- Email: [email protected]
ITERATIONS OF PERTURBED TENT MAPS WITH APPLICATIONS TO CHAOS CONTROL
ITERATIONS OF PERTURBED TENT MAPS WITH APPLICATIONS TO CHAOS CONTROL B.T.Polyak1 , E.N. Gryazina2 1
2
Institute for Control Science RAS, Moscow, Russia Moscow Physical-Technical Institute (State University), Moscow, Russia e-mail [email protected], [email protected]
Abstract: Iterations of 1D simple maps such as logistic, tent, cubic ones are very well studied. However perturbed versions of these maps (close in uniform norm but with strongly varying derivatives) can exhibit completely different behavior. We encounter such situation when dealing with chaos stabilization via small control. In this paper we present analytical investigation of this effect for one particular case — piecewise linear perturbation of the tent map. Surprisingly, iterations of this map converge to the unique fixed point very fast for all initial points. The result is in sharp contrast with iterations of the original tent map but explains fast stabilization of unstable periodic orbits by predictive control, proposed in Polyak & Maslov (2005); Polyak (2005). Keywords: Chaos, nonlinear maps, iterations, Markov chains, stabilization, predictive control.
1. INTRODUCTION Effects of bifurcations, chaotic behavior, random properties of iterations of simple one-dimensional maps are very well studied; e.g., see monographs Mira (1987); Elaydi (2000). The problem of interest is if all these effects are preserved under small perturbations of the maps. ”Small” means that the perturbed functions F (x) remain deterministic and close to the original ones f (x) in the uniform norm, i.e. max |F (x)−f (x)| is small, however x
the derivatives of F (x) and f (x) may be strongly different. Such situations arise naturally due to rounding errors in machine computations. In recent paper (Sauer (2005)) it was demonstrated that such errors change the behavior of long trajectories dramatically. Indeed for the classical logistic map f (x) = 4x(1 − x) there are no stable fixed points on [0, 1] under precise arithmetic, but 1
Corresponding author B.T.Polyak: [email protected]
rounding errors cause that 16% of initial points converge to zero. Everybody can check this fact on his computer; note that the number of iterations should be of order 106 . Another appearing effect is the existence of two stable orbits (of high period), and about 80% of the trajectories are attracted to them. We met the problem working with chaos stabilization by predictive control (Polyak & Maslov (2005); Polyak (2005)). The pioneering paper (Ott, Grebogi & Yorke (1990)) opened new direction of research related to chaos control, now it includes many monographs and survey papers; e.g., see Fradkov & Pogromsky (1998); Boccaletti, Grebogi & Lai (2000); Chen (2003). The main specification of the desired control is the low level of control efforts. Thus in the simplest case if the system has the form xn+1 = f (xn ) + un , xn ∈ R1 and one uses feedback control un = u(xn ) with u small, we are in the framework of the above
The paper is organized as follows. In Section 2 we analyze global behavior of iterations for f (x) being the tent map: f (x) = 1 − |2x − 1| and F (x) its piecewise linear approximation. The main technique originates to Kalman (1956), who proposed to use Markov chains for the analysis of nonlinear difference equations. On this way we manage to get analytical description of the stabilization time, which happens to be short enough. In Section 3 we consider the problem of stabilization of the unstable fixed point of the tent map by use of predictive control. We show that the problem is reduced to iterations with F (x) piecewise linear. Results of simulation confirm fast convergence effect. We conclude that the typical strategy in chaos control (going back to (Ott, Grebogi & Yorke (1990))) — to apply control only in the neighborhood of the desired orbit — can be ineffective if compared with permanent control action.
