ON STABILIZING N-DIMENSIONAL CHAOTIC SYSTEMS

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1 International Journal of Bifurcation and Chaos, Vol. 3, No. 2 (23) c World Scientific Publishing Company ON STABILIZING N-DIMENSIONAL CHAOTIC SYSTEMS LAURENT LAVAL and NACER K. M SIRDI Laboratoire de Robotique de Paris, UPMC-UVSQ-CNRS URA 778, -2, av. de l Europe 784 Velizy, France Received October, 2; Revised November 3, 2 This paper deals with the control of a class of n-dimensional chaotic systems. The proposed method consists in a Variable Structure Control approach based on system energy consideration for both controller design and system stabilization. First, we present some theoretical results related to the stabilization of global invariant sets included in a selected two-dimensional subspace of the state space. Then, we define some conditions, involving both system definition and control law design, under which the stabilized orbits can be maintained in a neighborhood of an invariant, nondegenerate, closed conic section (i.e. an ellipse or a circle). Finally, an example related to the chaotic circuit of Chua is given. Keywords: Control of chaos; nonlinear systems; Sliding Mode control; time-periodic orbits.. Introduction Since the seminal work of W. Hübler and coauthors [Hübler et al., 988], who first demonstrated that an appropriate external input is efficient to force a chaotic system to perform desired motions, dramatic progress has been made in the control of chaos. Indeed, emergence of many applications in various areas such as communication, electronics, physiology, biology, fluid mechanics, chemistry... (e.g. [Hayes et al., 993; Kim & Stringer, 992; Peillon, 998; Teodorescu, 998; Schroeder, 99; Strogatz, 994] and references therein) has motivated an increasing amount of research in control methods to steer trajectories of chaotic systems (e.g. [Ditto et al., 995; Chen, 999; Chen & Dong, 998] and references therein). A brief overview of existing methods then points out that, according to their intrinsic control schemes, several approaches to design targeted orbits are mainly considered. For instance, control methods derived from OGY method [Ott et al., 99] aim at stabilizing an unstable orbit (embedded in the strange attractor) by setting up a control mechanism when the system trajectory reaches a neighborhood of the desired orbit. For such methods, the design of the targeted orbits then consists in identifying an unstable periodic point of interest through experimental tests and/or nominal representation of the system dynamics. Control schemes derived from Pyragas method [Pyragas, 992] (referred to as time-delayed autosynchronization control strategies) aim at counteracting the effect of an exciting signal related to chaotic motion. To this end, these control schemes introduce a delay in the controller with respect to a given periodic orbit. Hence, such methods do not require any model of the targeted time-periodic orbit, except for its period. Methods coming from classical control theory (Adaptive control [Hübler, 989; González, 994; Petrov et al., 994], H control [Bhajekar et al., 994], Sliding Mode control [Yu, 996; Lenz & Berstecher, 997; Yau et al., 2]...), involving 473

2 474 L. Laval & N. K. M Sirdi both linear and nonlinear control approaches, aim at minimizing a criterion related to the tracking error between the actual system trajectory and the desired one. The targeted trajectories are then usually designed from a nominal (possibly linear) representation of the system dynamics. In this paper, we address the problem of stabilizing the orbits for a class of n-dimensional chaotic systems without the use of any mathematical description of the targeted orbits. For this purpose, the proposed method (referred to as ESMC approach [Laval & M Sirdi, 22]) consists in a sliding mode based control design that we extend by introducing an explicit consideration of system energy as the basis for both controller design and system stabilization. Instead of minimizing the tracking error between the system trajectory and a desired one, the control objective is