550 XU Hai-Bo, WANG Guang-Rui, and CHEN Shi-Gang Vol. 37 the denition of the domain. The map is a generalization of the standard map for which (J) = J

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1 Commun. Theor. Phys. (Beijing, China) 37 (2002) pp 549{556 c International Academic Publishers Vol. 37, No. 5, May 15, 2002 Controlling Strong Chaos by an Aperiodic Perturbation in Area Preserving Maps XU Hai-Bo, WANG Guang-Rui, and CHEN Shi-Gang Institute of Applied Physics and Computational Mathematics, Beijing , China (Received October 8, 2001) Abstract We demonstrate a method for controlling strong chaos by an aperiodic perturbation in two-dimensional Hamiltonian systems. The method has the advantages that the controlled system remains conservative property and the selection of the perturbation has a considerable diversity. We illustrate this method with two area preserving maps: the non-monotonic twist map which is a mixed system and the perturbed cat map which exhibits hard chaos. Numerical results show that the strong chaos can be eectively controlled into regular motions, and the nal states are always quasiperiodic ones. The method is robust against the presence of weak external noise. PACS numbers: a, b Key words: controlling chaos, area preserving map, regular motion 1 Introduction A wide variety of methods have been proposed for controlling chaos in nonlinear dynamical systems since the pioneering work of Ott, Grebogi and Yorke (OGY) in [1] However, a large number of works so far have concentrated on dissipative systems. [2;6] For conservative systems, controlling chaos is more dicult because there are not chaotic attractors and the search for chaotic behavior stretches to large areas of the phase space. The initial conditions are special controlling parameters and have important roles in describing chaotic behavior. So only a few methods [7;9] have been found, which are either the extensions of the OGY's method or validity for the special systems. Obviously, a general way is lack for controlling Hamiltonian chaos. In particular, no work has been contributed to control strong chaos by a perturbation in Hamiltonian systems, which does not change the conservative property of the system. In this paper, we demonstrate a method for controlling strong chaos by an aperiodic perturbation in two-dimensional Hamiltonian systems. Especially, by making use of this method the strong chaos can be controlled in highly chaotic systems, even in hard chaotic systems where no small stable regions exist. The method has the advantages that the controlled system remains conservative property and the selection of the perturbation has a considerable diversity. Our goal is to stabilize the strongly chaotic motions in phase space into regular motions by a small perturbation. Usually for controlling chaos one means a process achieved by a small perturbation less than a few percent. We also expect here that the intensity of the perturbation is as small as possible. However, the results show that controlling a highly chaotic system may require a moderately large perturbation. A fundamental problem of Hamiltonian dynamics is to understand the behavior of an integrable Hamiltonian system subject to perturbation. The Hamiltonian system can be studied with area preserving maps of the form [10] Ji+1 = Ji + f(i Ji+1) i+1 = i + (Ji+1) + g(i Ji+1) (1) where the area preserving = 0 : In this paper, we study the controlling chaos in two area preserving maps: the non-monotonic twist map which is a mixed system and the perturbed cat map which exhibits hard chaos. In order to distinguish the perturbation or the force of the periodically driven Hamiltonian system and the perturbation for controlling chaos, we call the latter a controlling signal. An outline of the rest of the paper is as follows. Numerical results for the non-monotonic twist map and the perturbed cat map are presented in Secs 2 and 3, respectively. In Sec. 4 we briey study the eect of external noise. Finally, the main results are discussed and concluded in Sec Controlling Chaos in the Non-monotonic Twist Map We choose the non-monotonic twist map introduced by Howard and Hohs [11] in order to model the behavior of particles in an accelerator device. The non-monotonic twist map is dened by Jn+1 = Jn ; k 2 sin(2 n) n+1 = n + Jn+1 ; J 2 n+1 (3) where k and are real positive numbers, the domain of the interest is D := f(j )jj 2 [;1 1] and 2 [;1 1]g. If jjj > 1 or jj > 1, we take the fractional part of them in the [;1 1] region. In this paper, we write \mod 0 1" for The project supported by the Special Funds for Major State Basic Research Projects, National Natural Science Foundation of China under Grant Nos and , and Science Foundation of China Academy of Engineering Physics under Grant No