2. GLOBAL BEHAVIOR OF ITERATIONS FOR PERTURBED TENT MAP The basic function f (x) is the standard tent map: f (x) = 1 − |2x − 1|, x ∈ [0, 1]. It is well known that its iterations xn+1 = f (xn ) exhibit chaotic behavior (in particular, they are uniformly distributed on [0, 1] if x0 has this distribution), and the fixed point x∗ = 2/3 is unstable. We are interested in the behavior of iterations xn+1 = F (xn ), where F is slightly perturbed tent map as follows. We divide [0, 1] into N = 3·2k−1 − 1, k ≥ 2 equal intervals (with numbers 1, . . . , N ) and F (x) is piecewise linear in every interval, i.e. for x < 1/2, α ∈ [ 21 , 1], m x m m 1−α f( ) + , x∈[ , + ] N 1−α N N N F (x) = 1 − α m +1 2m + x m 1 f( )+ , x∈[ + , ] 2N α N N N
for m = 0, . . . , M, M = 3 · 2k−2 and F (x) = f (x) for x ∈ [(M + 1)/N, 0.5]. The function F (x) is symmetrically continued for x ∈ [0.5, 1] : F (1 − x) = F (x) except the interval that contains x∗ (its number is i∗ = 2k ). For this interval we take F (x) = 2k /N − (1 − α)x (see Fig. 1). 1 10/11 9/11 8/11 7/11 6/11
F(x)
setup with F (x) = f (x) + u(x). In the papers (Polyak & Maslov (2005); Polyak (2005)) u(x) was constructed by prediction of the trajectory starting at the point x, and if f (x) had an unstable periodic orbit the function F (x) had the same orbit stable. Thus it is the local attractor of the trajectories of xn+1 = F (xn ). The global behavior of iterations was not clear. One can expect that chaotic properties of the trajectories of an unperturbed system are preserved for the perturbed one, and trajectories will reach the attraction basin. However this basin is small enough and it can take much time to get it. Surprisingly, computer simulation demonstrated unexpectedly fast stabilization. The attempts to explain this phenomena were the motivation for the present research.
5/11 4/11 3/11 2/11 1/11 0
0
1/11
2/11
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x
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8/11
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1
Fig. 1. Mapping F for k = 3, N = 11 and α = 0.75 The graph of F (x) for α > 0.5 reminds a ladder, α defines the slope of the footsteps (for α = 0.5 F (x) coincides with f (x) except of i∗ -th interval). It is obvious that F (x) is indeed the perturbation of f (x): max |F (x)−f (x)| ≤ 2/N for all α. However 0≤x≤1
the fixed point X ∗ of F (x) is stable (because |F 0 (X ∗ )| < 1), and iterations remain in the i∗ th interval if they get there (it is the absorbing interval). To analyze the iterations xn+1 = F (xn ) we exploit the technique of Markov chains, originally proposed in Kalman (1956). Let the initial points x0 be uniformly distributed in [0, 1], then xn are uniformly distributed in any interval [m/N, (m + 1)/N ] (its number is m+1). The transition probability pij from the i-th to the j-th interval is given by Markov chain with the matrix
1−α 0 ... 0 P = 0 0 ... 0 1−α
α 0 0 0 1−α α ... ... ... 0 0 0 0 0 0 0 0 0 ... ... ... 0 1−α α α 0 0
... 0 0 0 ... 0 0 0 ... ... ... ... ... 1 − α α 0 ... 0 0 1 , ... 1 − α α 0 ... ... ... ... ... 0 0 0 ... 0 0 0
where pi,2i = α, pi,2i−1 = 1 − α, i = 1, . . . , M, pi,2N −2i = α, pi,2N −2i−1 = 1 − α, i = M + 1, . . . , N, i 6= i∗ , pM +1,N = 1, pi∗ ,i∗ = 1 while all other entries of P are zeros. An element (n) pij of the matrix P n represents the probability for the i-th interval to map to the j-th interval after n iterations.