to regulate the energy with respect to a shaped nominal representation, implicitly related to system trajectories. In this paper, we establish some theoretical results related to the stabilization of global invariant sets for a class of n-dimensional systems, such as theses sets are included in a selected two-dimensional subspace of the state space. Then, we focus on some conditions, related to system definition and control law design, under which the stabilized orbits can be maintained in a neighborhood of an invariant, nondegenerate, closed conic section (i.e. an ellipse or a circle). A main feature of the proposed control method is that no analytical expression of the targeted orbit is required. Nevertheless, geometrical properties of the stabilized orbits can be tuned by means of control law parameters and the selected magnitude of system energy to target. This paper is organized as follows. Section 2 deals with the theoretical framework. Section 3 illustrates some capabilities of the proposed Energy based Sliding Mode control approach through examples related to the chaotic circuit of Chua. Finally, in Sec. 4, some concluding remarks are given. 2. Theoretical Framework First, this section presents a theoretical result (Lemma ) related to the control of a class of n-dimensional (chaotic) systems, such that the system trajectories converge to and are maintained in a neighborhood of a selected 2D-subspace of the state space. Then, we discuss about some consequences of the control law design on the dynamical behavior of the stabilized system. Finally, we derive a second result (Corollary ) involving some conditions for both system definition and control law design, and related to the steering of systems trajectories towards a neighborhood of an invariant, closed conic section (such as an ellipse or a circle). First, let us consider the class of n-dimensional nonlinear autonomous systems of the form, Ẋ = F (X, U) = f(x) + U () where X R n is the state vector partitioned as X = [X T X2 T ]T with X R n 2, X 2 = [x 2 x 22 ] T R 2. X() R n is the vector of initial conditions, f(x) = [f T (X, X 2 ) f2 T (X, X 2 )] T with f (X, X 2 ) C (R n 2 ) and f 2 (X, X 2 ) C (R 2 ), U R n is the vector of control inputs. Moreover let us consider the following assumptions. Assumption A. The system is at least locally observable and controllable. Assumption A2. The energy of the system can be represented by a Lyapunov function V which can be split into two parts V and V 2 (i.e. V = V + V 2 ) related to scalar positive functions V T (X ) and V IS (X 2 ) respectively (i.e. V = g (V T (X )) and V 2 = g 2 (V IS (X 2 ))). Assumption A3. Positive functions V T (X ) and V IS (X 2 ) have continuous first derivatives which can be expressed as, V T = ẊT Ψ X (2) V IS = ẊT 2 Ψ 2 X 2 (3) where Ψ R n 2 n 2 and Ψ 2 R 2 2 are diagonal matrices with strictly positive real values. Now we state the first theoretical result. Lemma. Consider the autonomous system () and Assumptions A A3. Moreover, consider the following control structure, U = [ u T T u T IS] T (4) Energy based Sliding Mode control.

3 On Stabilizing N-Dimensional Chaotic Systems 475 Substituting () into (7) leads to, u T = Γ sign(v T )X ˆf (X, X 2 ) (5) u IS = Γ 2 sign(v IS V IS)X 2 ˆf 2 (, X 2 ) (6) where u T R n 2, u IS R 2, ˆf (X, X 2 ) and ˆf 2 (, X 2 ) are vectors of continuous functions which represent (local ) equivalent system dynamics, 2 Γ R n 2 n 2 and Γ 2 R 2 2 are diagonal matrices with strictly positive real values, and VIS is a positive constant which characterizes a desired magnitude of energy. Then, (i) all solutions of the controlled system asymptotically converge to and are maintained in a global invariant set Ω IS included in the same subspace as X 2 and defined by 3 V IS VIS =. (ii) the energy of the controlled system converges to a neighborhood ε IS of VIS. Proof (of lemma). With respect to Assumption A2, let us consider a Lyapunov function candidate V = V + V 2 with V = 2 V T 2 and V 2 = 2 (V IS VIS )2. First, let us focus on V = 2 V T 2. Then, V = V T V T (7) From Assumption A3 and system definition (), it appears