VOLUME 84, NUMBER 1 PHYSICAL REVIEW LETTERS 3JANUARY 2000 Controlling Hyperchaos Ling Yang,* Zengrong Liu, † and Jian-min Mao ‡ Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (Received 15 March 1999) For a finite-dimensional dynamical system, whose governing equations may or may not be analytically available, we show how to stabilize an unstable orbit in a neighborhood of a “fully”unstable fixed point (i.e., a fixed point at which all eigenvalues of the Jacobian matrix have modulus greater than unity). Only one of the unstable directions is to be stabilized via time-dependent adjustments of control parameters. The parameter adjustments can be optimized. PACS numbers: 05.45.Gg, 05.45.Pq In many engineering and other practical problems, chaos is undesirable and therefore needs to be controlled. The is- sue of controlling chaos, however, had not been actively studied until the year of 1990 when Ott, Grebogi, and Yorke (OGY) gave a method for controlling chaos [1]. Experimental works on the feedback stabilization have been carried out [2,3]. The method has been extended and modified [4–7], among which the method proposed by Romeiras, Grebogi, Ott, and Dayawansa (RGOD) [4] has attracted much attention. As a related topic, “using chaos to direct trajectories to targets” has also been investi- gated [8]. From a practical point of view, controlling a “fully” un- stable system (i.e., a system having a fixed point with no preexisting stable manifold in its neighborhood) is as in- teresting and important as the one with a stable manifold. However, the issue of controlling such hyperchaos has not been particularly addressed. Actually, the OGY method [1] is to stabilize an unstable orbit in the neighborhood of a hyperbolic fixed point by forcing the orbit onto the stable manifold. Therefore the method cannot be used to control hyperchaos. The RGOD method [4] uses a feedback ma- trix which makes the fixed point under consideration fully stable. Therefore, in our opinion, the method is not suit- able for controlling hyperchaos since the method changes the stability property of the fixed point completely: it is fully unstable originally but fully stable after the parame- ter adjustment. In this Letter we use a new idea to stabilize unstable orbits even if there is no preexisting stable mani- fold nearby. The systems under our study may or may not be given analytically. In other words, a priori analytical knowledge of system dynamics is not necessary. The unstable mani- fold can be detected from (chaotic) experimental data by using the embedding procedure (also known as technique of reconstruction) [1,9]. The embedding theorem asserts that if an orbit is in an attractor in the phase space, then the corresponding orbit in the embedding space is also in an attractor (but in the embedding space). The theorem fur- ther asserts that the two attractors have the same dimension and the same Lyapunov exponents. Let the unstable orbit to be controlled be in an embedding space of dimension N , where N $ 1 is a finite integer, and be near a fixed point at which the dimension of the unstable manifold is N u , where N u # N is an integer. (N u  N is the case we are particularly interested in.) We formulate the system under consideration by map T : j n ! j n11 , where j n11  F e j n  . (1) Here j[ R N is the dynamical variable, e[ R N u is a small control parameter vector, and F e j n  is a vector- valued function of j n with e as parameter. For flows, map (1) is a Poincaré map. We emphasize that this map is for- mally written (i.e., the map may be given not analytically but given by a data set) and that properties of the vector field F e are experimentally accessible. Let j 0  be the fixed point of map (1) with e  0. We want, by slightly adjusting parameter e, to control an orbit of the map that runs away from the fixed point if e  0. Let J be the Jacobian matrix of the map with e  0 evaluated at the fixed point, i.e., J  μ ≠F 0 ≠j n ∂ j n j 0  . (2) The elements of the Jacobian matrix can be determined experimentally in practical problems. Without loss of gen- erality, it is assumed that proper coordinate changes have been made so that j 0  is the origin of the N -dimensional space and that the eigenvectors of the Jacobian matrix are along the coordinate axes of the space. The eigenvalues of the Jacobian matrix J in Eq. (2) are all with modu- lus greater than unity for the case we study in this Let- ter. Therefore the determinant of the Jacobian matrix is not equal to one, and the implicit function theorem can be used to assert that map (1) with small parameter e has a fixed point in the neighborhood of j 0  . Denote this fixed point by j  and define the following matrix: M  μ ≠j  ≠e ∂ e0 . (3) Again, the elements of this matrix can be obtained experi- mentally for a practical problem under study. When e is small, consider a neighborhood of j 0  , W , that is large enough to also include a neighborhood of 0031-9007 00  84(1)  67(4)$15.00 © 1999 The American Physical Society 67