For α = 1 the behavior of iterations xn+1 = F (xn ) is determinate, any interval maps to the other interval. Theorem 1. For α = 1 it takes no more than 2k−1 iterations to get to the the i∗ -th interval from any (2k−1) initial interval, i.e. pii∗ = 1, ∀i. Moreover, if s t i = 2 3 r, where s, t, r are integer and r does not have 2 and 3 among divisors, then ½ k−s t=0 (m) pii∗ = 1, where m = (1) 2k − s, t ≥ 1 Proof. Denote the i-th interval as an interval of (R) (m) rank R = R(i) such that pii∗ = 1, pii∗ = 0, m < R. Let us show that it is possible to specify the rank for every interval and it is given by (1). Indeed, R(2k ) = 0, R(2k−1 ) = 1, then for any interval i of rank R ≤ k there are two intervals that lead to it. Moreover, the numbers of these intervals are i/2 and 3 · 2k−1 − i/2. Note that these are the numbers of the form 2k−R r, where r is an odd indivisible by 3 and r < 3 · 2R (r = 1, 5, 7, 11, 13...). Thus there are 1 + 1 + 2 + . . . + 2R−1 = 2R intervals of rank R ≤ k. There is the only interval among these of rank k that is linked with the higher rank, it is the last interval with number i = N = 3 · 2k−1 − 1. The only way to get there is from the central interval M = 3 · 2k−2 of rank k + 1. Further the intervals of higher rank double again, the structure of the tree is the same as for R ≤ k, but all the numbers are divisible by 3. The total number of intervals of ranks k < R ≤ 2k − 1 is equal to 1 + 2 + . . . 2k−2 = 2k−1 − 1. Hence the number of all intervals of all ranks 0 ≤ R ≤ 2k − 1 is 2k + 2k−1 − 1 = N . Thus we specified ranks for all intervals, it means that there are no cycles and all the intervals are linked with the absorbing interval. The highest possible rank 2k − 1 corresponds to the longest path of 2k − 1 iterations. ¦ Example 1. For k = 3, N = 11 let us specify a rank of every interval. The procedure proposed in the proof of Theorem 1 is shown in Fig. 2. The lower circle correspond to the absorbing interval — the interval of rank 0. Numbers in the circles correspond to the interval number, a number beside is its rank. Note that every interval (except N -th) is linked with two intervals i/2 and N + 1 − i/2, i.e. the sum of the numbers is always N + 1. Now we proceed to the general case 0.5 < α < 1. It was shown recently Sauer (2005) that in finite arithmetics the iterations of the unperturbed tent map converge to zero rather quickly. However this effect takes place only for one particular α = 0.5 for the map F , disregarding this α we avoid undesirable convergence to zero in our
5 4 3 2 1 0
²¯ ²¯ 3 9 ±° ±° X » XXX»»» ²¯ 6 ±° » »»» ²¯ ²¯ ²¯ ²¯ 1 11 5 7 ±° ±° ±° ±° HH © H © © H © ²¯ ²¯ 2 10 ±° ±° P ³ PP ³ PP ³³ ³ P ³ ²¯ 4 ±° ²¯ 8 ±°
Fig. 2. Rank tree for k = 3, N = 11 experiments. Denote by si the mean number of iterations required to get from the i-th interval to the absorbing one i∗ . To estimate this quantity we apply the technique of recurrent relations which is used in the theory of random walks; e.g., see Feller (1950). Theorem 2. For 1/2 < α < 1 and all 1 ≤ i ≤ N si ≤
1 α2k−1 (1
− α)
.
(2)
Proof. Let qR = max si , 0 ≤ R ≤ K = 2k − 1. R(i)=R
For every interval of rank R 6= 0, R 6= k + 1 the transition probability to the interval of rank R−1 is α and transition probability to the nearby interval is 1 − α. The worst case is that the nearby interval has the highest possible rank K. Since qR ≤ qR+1 ≤ RK , ∀R, then we have qR ≤ αqR−1 +(1−α)qK +1, 1 ≤ R ≤ K, R 6= k+1, q0 = 0 for rank 0 and qk+1 = qk + 1. It is easy to check that the solutions of this recurrent inequality satisfy the estimate qR ≤ α2k−11(1−α) , 0 ≤ R ≤ K. ¦ Theorem 1 ensures that for α = 1 all the trajectories are finite, i.e. iterations xn+1 = F (xn ) terminate at the i∗ -th interval (“fixed point”) after no more than 2k−1 iterations, while the total number of intervals is exponentially greater: N = 3·2k−1 − 1. For α < 1 the average number of iterations to achieve the fixed point is s = O(α−2k+1 ) for all initial points. It is interesting to compare this result with the case α = 0.5, then F (x) coincides with f (x) everywhere except the i∗ -th interval. For this case the number of steps to achieve this interval (averaged over initial points on [0, 1]) is O(N ). But s/N = O(q k ), q = 1/2α2 < 1 for α being close enough to 1. Thus we conclude that convergence to the fixed point is much faster if we perturb the original function everywhere, not only in the neighborhood of the fixed point.
Results of numerical simulation are given in Table 1, where the average number of iterations to achieve the absorbing interval is presented Table 1. Numerical results
0.501
0.8
0.9
k
si
5 10 20 5 10 20 5 10 20
65 2335 2373906 13 56 615 8 24 104
1 0.9375 0.875 0.8125 0.75 0.6875 0.625 0.5625
F(x)
α
f (fm−1 (x)). Then F (x) is a piecewise linear function on [0, 1]: the unit interval of the x-axis is divided into 2p+2 equal intervals and the function F (x) is linear in every interval; e.g., Fig. 3 depicts F (x) for p = 2.