that V T can be expressed as, V T = ẊT Ψ X V T = (f (X, X 2 ) + u T ) T Ψ X (8) Substituting (5) into (8) leads to, V T = [f (X, X 2 ) Γ sign(v T )X ˆf (X, X 2 )] T Ψ X (9) From the analysis of the actual system (i.e. according to f (X, X 2 )), we can find a local 4 equivalent system representation ˆf (X, X 2 ) such that the averaged value f (X, X 2 ) ˆf (X, X 2 ) is zero. Therefore, in the mean, V T Γ sign(v T )X T Ψ X () V = V T V T Γ sign(v T )X T Ψ X V T () Therefore, according to positiveness of Γ and Ψ, V is negative semi-definite and V is positive definite. Consequently, from sliding mode theory (e.g. [Utkin, 977; De Carlo et al., 996; Slotine & Li, 99]), V T converges to a vicinity of in a finite time t >. Moreover, from (), X is bounded and also goes to a vicinity of zero (as V T and V T go to zero). Now, let us focus on V 2 = 2 (V IS VIS )2. Then, V 2 = V IS (V IS V IS ) (2) From assumption A3 and system definition (), it appears that V IS can be expressed as, V IS = ẊT 2 Ψ 2X 2 V IS = (f 2 (X, X 2 ) + u IS ) T Ψ 2 X 2 (3) Substituting (6) into (3) leads to, V IS = [f 2 (X, X 2 ) Γ 2 sign(v IS V IS )X 2. ˆf 2 (, X 2 )] T < Ψ 2 X 2 (4) V IS = (f 2 (X, X 2 ) ˆf 2 (, X 2 )) T Ψ 2 X 2 Recalling that: (Γ 2 sign(v IS V IS)X 2 ) T Ψ 2 X 2 (5) ˆf 2 (, X 2 ) represents equivalent system dynamics 5 when X is zero (in the mean), V T goes to a vicinity of in a finite time t (as a result of V T analysis). Thus, for t t, the averaged value f 2 (X, X 2 ) ˆf 2 (, X 2 ) is zero. Therefore, in the mean, V IS (Γ 2 sign(v IS V IS)X 2 ) T Ψ 2 X 2 ( t t ) (6) Substituting (6) into (2) leads to, V 2 = V IS (V IS V IS) (Γ 2 sign(v IS V IS)X 2 ) T Ψ 2 X 2 (V IS V IS) (7) 2 Deduced from analysis of the actual system. 3 : averaged value of. 4 Near the neighborhood of the invariant set to be stabilized. 5 According to f 2(X, X 2)

4 476 L. Laval & N. K. M Sirdi Therefore, according to positiveness of Γ 2 and Ψ 2, V2 is negative semi-definite and V 2 is positive definite ( t t ). Consequently, from sliding mode theory, the sliding surface defined by S 2 (X) = V IS VIS = is attractive and V IS converges to a vicinity of VIS in a finite time t 2 t >. Thus, we can conclude that, in a finite time t 2 t >, the controlled system trajectories are driven to and maintained in an invariant set Ω IS defined by V IS VIS = and included in the same subspace as X 2. Moreover, as the system energy is bounded and its representation (related to V T (X ) and V IS (X 2 )) goes to a vicinity of V (in a finite time t 2 ), this energy converges to a neighborhood ε IS of VIS (where ε IS is related to the discrepancy between the true system energy and its nominal representation). 2.. Discussion By considering Lemma and control law (5) and (6), let us remark that terms ˆf (X, X 2 ) and ˆf 2 (, X 2 ) (related to the equivalent system representation) enable some design features 6 to perform compensation of the true system dynamics of interest (such as nonlinearities...). Therefore, behavior of the stabilized system depends directly on the design of ˆf (X, X 2 ) and ˆf 2 (, X 2 ). As an illustration, let us consider the following example related to Lorenz chaotic system (e.g. [Yu, 996]). This system can be expressed as, ẋ = ay ax ẏ = bx y xz ż = xy cz where a, b and c are system parameters. Moreover, let us consider the state vector X [ = ] [x y z] T y partitioned as X = [x] and X 2 = z, and control law (5) and (6). Now, by choosing an equivalent system representation of the form, ˆf (X, X 2 ) = ây xz ˆf 2 (, X 2 ) = where â is an estimated value of the true system parameter a, we obtain the following (local) representation of the stabilized system (when x and 6 According to the knowledge of the true system dynamics and the design purposes. IS (V IS VIS ) have converged to some neighborhoods of zero and sign(v IS VIS ) is zero in the mean), ẏ = y ż = c z (where denotes the averaged value of ). Therefore, by choosing an appropriate equivalent system representation ( ˆf (X, X 2 ) and ˆf 2 (, X 2 )), the behavior of the stabilized system (under control) is equivalent to those of a (stable) linear one. This remark leads