2 550 XU Hai-Bo, WANG Guang-Rui, and CHEN Shi-Gang Vol. 37 the denition of the domain. The map is a generalization of the standard map for which (J) = J. n+1 = n + Jn+1 ; J 2 n+1 (mod0 1) : (4) We select three types of controlling signals which represent aperiodic perturbations of linear function, odd and even nonlinear functions, respectively. The controlling signals are given as p() = 8 >< >: ; 2 sinh(2 n) ; 2 cosh(2 n) ;n (5) where is the perturbation parameter. The controlling signals do not change the conservative property of the system. Fig. 1 The evolution of J as a function of n for different controlling signals with k = 1:5 and the same initial value. (a) p() = ;(=2) sinh(2n) = 0:005 (b) p() = ;(=2) cosh(2n) = 0:005 (c) p() = ;n = 0:08 (d) p() = ;(=2) sinh(2n) = 0:005, and the domain is D :=f(j )jj 2 (;1 1) and 2 [;1 1] (mod 0 1)g. The controlling signal is switched on at n = For conservative system, no trajectory is attractive. The system starting from an arbitrary initial point in the phase space will soon or later return to the region innitely close to the initial point. In order to stabilize the chaotic motion into a regular one and not change the conservative property of the system in the meantime, we act a proper controlling signal on the system in the following form Jn+1 = Jn ; k 2 sin(2 n) + p() (mod 0 1) Fig. 2 The changes of the controlling signal p() ((a) (d)) and the force f() ((e) (h)) of the periodically driven system as functions of n corresponding to the cases of Figs 1(a) 1(d), respectively. Numerical results show that the strong chaos can be controlled into regular motions, and the nal states are

3 No. 5 Controlling Strong Chaos by an Aperiodic Perturbation in Area Preserving Maps 551 always quasi-periodic ones. Figure 1 presents the evolution of J as a function of iteration number n with k = 1:5 and the same initial value. The controlling signal is switched on at n = Figures 1(a) 1(c) correspond to three dierent controlling signals described in Eq. (5) with dierent perturbation parameters, where the perturbation parameters have been carefully chosen in order to avoid spurious behavior. Figure 1(d) corresponds to Fig. 1(a), but the domain is D :=f(j )jj 2 (;1 1) and 2 [;1 1](mod 0 1)g. For the domain J 2 (;1 1), if we select ;(=2)cosh(2n) to be the controlling signal, it may cause J to increase monotonically and may not stabilize the system. In this case, we should better select an odd function to be the controlling signal. The relaxation times for stabilizing the system are rather short, and this is of some importance in practice. Fig. 3 Typical orbits with k = 1:5 after 10 4 iterations. (a) = 0 (b) p() = ;(=2) sinh(2n) = 0:005 (c) p() = ;(=2) cosh(2n) = 0:005 (d) p() = ;n = 0:08 (e) = 0 J 2 (;1 1) (f) p() = ;(=2) sinh(2n) = 0:005 J 2 (;1 1). Now we compare the intensity of the controlling signal p() and that of the nonlinear force f() of the periodically driven system, f(n) = ; k 2 sin(2 n) : (6) Figure 2 shows that the changes of the controlling signal p() and the force f() of the periodically driven system as functions of n corresponding to the cases of Figs 1(a) 1(d), respectively. Figures 2(a) 2(d) present p() vs. n, and 2(e) 2(h) present f() vs. n corresponding to Figs 2(a) 2(d). Generally, the controlling signal p() keeps on an oscillation with an amplitude which is incomparably smaller than that of the nonlinear force f() after large jumps in a few steps. Once p() exhibits stable oscillation, the controlled system can be constrained into a local quasiperiodic one. Figures 3(a) and 3(e) show some typical chaotic or-

4 552 XU Hai-Bo, WANG Guang-Rui, and CHEN Shi-Gang Vol. 37 bits for J 2 [;1 1] (mod 0 1) and J 2 (;1 1) without the controlling signal, respectively. After the controlling signal is acted on them, they all turn into quasiperiodic motions in the limited region. The results are shown in Figs 3(b), 3(c), 3(d) and 3(f). Numerical simulations show that we can perfectly stabilize strong chaos to localized regular motions from arbitrary initial conditions by applying a simple controlling signal described in Eq. (5) with a small controlling signal. It is interesting to see that the controlling signal always removes the system to the island near the point (0 0). In this region a small perturbation is enough to overcome the instability or to keep stability. 3 Controlling Chaos in Perturbed Cat Map To check the generality of this method we turn now to consider a highly chaotic system, the perturbed cat map, [12] which is an area-preserving map from the unit torus to itself, Jn+1 = Jn + n + k 2 [cos(2 n) ; cos(4n)] (mod 0 1) n+1 = n + Jn+1 (mod 0 1) : (7) At k = 0 the map, known as Arnold's cat map, is highly chaotic. Namely, time correlation functions decay faster than exponential. As a result, at nite values of k, relaxation is exponential. It has also been proved that for k < 0:11, all the xed points of Eq. (7) are in one-to-one correspondence with the original cat map. Consequently, all orbits are unstable and the system can be characterized as \hard chaotic". Fig. 4 The evolution of J as a function of n for different controlling signals with the same initial value. (a) k = 1:5, p() = ;(=2) sinh(2n) = 0:1 (b) k = 1:5, p() = ;(=2) cosh(2n) = 0:048 (c) k = 1:5, p() = ;n = 2:3 (d) k = 0, p() = ;(=2) sinh(2n) = 0:1. The controlling signal is switched on at n = Fig. 5 The changes of the controlling signal p() ((a) (d)) and the force f() ((e) (h)) of the periodically driven system as functions of n corresponding to the cases of Figs 1(a) 1(d), respectively.