0.5 0.4375 0.375 0.3125 0.25 0.1875
We see that the convergence for α close to 1 is indeed much faster than for α = 0.501 (i.e. nearly for unperturbed tent map). Note that the construction of the perturbation F (x) is very sensitive. If one changes it, new features may happen. For instance if we take N even, some cycles can arise for α = 1 and the global convergence is lacking. An example is 8 × 8 matrix P with entries p12 = p24 = p36 = p48 = p58 = p66 = p74 = p82 = 1, pij = 0 otherwise. Then there is the cycle 2 → 4 → 8 → 2 and iterations starting in the intervals 1, 2, 4, 5, 7, 8 do not converge to the absorbing interval 6.
0.125 0.0625 0
0
0.125
0.25
0.375
0.5 x
0.625
0.75
0.875
1
Fig. 3. Mapping F for p = 2 The function F (x) reminds the function analyzed in Section 2, (compare Fig. 1 and Fig. 3). Thus we can expect that the same effect of fast global convergence holds. Indeed, the results of simulation confirm this conclusion. The mean numbers of iterations needed for the stabilization of the fixed point, 2-cycle (µ = −4) and 3-cycle (µ = −8) are given in Table 2. Table 2. Mean stabilization time
3. APPLICATIONS TO CHAOS CONTROL Recently (Polyak & Maslov (2005); Polyak (2005)) a new approach to stabilization of unstable periodic orbits of multidimensional maps has been proposed. It is based on the prediction of the trajectory and the use of difference of two predictions for control synthesis. By increasing the prediction horizon control level can be made as small as we like. Local stability has been proved in these publications: if a trajectory happens to be close enough to the cycle to be stabilized, then under control it is attracted to the cycle. Numerical simulation demonstrated very fast global stabilization for numerous examples; the origins of this effect were not clear. Below we try to explain them for the case of the tent map. Consider the system xn+1 = F (xn ) = f (xn ) + u(xn ), where f (x) = 1 − |2x − 1|, x ∈ R1 is the tent map. We use prediction control proposed in (Polyak & Maslov (2005); Polyak (2005)) for the stabilization of the unstable fixed point x∗ = 2/3 of f (x), for this particular case it reads as: u(x) = ε(fp+2 (x) − fp+1 (x)), ε =
1 , − 1)
µp (µ
where µ = f 0 (x∗ ) = −2 is the multiplicator and p ≥ 0 is a prediction horizon. Notation fm denotes m-th iteration of f , i.e. f1 (x) = f (x), fm (x) =
p
si for the fixed point
si for 2-cycle
si for 3-cycle
2 5 10 12 15 20 30
7 10 12 13 16 23 35
8 15 25 28 36 44
12 21 37 42 57
Of course the model and results of Section 2 is not the rigorous validation of the fast global convergence effect in stabilizing chaos by predictive control. Nevertheless it provides some possible explanation of the phenomena. REFERENCES Boccaletti, S., Grebogi, C., & Lai, Y.-C. (2000). The control of chaos: theory and applications, Phys. Rep., 329, 103–197. Chen, G., Yu, X. (eds) (2003). Chaos control, Lect. Notes Contr. Inf. Sci., No. 292. Elaydi, S.N. (2000). Discrete chaos, Chapman & Hall/CRC Press, Boca Raton. Feller, W. (1950). An introduction to probability theory and its applications, Vol. I, New York John Wiley & Sons. Fradkov, A.L., & Pogromsky, A.Yu. (1998). Introduction to control of oscillations and chaos, Singapore: World Scientific.
Mira, Ch. (1987). Chaotic dynamics, Singapore: World Scientific. Ott, E., Grebogi, C., & Yorke, J.A. (1990). Controlling chaos, Phys. Rev. Lett., 64, 1196–1199. Polyak, B.T. (2005). Stabilization of chaos by prediction control, Autom. and Remote Control, 66, No.11, 1791–1804. Polyak, B.T., & Maslov, V.P. (2005). Controlling chaos by predictive control, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic. Kalman, R.E. (1956). Nonlinear aspects of sampled-data control systems, Proceedings of the symposium on nonlinear circuit analysis, VI, April 25,26,27; 273-313. Sauer, T. (2005). Computer arithmetic and sensitivity of natural measure, Journal of Difference Equations and Applications, 11, No.7, 669-676.