us to propose the following result related to the particular class of systems for which the equivalent system representation ( ˆf (X, X 2 ) and ˆf 2 (, X 2 )) can be designed such as, f 2 (, X 2 ) ˆf 2 (, X 2 ) = ΘX 2 (8) where Θ is a 2 2 matrix with real values. Corollary. Consider the autonomous system () and Assumptions A A3. Moreover, consider the control structure (5) and (6) and a representation ˆf 2 (, X 2 ) of equivalent system dynamics such as, f 2 (, X 2 ) ˆf 2 (, X 2 ) = ΘX 2 (Θ R 2 2 ). (9) Then, there exists a 2 2 diagonal matrix Γ 2 with strictly positive scalar values such that all solutions of the controlled system asymptotically converge to and are maintained in a neighborhood of a closed conic section (i.e. an ellipse or a circle) included in the same subspace as X 2. Proof (of corollary). With respect to Assumption A2, let us consider a Lyapunov function candidate V = V + V 2 with V = 2 V T 2 and V 2 = 2 (V IS VIS )2. Following the same way as the proof of Lemma (dedicated to the analysis of V T, VT, V and V ), we can establish that V is always negative semi-definite, V is positive definite, VT and V T converge to a vicinity of in a finite time t >, and that X is bounded and also goes to a vicinity of zero. Then, let us focus on V 2 = 2 (V IS V IS )2 and its derivative, V 2 = V IS (V IS V IS ) (2)

5 On Stabilizing N-Dimensional Chaotic Systems 477 From (4), VIS can be expressed as, V IS = [f 2 (X, X 2 ) Γ 2 sign(v IS V IS)X 2 ˆf 2 (, X 2 )] T Ψ 2 X 2 (2) For t t, the averaged value of X is zero. Thus, and, f 2 (X, X 2 ) f 2 (, X 2 ) ( t t ) V IS [f 2 (, X 2 ) Γ 2 sign(v IS V IS) X 2 Then, from (9), ˆf 2 (, X 2 )] T Ψ 2 X 2 ( t t ) (22) V IS [(Θ Γ 2 sign(v IS V IS )) X 2] T Ψ 2 X 2 ( t t ) (23) Substituting (23) into (2) leads to (in the mean), V 2 [(Θ Γ 2 sign(v IS V IS))X 2 ] T Ψ 2 X 2 (V IS V IS) ( t t ) (24) Now by assuming, [ ] γ Γ 2 = γ 22 [ ] ψ Ψ 2 = ψ 22 and [ ] θ θ 2 Θ = θ 2 θ 22 (γ R +, γ 22 R + ) (ψ R +, ψ 22 R + ) (θ ij R, i =,..., 2, j =,..., 2) Then, (24) can be expressed as, V 2 ψ [θ γ sign(v IS V IS)](V IS V IS)x (ψ 22 θ 2 + ψ θ 2 )(V IS V IS )x 22x 2 + ψ 22 [θ 22 γ 22 sign(v IS V IS )](V IS V IS )x2 22 (25) As some elements of θ may be positive and (V IS VIS ) may be also positive valued, the problem is to find some parameters γ and γ 22 so that negativeness of V 2 is always guaranteed. To this end, let us consider the following worst condition to fulfill, V 2 η( ) (26) η = ψ [ θ γ ] V IS V IS x2 2 + (ψ 22 θ 2 + ψ θ 2 ) V IS V IS x 22x 2 + ψ 22 [ θ 22 γ 22 ] V IS V IS x 2 22 (27) Thus, in any case, the condition to fulfill can be written as, ψ [ θ γ ] V IS V IS x2 2 + (ψ 22 θ 2 + ψ θ 2 ) V IS V IS x 22x 2 + ψ 22 [ θ 22 γ 22 ] V IS V IS x 2 22 ψ θ x (ψ 22 θ 2 + ψ θ 2 ) x 22 x 2 + ψ 22 θ 22 x 2 22 ψ γ x ψ 22γ 22 x 2 22 (28) Rewriting (28) leads to, X 2 T P X 2 X 2 T Ψ 2 Γ 2 X 2 Ψ θ P = 2 (ψ 22 θ 2 + ψ θ 2 ) 2 (ψ 22 θ 2 + ψ θ 2 ) ψ 22 θ 22 As Ψ 2 R 2 2 is a diagonal matrix with strictly positive real values (Cf. Assumption 3), then Ψ 2 is invertible. Thus, we can always find a matrix Γ 2 so that Γ 2 Ψ 2 P (29) By designing Γ 2 with respect to Eq. (29), η is always negative semi-definite. Then, from Eq. (26), V 2 is always negative semi-definite (and V 2 is positive definite). Therefore, for t t, V IS is bounded and converges to a vicinity of VIS in a finite time. Moreover, from Eq. (2), it appears that X 2 is also bounded. Now, let us focus on the geometrical properties of the stabilized orbits. For this purpose, let us consider Eqs. (2) and (25). Then, in the mean, the (stabilized) system trajectories verify the following relation, V IS ψ [θ γ sign(v IS V IS )]x2 2 + (ψ 22 θ 2 + ψ θ 2 )x 22 x 2 + ψ 22 [θ 22 γ 22 sign(v IS V IS)]x 2 22 (3)