5 No. 5 Controlling Strong Chaos by an Aperiodic Perturbation in Area Preserving Maps 553 Applying the same procedure as described in Sec. 2, we act three types of controlling signals on the perturbed cat map. Numerical simulations show that the strong chaos can be eectively controlled into the regular motion by deliberate selection of the perturbation parameter, and the nal states are always quasiperiodic ones. Figure 4 presents the evolution of J as a function of iteration number n with the same initial condition. Figures 4(a) 4(c) correspond to three dierent controlling signals described in Eq. (5) with k = 1:5 and dierent perturbation parameters. Figure 4(d) corresponds to Fig. 4(a) with k = 0 (i.e., the Arnold's cat map). The controlling signal is switched on at n = The relaxation times for stabilizing the system are rather short, and this is of some importance in practice. Fig. 6 Typical orbits after 10 4 iterations. (a) k = 1:5 = 0 (b) k = 1:5, p() = ;(=2) sinh(2n) = 0:1 (c) k = 1:5, p() = ;(=2) cosh(2n) = 0:048 (d) k = 1:5, p() = ;n = 2:3 (e) k = 0 = 0 (f) k = 0, p() = ;(=2) sinh(2n) = 0:1. Now we compare the intensity of the controlling signal p() and that of the nonlinear force f() of the periodically driven system, f(n) = n + k 2 [cos(2 n) ; cos(4n)] : (8) Figure 5 shows that the changes of the controlling signal p() and the nonlinear force f() of the periodically driven system as functions of n corresponding to the cases of Figs 4(a) 4(d), respectively. Figures 4(a) 4(d) present p() vs. n, and 4(e) 4(h) present f() vs. n corresponding to Figs 4(a) 4(d). The controlling signal p() keeps on an oscillation after large jumps in a few steps. However, the amplitude is of the same order of magnitude as that of

6 554 XU Hai-Bo, WANG Guang-Rui, and CHEN Shi-Gang Vol. 37 f(), which is dierent from that in the non-monotonic twist map. Once p() exhibits stable oscillation, the controlled system can be constrained into a local quasiperiodic one. Figures 6(a) and 6(e) show some typical chaotic orbits for k = 1:5 and k = 0 without the controlling signal, respectively. After the controlling signal is acted on them, they all turn into quasiperiodic motions in the limited region. The results are shown in Figs 6(b), 6(c), 6(d) and 6(f). Numerical simulations show that we can perfectly stabilize strong chaos to localized regular motions from arbitrary initial conditions by applying a simple controlling signal described in Eq. (5). The perturbed cat map is a highly chaotic system. Especially for Arnold's cat map, there are no stable regions. Indeed, the above limited regions are not inside the stable regions of the original system in these systems. 4 Eect of Noise An important issue whether a control method is useful in experiments is its robustness against the application of external noise. In this section, we consider the Gaussian white noise generated by using the Box{Muller method, [13] and introduce additive noise in the form where denotes the intensity of external noise, J 0 n = Jn + n 0 n = n + n (9) hni = hni = 0 hnn 0i = 0 h nn 0i = h nn 0i = nn 0 : (10) Fig. 7 The eect of noise for the nonmonotonic twist map. (a) (d) correspond to Figs 3(b), 3(c), 3(d) and 3(f) respectively. The intensity of the perturbation is 1:0 10 ;4. It is to be noticed that additive noise does not change the conservative property of the system, but it is not the case for multiplicative noise. Figure 7 shows the result for the non-monotonic twist map by considering the presence of additive noise. Figures 7(a) 7(d) correspond to Figs 3(b), 3(c), 3(d) and 3(f), respectively. The intensity of noise acting on the system variables J and is = 1:0 10 ;4. Figure 8 shows the result for the perturbed cat map by considering the presence of additive noise. Figure 8(a) corresponds to Fig. 6(b) with = 5:0 10 ;5 Figure 8(b) corresponds to Fig. 6(c) with = 2:0 10 ;5 Figure 8(c) corresponds to Fig. 6(d) with = 2:0 10 ;5 Figure 8(d) corresponds to Fig. 6(f) with = 1:0 10 ;4. Comparing Fig. 7 with Fig. 3 and Fig. 8 with Fig. 6, respectively,

