractals 11 (2000) 297±302

We refer to the term noise in conjunction with the mathematical considerations presented in [5]. When a noise interferes with the evolution of a dynamic system it is called dynamic noise. Its general formulation for the case of a continuous or discrete in time dynamic systems is x_ ˆ f …x; l; t; w†

or

xn‡1 ˆ F…xn ; k; n; wn †;

…2†

where w is a noise function and wn is a vector with components of elements of noise. Such a noise may take the form of an additive or a multiplicative expression with illustrate the kind of parameters by which noise may appear in the equation of a dynamic system. The general form of an additive dynamic noise is given for the case of a continuous or discrete in time dynamic system by x_ ˆ f …x; l; t† ‡ w

or xn‡1 ˆ F…xn ; k; n† ‡ wn :

…3†

The general form of a multiplicative dynamic noise for the case of a continuous or a discrete in time dynamic system is given by x_ ˆ f …x; g…l; w†; t†

or

xn‡1 ˆ F…xn ; c…k; wn †; n†;

…4†

where g is a vectorial function and c is a matricial function. We also consider the case of an output noise which does not in¯uence the evolution of the dynamic system (1). Its general form X…t† (resp. X n ) for the case of a continuous (resp. discrete) in time dynamic system describes the evolution of which is de®ned by x…t† (resp. xn‡1 ) (solution of the system (1)), t 2 R (resp. n 2 N ) and is expressed by the general formalism X…t† ˆ K…x…t†; t; w† or

X n ˆ G…xn ; n; wn †;

…5†

where K is a functional operator and G is a function. The general form of an additive output noise for the case of a continuous (resp. discrete) in time dynamic system is given by X…t† ˆ x…t† ‡ w

or

X n ˆ xn ‡ wn :

…6†

The general form of an multiplicative output noise for the case of a continuous in time dynamic system is given by X…t† ˆ h…w; x…t††

or X n ˆ H…xn ; wn †;

…7†

where H is a mapping function, and h is a map. Some of the entries in the vector w may be one. In some special cases the map h appears in the form h…w; x…t†† ˆ fw1 x1 ; . . . ; wn xn g. We o€er bibliographical comments concerning these notions. The term dynamic noise appears in [5]. The additive (resp. multiplicative) dynamic noise for discrete in time systems is cited there as ``discrete in time systems with stochastic (resp. multiplicative) perturbation''. The case of an additive noise for the 2-Torus map appears in the work of Ott et al. [6] in 1985. The same case for the Henon-map appears in the work done by Cheng et al. [7] in 1992, which is referred as dynamic noise. The case of a multiplicative noise for the logistic map appears already in 1981 in the works done by Mayer-Kress and Haken [8], see also the work in 1982 by Crutch®eld et al. [9]. This case is also considered in [10]. We note also the paper published by Haken and Wunderlin [11] concerning the additive and multiplicative noise on discrete maps. The case of an output noise is familiar in the bibliography of the electrical circuit theory [12]. The case of an additive output noise for the discrete in time dynamic systems appears in the work by Hammel [13] in 1990 which is referred as shadowing property, or a noisy deviation of an orbit. This noise appears also in the numerical investigation of [14] concerning the in¯uence of noise on the geometric structure of the Henon-map.

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3. The in¯uence of noise on the correlation dimension of attractors Lets us recall the results obtained in [1,15] concerning the in¯uence of noise on the correlation dimension of attractors. Assumption 1. The correlation dimension of a white noise is in®nite. This assumption is in accordance with the algorithm of Grassberg and Procacia [16] remembering the correlation dimension of a white noise is always equal to the embedding dimension on which we reconstruct the attractor. Using the previous assumption we have proved the following theorems in [1] (For the Wold decomposition of a time series see [17]). Theorem 1 [1]. The existence of a small correlation dimension for a time series implies that the deterministic part of its Wold decomposition is not zero. Theorem 2 [1]. The magnitude of the correlation dimension of an attractor of a chaotic dynamic system continuous in time is increasing due to an additive dynamic noise. Let us present brie¯y, the main idea of this proof. Following a reduction to an hyperbolic stochastic system, we adopt the hypothesis that the correlation dimension of this system is equal to d. Let us denote by x…t† its general solution. Our hypothesis implies that, the consideration of N points xj ˆ x…tj † in a m-dimensional space around the xi point with kxj ÿ xi k 6 r where k . . . k de®nes the Euclidean norm and i ˆ 1; . . . ; N , instructs us that the correlation integral C…m; r† may be expressed, for some positive r, as a power of radius r to the power d as r ! 0, i.e. C…m; r† / rd ; as r ! 0:

…8†

Let us now consider the in¯uence of a noise b…t† applied to this system and denote by y…t† its general solution. Applying the theory of distributions, we prove that within a sphere of radius r around the point yi there exist at most N points. Hence, the correlation integral Cb …m; r† of the trajectory y…t† is smaller or equal to the correlation integral C…m; r† of the trajectory of x…t†, Cb …m; r† 6 C…m; r†:

…9†

We deduce that the correlation integral Cb …m; r† may be expressed as a power of the radius r to a correlation dimension db (of the system if perturbed by noise) as r ! 0. As the function rd is decreasing when r is smaller than 1, Eq. (9) implies that db is larger or equal to d. Thus the correlation dimension of the perturbed noisy system will be greater or equal to the correlation dimension d of the system. The Theorem 2 is in accordance with the work done by Baddi et al. [18]. The ®rst two authors (JA and IA) have pursued further the work of [1] in [15]. Theorem 3 [15]. The magnitude of the correlation dimension of an attractor of a chaotic dynamic system continuous or discrete in time is increasing due to an additive output noise.

4. The in¯uence of noise on largest Lyapunov exponent of attractors We have also studied the in¯uence of noise on the largest Lyapunov exponent. The numerical results which we have obtained and presented in [3], indicate an increase of the magnitute of the largest Lyapunov exponent. These results are part of a Ph.D. thesis of Athanasiou and will be appear in future in nascendi article by Pavlos et al. [19]. There are also numerical results concerning the in¯uence of a multiplicative dynamic noise on the largest Lyapunov exponent for the logistic map [9] and of an additive dynamic noise for the logistic map [20]. These results indicate also an increase of the magnitude of the largest Lyapunov exponent due to noise.

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J. Argyris et al. / Chaos, Solitons and Fractals 11 (2000) 297±302

The ®rst two authors (JA and IA) have proved the following results, presented in [15], based on the assumption. Assumption 2. The largest Lyapunov exponent of a white noise is in®nite. This assumption is obvious because the white noise always ®lls densely the space on which we reconstruct the attractor. Thus the divergence between initially close points becomes in®nite. Theorem 4 [15]. The magnitude of the largest Lyapunov exponent of an attractor of a chaotic dynamic system continuous in time is increasing due to an additive dynamic noise. Let us now present an outline of this proof. Following a reduction to a stochastic hyperbolic system, we adopt the hypothesis that the largest Lyapunov exponent of this system is equal to kmax;1 . Thus, if we consider two initially close orbits for this system we deduce from this hypothesis (see also [15]), that the distance L…ti † between two initially close orbits satis®es after an iteration step ts the relation L…ti † / exp…kmax;1 ti †:

…10†

We consider next the in¯uence of an additive noise on this system. If we consider a reference orbit associated with the previous system and denote by Lb …ti † the distance between two initially close orbits we deduce, following the same iteration step ts , [15] L…ti † 6 Lb …ti †

…11†

Taking the limit ts to in®nity we establish from relation (11), the existence of the largest Lyapunov exponent kb;max;1 associated with this system with noise and con®rm that Lb …ti † / exp…kb;max;1 ti †

…12†

We deduce immediately from relation (12) and the fact that the exponential function is increasing that kb;max;1 P kmax;1 . Theorem 5 [15]. The magnitude of the largest Lyapunov exponent of an attractor of a chaotic dynamic system continuous or discrete in time is increasing due to an additive output noise.

5. Models for prediction for a time series with a ®nite correlation dimension In this section, we recall the method of [1] concerning the application of selective dynamic systems subject to noise as models for the prediction of a time series T which possesses a small correlation dimension Dc . We propose as model of prediction a new time series Z de®ned by the equation Z…t† ˆ MfT …t ÿ 1†; T …t ÿ 2†; . . . ; T …t ÿ d†g ‡ e…t†;

…13†

where M is a smooth matricial operator and e…t† a white noise [5]. In most cases M is a matricial polynomial (e.g. an ARMA model [21]) or a nonlinear function. This model, without the term of noise, appears in the paper by Farmer and Sidorowich [22]. There exists also a statistical model which was discussed at the meeting of the Statistical-Royal Society [7]. The model de®ned in (13) is a dynamic system with noise. We choose as M a function such that the time series Z possesses a close dimensionality with the original time series T. This speci®cation should yield, in accordance with the Takens theorem [4], a good candidate for generating phenomena leading to a similar structure of the attractor. We intend to apply the method of prediction to the case of the time series emanating from the study of attractors as they appear in the area of applied mechanics, in a joint publication with Tenek [23].

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Fig. 1. The Henon attractor.

Fig. 2. The in¯uence of an additive noise on the Henon map [1] for values of parameters: (a) a1 ˆ a2 ˆ 0:001; (b) a1 ˆ a2 ˆ 0:002; and (c) a1 ˆ a2 ˆ 0:005.