2
Institute for Control Science RAS, Moscow, Russia Moscow Physical-Technical Institute (State University), Moscow, Russia e-mail [email protected], [email protected]
Abstract: Iterations of 1D simple maps such as logistic, tent, cubic ones are very well studied. However perturbed versions of these maps (close in uniform norm but with strongly varying derivatives) can exhibit completely different behavior. We encounter such situation when dealing with chaos stabilization via small control. In this paper we present analytical investigation of this effect for one particular case — piecewise linear perturbation of the tent map. Surprisingly, iterations of this map converge to the unique fixed point very fast for all initial points. The result is in sharp contrast with iterations of the original tent map but explains fast stabilization of unstable periodic orbits by predictive control, proposed in Polyak & Maslov (2005); Polyak (2005). Keywords: Chaos, nonlinear maps, iterations, Markov chains, stabilization, predictive control.
1. INTRODUCTION Effects of bifurcations, chaotic behavior, random properties of iterations of simple one-dimensional maps are very well studied; e.g., see monographs Mira (1987); Elaydi (2000). The problem of interest is if all these effects are preserved under small perturbations of the maps. ”Small” means that the perturbed functions F (x) remain deterministic and close to the original ones f (x) in the uniform norm, i.e. max |F (x)−f (x)| is small, however x
the derivatives of F (x) and f (x) may be strongly different. Such situations arise naturally due to rounding errors in machine computations. In recent paper (Sauer (2005)) it was demonstrated that such errors change the behavior of long trajectories dramatically. Indeed for the classical logistic map f (x) = 4x(1 − x) there are no stable fixed points on [0, 1] under precise arithmetic, but 1
Corresponding author B.T.Polyak: [email protected]
rounding errors cause that 16% of initial points converge to zero. Everybody can check this fact on his computer; note that the number of iterations should be of order 106 . Another appearing effect is the existence of two stable orbits (of high period), and about 80% of the trajectories are attracted to them. We met the problem working with chaos stabilization by predictive control (Polyak & Maslov (2005); Polyak (2005)). The pioneering paper (Ott, Grebogi & Yorke (1990)) opened new direction of research related to chaos control, now it includes many monographs and survey papers; e.g., see Fradkov & Pogromsky (1998); Boccaletti, Grebogi & Lai (2000); Chen (2003). The main specification of the desired control is the low level of control efforts. Thus in the simplest case if the system has the form xn+1 = f (xn ) + un , xn ∈ R1 and one uses feedback control un = u(xn ) with u small, we are in the framework of the above
The paper is organized as follows. In Section 2 we analyze global behavior of iterations for f (x) being the tent map: f (x) = 1 − |2x − 1| and F (x) its piecewise linear approximation. The main technique originates to Kalman (1956), who proposed to use Markov chains for the analysis of nonlinear difference equations. On this way we manage to get analytical description of the stabilization time, which happens to be short enough. In Section 3 we consider the problem of stabilization of the unstable fixed point of the tent map by use of predictive control. We show that the problem is reduced to iterations with F (x) piecewise linear. Results of simulation confirm fast convergence effect. We conclude that the typical strategy in chaos control (going back to (Ott, Grebogi & Yorke (1990))) — to apply control only in the neighborhood of the desired orbit — can be ineffective if compared with permanent control action.