7 No. 5 Controlling Strong Chaos by an Aperiodic Perturbation in Area Preserving Maps 555 we observe that the noisy orbits are neither the regular structure of island chains nor the strongly chaotic structure, but they remain within a small neighborhood of the noise-free orbits and do not wander over the whole phase space. Therefore we conclude that the method is robust against weak external noise. Fig. 8 The eect of noise for the perturbed cat map. (a) (d) correspond to Figs 6(b), 6(c), 6(d) and 6(f) respectively. The intensities of the perturbations are (a) = 5:0 10 ;5 (b) = 2:0 10 ;5 (c) = 2:0 10 ;5 (d) = 1:0 10 ;4. 5 Discussions and Conclusions In this paper, we demonstrate a method for controlling strong chaos by an aperiodic perturbation in two area preserving maps: the non-monotonic twist map which is a mixed system and the perturbed cat map which exhibits hard chaos. The method has the advantages that the controlled system remains conservative property and the selection of controlling signal has a considerable diversity. Provided that the controlling signal p() depends only on, it does not change the conservative property of the Hamiltonian system. We select three types of controlling signals which represent aperiodic perturbations of linear function, odd and even nonlinear functions, respectively. Numerical results show that our method is quite eective. The perturbation parameter has an important role in controlling strong chaos. If we choose a proper perturbation parameter, the strong chaos for any Hamiltonian system can be stabilized to localized regular motions from arbitrary initial conditions. Furthermore the method is robust against the presence of weak external noise. The perturbation may be extremely small in most chaotic systems that have some small stable regions. However, for a highly chaotic system, the perturbation may be moderately large. We now discuss the mechanism how the controlling signal drives the conservative system to an \attractive" region. We know that the controlled Hamiltonian system is still a conservative one, in other words, the controlling signal does not change the conservative property. Because of Hamiltonian feature of the system, no trajectory and phase space is attractive without the controlling signal. The system starting from an arbitrary initial point in the phase space will soon or later return to the region innitely close to the initial point. However, the controlling signal is an aperiodic function, it causes the overlap of the iterated regions under the \mod 0 1". So the system may be changed from the invertible to non-invertible. Consequently, an \attractive region" in the controlled Hamiltonian system may appear if a proper perturbation is acted. The dissipative-like behavior in some sense is called quasidissipative property. Nevertheless, no trajectory is attractive even for the controlled system, therefore, innitely many nal regular states can be realized from dierent initial states.

8 556 XU Hai-Bo, WANG Guang-Rui, and CHEN Shi-Gang Vol. 37 Our method not only gives a useful way for controlling chaos in two-dimensional Hamiltonian systems but also provides an intuitive understanding for the complex dynamics of a driven Hamiltonian system under two external perturbations. The extension of the present approach to high-dimensional systems and quantum systems needs further investigation. Acknowledgment XU Hai-Bo would like to thank Dr. ZHANG Ying for her initial idea and discussion. References [1] E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. 64 (1990) [2] Y. Braiman and I. Goldhirsch, Phys. Rev. Lett. 66 (1991) [3] E.R. Hunt, Phys. Rev. Lett. 67 (1991) [4] E.A. Jackson, Phys. Lett. A151 (1990) 478. [5] M.A. Matias and J. Guemez, Phys. Rev. Lett. 72 (1994) [6] N.P. Chau, Phys. Rev. E57 (1998) 378. [7] Y.C. Lai, M. Ding, and C. Grebogi, Phys. Rev. E47 (1993) 86. [8] Z. Wu, Z. Zhu, and C. Zhang, Phys. Rev. E57 (1998) 366. [9] A. Oloumi and D. Teycheme, Phys. Rev. E60 (1999) R6279. [10] D. del-castillo-negrette, J.M. Greene, and P.J. Morrison, Physica D100 (1997) 311. [11] J.E. Howard and S.M. Hohs, Phys. Rev. A29 (1984) 418. [12] G. Blum and O. Agam, Phys. Rev. E62 (2000) [13] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes: The Art of Scientic Computing, Cambridge University Press, New York (1986).