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J. Argyris et al. / Chaos, Solitons and Fractals 11 (2000) 297±302

6. Further extension of this work We are interested in investigating the in¯uence of noise on the geometric structure of the attractor and intend to use the results proposed in [6]. Some preliminary results have been announced in [3]. Here we present numerical results towards to a bifurcation analysis of the in¯uence of an additive noise to the geometric structure of attractors. We consider the Henon attractor, Fig 1. Afterwards we apply various levels of noise, Figs. 2(a)±(c). We draw attention to the intensity of the noise which may imply a catastrophic transformation in the standard form of this attractor. Following an illustrating commentary of Prof. El Naschie to Prof. Argyris we intend to investigate the in¯uence of noise on the coexistence of more than one attractor. Ultimately, we indeed to investigate the in¯uence of a multiplicative noise on the Lyapunov exponent and on the correlation dimension for a multiplicative noise. First results in this direction will appear in [24]. References [1] J. Argyris, I. Andreadis, G. Pavlos, M. Athanasiou, On the in¯uence of noise in the correlation dimension of chaotic attractors, Chaos, Solitons and Fractals 9 (1998) 343±361. [2] J. Argyris, G. Faust, M. Haase, An Exploration of Chaos, North Holland, Elsevier, Amsterdam, 1994. [3] J. Argyris, I. Andreadis, G. Pavlos, M. Athanasiou, On the in¯uence of noise on the largest Lyapunov exponent and on the geometric structure of attractors, Chaos, Solitons and Fractals 9 (1998) 947±958. [4] F. Takens, Detecting strange attractors in turbulence, in: D. Rand, L.S. Young (Eds.), Dynamical Systems and Turbulence, Springer, New York, 1980, pp. 366±381. [5] A. Lasota, M.C. Makey, Chaos Fractals and Noise, Stochastic Aspects of Dynamical Systems, Springer, New York, 1994. [6] E. Ott, E.D. Yorke, J.A. Yorke, A scaling law: How an attractor's volume depends on noise level, Physica D 16 (1985) 62±78. [7] B. Cheng, H. Tong, On consistent nonparametric order determination and chaos, J. R. Statist. Soc. B. 54 (1992) 427±449. [8] G. Mayer-Kress, H. Haken, The in¯uence of noise on the logistic model, J. Statist. Phys. 26 (1981) 1±187. [9] J.P Crutch®eld, J.D. Farmer, B.A. Huberman, Fluctuations and simple chaotic dynamics, Phys. Rep. 92 (1992) 45±82. [10] I. Kapitaniak, Chaos in Systems with Noise, World Scienti®c, New York, 1990. [11] H. Haken, A. Wunderlin, Slaving principle for stochastic di€erential equations with additive and multiplicative noise and for discrete noisy maps, Z. Phys. B 47 (1982) 179±187. [12] K. Karoumpalos, Introduction to the Theory of Noise and Applications (in greek), Athens, 1979. [13] S.M. Hammel, A noise reduction method for chaotic systems, Phys. Lett. A 148 (1990) 421±428. [14] A. Ben-Mizrachi, I. Procaccia, P. Grassberger, Characterization of experimental noisy attractors, Phys. Rev. A 29 (1984) 975±977. [15] J. Argyris, I. Andreadis, On the in¯uence of noise on the largest Lyapunov exponent of attractors of stochastic dynamic systems, Chaos, Solitons and Fractals 9 (1998) 959±963. [16] P. Grassberg, I. Procaccia, Measuring the strangeness of a strange attractor, Physica D 9 (1983) 189±208. [17] G. Janacek, L. Swift, Time Series Forecasting, Simulation, Applications, Ellis Horwood, New York, 1993. [18] R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, M.A. Rubio, Dimension increase in ®ltered chaotic signals, Phys. Rev. Lett. 60 (1998) 979±982. [19] G. Pavlos, M. Athanasiou, P. Sotiropoulos, The in¯uence of white noise on the largest Lyapunov exponent of dynamic systems, in preparation. [20] G. Malescio, E€ects of noise on chaotic one-dimensional maps, Phys. Lett. A 218 (1996) 25±96. [21] T. Ozaki, Non-Linear time series models and dynamical systems, in: E.J. Harman, P.R. Vrishnaiah, W.M. Rand (Eds.), Handbook of Statistics, Elsevier, New York, 1985 pp. 25±83. [22] J.D. Farmer, J. Sidorowich, Predicting chaotic time series, Phys. Rev. Lett. 59 (1987) 845±848. [23] J. Argyris, L. Tenek, I. Andreadis, M. Athanasiou, G. Pavlos, On chaotic oscillations of a laminated composite cylinder subject to periodic application of temperature, Chaos, Solitons and Fractals 9 (1998) 1529±1554. [24] J. Argyris, I. Andreadis, On linearizable noisy systems, Chaos, Solitons and Fractals 9 (1998) 895±899.