2. GLOBAL BEHAVIOR OF ITERATIONS FOR PERTURBED TENT MAP The basic function f (x) is the standard tent map: f (x) = 1 − |2x − 1|, x ∈ [0, 1]. It is well known that its iterations xn+1 = f (xn ) exhibit chaotic behavior (in particular, they are uniformly distributed on [0, 1] if x0 has this distribution), and the fixed point x∗ = 2/3 is unstable. We are interested in the behavior of iterations xn+1 = F (xn ), where F is slightly perturbed tent map as follows. We divide [0, 1] into N = 3·2k−1 − 1, k ≥ 2 equal intervals (with numbers 1, . . . , N ) and F (x) is piecewise linear in every interval, i.e. for x < 1/2, α ∈ [ 21 , 1], m x m m 1−α f( ) + , x∈[ , + ] N 1−α N N N F (x) = 1 − α m +1 2m + x m 1 f( )+ , x∈[ + , ] 2N α N N N
for m = 0, . . . , M, M = 3 · 2k−2 and F (x) = f (x) for x ∈ [(M + 1)/N, 0.5]. The function F (x) is symmetrically continued for x ∈ [0.5, 1] : F (1 − x) = F (x) except the interval that contains x∗ (its number is i∗ = 2k ). For this interval we take F (x) = 2k /N − (1 − α)x (see Fig. 1). 1 10/11 9/11 8/11 7/11 6/11
F(x)
setup with F (x) = f (x) + u(x). In the papers (Polyak & Maslov (2005); Polyak (2005)) u(x) was constructed by prediction of the trajectory starting at the point x, and if f (x) had an unstable periodic orbit the function F (x) had the same orbit stable. Thus it is the local attractor of the trajectories of xn+1 = F (xn ). The global behavior of iterations was not clear. One can expect that chaotic properties of the trajectories of an unperturbed system are preserved for the perturbed one, and trajectories will reach the attraction basin. However this basin is small enough and it can take much time to get it. Surprisingly, computer simulation demonstrated unexpectedly fast stabilization. The attempts to explain this phenomena were the motivation for the present research.
5/11 4/11 3/11 2/11 1/11 0
0
1/11
2/11
3/11
4/11
5/11
x
6/11
7/11
8/11
9/11
10/11
1
Fig. 1. Mapping F for k = 3, N = 11 and α = 0.75 The graph of F (x) for α > 0.5 reminds a ladder, α defines the slope of the footsteps (for α = 0.5 F (x) coincides with f (x) except of i∗ -th interval). It is obvious that F (x) is indeed the perturbation of f (x): max |F (x)−f (x)| ≤ 2/N for all α. However 0≤x≤1
the fixed point X ∗ of F (x) is stable (because |F 0 (X ∗ )| < 1), and iterations remain in the i∗ th interval if they get there (it is the absorbing interval). To analyze the iterations xn+1 = F (xn ) we exploit the technique of Markov chains, originally proposed in Kalman (1956). Let the initial points x0 be uniformly distributed in [0, 1], then xn are uniformly distributed in any interval [m/N, (m + 1)/N ] (its number is m+1). The transition probability pij from the i-th to the j-th interval is given by Markov chain with the matrix
1−α 0 ... 0 P = 0 0 ... 0 1−α
α 0 0 0 1−α α ... ... ... 0 0 0 0 0 0 0 0 0 ... ... ... 0 1−α α α 0 0
... 0 0 0 ... 0 0 0 ... ... ... ... ... 1 − α α 0 ... 0 0 1 , ... 1 − α α 0 ... ... ... ... ... 0 0 0 ... 0 0 0
where pi,2i = α, pi,2i−1 = 1 − α, i = 1, . . . , M, pi,2N −2i = α, pi,2N −2i−1 = 1 − α, i = M + 1, . . . , N, i 6= i∗ , pM +1,N = 1, pi∗ ,i∗ = 1 while all other entries of P are zeros. An element (n) pij of the matrix P n represents the probability for the i-th interval to map to the j-th interval after n iterations.
For α = 1 the behavior of iterations xn+1 = F (xn ) is determinate, any interval maps to the other interval. Theorem 1. For α = 1 it takes no more than 2k−1 iterations to get to the the i∗ -th interval from any (2k−1) initial interval, i.e. pii∗ = 1, ∀i. Moreover, if s t i = 2 3 r, where s, t, r are integer and r does not have 2 and 3 among divisors, then ½ k−s t=0 (m) pii∗ = 1, where m = (1) 2k − s, t ≥ 1 Proof. Denote the i-th interval as an interval of (R) (m) rank R = R(i) such that pii∗ = 1, pii∗ = 0, m < R. Let us show that it is possible to specify the rank for every interval and it is given by (1). Indeed, R(2k ) = 0, R(2k−1 ) = 1, then for any interval i of rank R ≤ k there are two intervals that lead to it. Moreover, the numbers of these intervals are i/2 and 3 · 2k−1 − i/2. Note that these are the numbers of the form 2k−R r, where r is an odd indivisible by 3 and r < 3 · 2R (r = 1, 5, 7, 11, 13...). Thus there are 1 + 1 + 2 + . . . + 2R−1 = 2R intervals of rank R ≤ k. There is the only interval among these of rank k that is linked with the higher rank, it is the last interval with number i = N = 3 · 2k−1 − 1. The only way to get there is from the central interval M = 3 · 2k−2 of rank k + 1. Further the intervals of higher rank double again, the structure of the tree is the same as for R ≤ k, but all the numbers are divisible by 3. The total number of intervals of ranks k < R ≤ 2k − 1 is equal to 1 + 2 + . . . 2k−2 = 2k−1 − 1. Hence the number of all intervals of all ranks 0 ≤ R ≤ 2k − 1 is 2k + 2k−1 − 1 = N . Thus we specified ranks for all intervals, it means that there are no cycles and all the intervals are linked with the absorbing interval. The highest possible rank 2k − 1 corresponds to the longest path of 2k − 1 iterations. ¦ Example 1. For k = 3, N = 11 let us specify a rank of every interval. The procedure proposed in the proof of Theorem 1 is shown in Fig. 2. The lower circle correspond to the absorbing interval — the interval of rank 0. Numbers in the circles correspond to the interval number, a number beside is its rank. Note that every interval (except N -th) is linked with two intervals i/2 and N + 1 − i/2, i.e. the sum of the numbers is always N + 1. Now we proceed to the general case 0.5 < α < 1. It was shown recently Sauer (2005) that in finite arithmetics the iterations of the unperturbed tent map converge to zero rather quickly. However this effect takes place only for one particular α = 0.5 for the map F , disregarding this α we avoid undesirable convergence to zero in our
5 4 3 2 1 0
²¯ ²¯ 3 9 ±° ±° X » XXX»»» ²¯ 6 ±° » »»» ²¯ ²¯ ²¯ ²¯ 1 11 5 7 ±° ±° ±° ±° HH © H © © H © ²¯ ²¯ 2 10 ±° ±° P ³ PP ³ PP ³³ ³ P ³ ²¯ 4 ±° ²¯ 8 ±°
Fig. 2. Rank tree for k = 3, N = 11 experiments. Denote by si the mean number of iterations required to get from the i-th interval to the absorbing one i∗ . To estimate this quantity we apply the technique of recurrent relations which is used in the theory of random walks; e.g., see Feller (1950). Theorem 2. For 1/2 < α < 1 and all 1 ≤ i ≤ N si ≤
1 α2k−1 (1
− α)
.
(2)
Proof. Let qR = max si , 0 ≤ R ≤ K = 2k − 1. R(i)=R
For every interval of rank R 6= 0, R 6= k + 1 the transition probability to the interval of rank R−1 is α and transition probability to the nearby interval is 1 − α. The worst case is that the nearby interval has the highest possible rank K. Since qR ≤ qR+1 ≤ RK , ∀R, then we have qR ≤ αqR−1 +(1−α)qK +1, 1 ≤ R ≤ K, R 6= k+1, q0 = 0 for rank 0 and qk+1 = qk + 1. It is easy to check that the solutions of this recurrent inequality satisfy the estimate qR ≤ α2k−11(1−α) , 0 ≤ R ≤ K. ¦ Theorem 1 ensures that for α = 1 all the trajectories are finite, i.e. iterations xn+1 = F (xn ) terminate at the i∗ -th interval (“fixed point”) after no more than 2k−1 iterations, while the total number of intervals is exponentially greater: N = 3·2k−1 − 1. For α < 1 the average number of iterations to achieve the fixed point is s = O(α−2k+1 ) for all initial points. It is interesting to compare this result with the case α = 0.5, then F (x) coincides with f (x) everywhere except the i∗ -th interval. For this case the number of steps to achieve this interval (averaged over initial points on [0, 1]) is O(N ). But s/N = O(q k ), q = 1/2α2 < 1 for α being close enough to 1. Thus we conclude that convergence to the fixed point is much faster if we perturb the original function everywhere, not only in the neighborhood of the fixed point.
Results of numerical simulation are given in Table 1, where the average number of iterations to achieve the absorbing interval is presented Table 1. Numerical results
0.501
0.8
0.9
k
si
5 10 20 5 10 20 5 10 20
65 2335 2373906 13 56 615 8 24 104
1 0.9375 0.875 0.8125 0.75 0.6875 0.625 0.5625
F(x)
α
f (fm−1 (x)). Then F (x) is a piecewise linear function on [0, 1]: the unit interval of the x-axis is divided into 2p+2 equal intervals and the function F (x) is linear in every interval; e.g., Fig. 3 depicts F (x) for p = 2.
0.5 0.4375 0.375 0.3125 0.25 0.1875
We see that the convergence for α close to 1 is indeed much faster than for α = 0.501 (i.e. nearly for unperturbed tent map). Note that the construction of the perturbation F (x) is very sensitive. If one changes it, new features may happen. For instance if we take N even, some cycles can arise for α = 1 and the global convergence is lacking. An example is 8 × 8 matrix P with entries p12 = p24 = p36 = p48 = p58 = p66 = p74 = p82 = 1, pij = 0 otherwise. Then there is the cycle 2 → 4 → 8 → 2 and iterations starting in the intervals 1, 2, 4, 5, 7, 8 do not converge to the absorbing interval 6.
0.125 0.0625 0
0
0.125
0.25
0.375
0.5 x
0.625
0.75
0.875
1
Fig. 3. Mapping F for p = 2 The function F (x) reminds the function analyzed in Section 2, (compare Fig. 1 and Fig. 3). Thus we can expect that the same effect of fast global convergence holds. Indeed, the results of simulation confirm this conclusion. The mean numbers of iterations needed for the stabilization of the fixed point, 2-cycle (µ = −4) and 3-cycle (µ = −8) are given in Table 2. Table 2. Mean stabilization time
3. APPLICATIONS TO CHAOS CONTROL Recently (Polyak & Maslov (2005); Polyak (2005)) a new approach to stabilization of unstable periodic orbits of multidimensional maps has been proposed. It is based on the prediction of the trajectory and the use of difference of two predictions for control synthesis. By increasing the prediction horizon control level can be made as small as we like. Local stability has been proved in these publications: if a trajectory happens to be close enough to the cycle to be stabilized, then under control it is attracted to the cycle. Numerical simulation demonstrated very fast global stabilization for numerous examples; the origins of this effect were not clear. Below we try to explain them for the case of the tent map. Consider the system xn+1 = F (xn ) = f (xn ) + u(xn ), where f (x) = 1 − |2x − 1|, x ∈ R1 is the tent map. We use prediction control proposed in (Polyak & Maslov (2005); Polyak (2005)) for the stabilization of the unstable fixed point x∗ = 2/3 of f (x), for this particular case it reads as: u(x) = ε(fp+2 (x) − fp+1 (x)), ε =
1 , − 1)
µp (µ
where µ = f 0 (x∗ ) = −2 is the multiplicator and p ≥ 0 is a prediction horizon. Notation fm denotes m-th iteration of f , i.e. f1 (x) = f (x), fm (x) =
p
si for the fixed point
si for 2-cycle
si for 3-cycle
2 5 10 12 15 20 30
7 10 12 13 16 23 35
8 15 25 28 36 44
12 21 37 42 57
Of course the model and results of Section 2 is not the rigorous validation of the fast global convergence effect in stabilizing chaos by predictive control. Nevertheless it provides some possible explanation of the phenomena. REFERENCES Boccaletti, S., Grebogi, C., & Lai, Y.-C. (2000). The control of chaos: theory and applications, Phys. Rep., 329, 103–197. Chen, G., Yu, X. (eds) (2003). Chaos control, Lect. Notes Contr. Inf. Sci., No. 292. Elaydi, S.N. (2000). Discrete chaos, Chapman & Hall/CRC Press, Boca Raton. Feller, W. (1950). An introduction to probability theory and its applications, Vol. I, New York John Wiley & Sons. Fradkov, A.L., & Pogromsky, A.Yu. (1998). Introduction to control of oscillations and chaos, Singapore: World Scientific.
Mira, Ch. (1987). Chaotic dynamics, Singapore: World Scientific. Ott, E., Grebogi, C., & Yorke, J.A. (1990). Controlling chaos, Phys. Rev. Lett., 64, 1196–1199. Polyak, B.T. (2005). Stabilization of chaos by prediction control, Autom. and Remote Control, 66, No.11, 1791–1804. Polyak, B.T., & Maslov, V.P. (2005). Controlling chaos by predictive control, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic. Kalman, R.E. (1956). Nonlinear aspects of sampled-data control systems, Proceedings of the symposium on nonlinear circuit analysis, VI, April 25,26,27; 273-313. Sauer, T. (2005). Computer arithmetic and sensitivity of natural measure, Journal of Difference Equations and Applications, 11, No.7, 669-676.