6 478 L. Laval & N. K. M Sirdi As V IS and X 2 are bounded (due to control effects) then x 22 x 2 is obviously bounded. Thus, the (stabilized) system trajectories can be characterized by a set of conic sections included in a closed 3D subspace, and deduced from a parametrized equation of the form, a(ψ 2, Θ, Γ 2 )x b(ψ 2, Θ)x 22 x 2 + c(ψ 2, Θ, Γ 2 )x ξ( V IS ) = (3) where ξ( V IS ) is related to the time derivative of (V IS VIS ). If VIS is zero then the system trajectories converge towards a neighborhood of a point (namely the origin (,, )). Else, the system trajectories converge towards a neighborhood of a closed conic section (i.e. an ellipse or a circle) defined by (3), as x 2 and x 22 are continuous and bounded. Remarks As Eq. (3) depends on both Γ 2 and VIS, then geometrical properties of the targeted conic section can be tuned by modifying these two (control) parameters. ( [ ]) If Θ is diagonal i.e. Θ = θ then negativeness of V 2 is always effective if Γ 2 is chosen so θ 22 that γ θ and γ 22 θ Example: Control of Chua s Circuit This section aims at providing an example of Energy based Sliding Mode control, related to the steering of trajectories of a chaotic system towards the neighborhood of a stable elliptic orbit. For this purpose, we consider the chaotic circuit of Chua as defined in [Madan, 993; Yang & Chua, 998]. This system can be formulated as a dimensionless model of the form [Yang & Chua, 998], ẋ = α(y x f d (x)) Ẋ = f(x) ẏ = x y + z (32) ż = βy where X = [x y z] T is the state vector and f d (x) = bx + 2 (a b)( x + x ). Using dimensionless notations, the energy of the system can be expressed as [Laval & M Sirdi, 22], E = 2 x2 + 2 αy2 + α 2 β z2 (33) 7 And its split form components V (X ) and V 2(X 2). Then, according to possible transfers of energy between Chua s circuit, let us consider a statevector partitioning of the form: X = x and X 2 = [ y z ] T. Moreover, with respect to Eq. (33) and Assumptions A2 and A3, let us consider the following Lyapunov function 7 V (X) to represent the energy of the system: V (X) = V (X ) + V 2 (X 2 ) V (X ) = 2 V 2 T with V T = 2 x2 (34) V 2 (X 2 ) = 2 (V IS V IS )2 (35) with V IS = 2 αy2 + α 2 β z2 (36) V IS is a constant (37) Then, from Lemma, the controlled chaotic system of Chua is given by ẋ = α(y x f d (x)) + u T ẏ = x y + z + u IS (38) ż = βy + u IS2 u T = Γ sign(v T )x ˆf (x, y, z) u IS = Γ sign(v IS VIS)y ˆf 2 (, y, z) u IS2 = Γ 22 sign(v IS VIS )z ˆf 22 (, y, z) (39) Assuming that the (local) equivalent system dynamics can be defined as, ˆf (x, y, z) = ˆα(y x ˆf d (x)) corresponding to, ˆf 2 (, y, z) = ˆf 22 (, y, z) = f 2 (, X 2 ) ˆf 2 (, X 2 ) [ ] X 2 β

7 On Stabilizing N-Dimensional Chaotic Systems X() z y(t) Fig y x Controlled system trajectory Fig. 4. State y trajectory versus time. 3 z(y) z z(t) y Fig. 2. State z trajectory versus state y. Fig. 5. State z trajectory versus time. x(t) Function V t Function V I S Fig. 3. State x trajectory versus time. Fig. 6. Magnitude of V T and V IS versus time.

8 48 L. Laval & N. K. M Sirdi therefore, technical condition (9) is fulfilled, meaning that Corollary 2 can be considered. Then, by considering: system parameters: α =, β = 4.87, a =.27 and b =.68, initial conditions: x() =.2, y() =. and z() =., control law gains: Γ = 4, Γ 2 = I 2, a targeted energy: VIS = 3 we obtain the simulation results of Figs. 6. As shown through Fig. 3, the system state x first converges to a neighborhood of zero in a finite time t.4 s. On the other side, system states y and z converge to a periodic behavior, in a finite time t 2 > t (with t 2.4 s), as shown in Figs. 4 and 5. This means that system trajectory first goes to (y, z) plane (in a finite time t ) before to perform a periodic trajectory (see Figs. and 2). These results are confirmed by the behaviors of the system energy representations related to V T and V IS, as shown in Fig. 8. This figure also shows that V IS converges effectively to a neighborhood of the targeted energy VIS = 3 (in a finite time). 4. Conclusion This paper has focused on a new control approach for a class of n-dimensional chaotic systems (fulfilling Assumptions A A3). The proposed approach considers the system energy as basis for both controller design and system stabilization. As a result, application of this method leads to extended capabilities compared to classical VSS controllers, as the system trajectories can be driven towards and maintained into global invariant sets without explicit definition of the targeted orbits. Moreover, it has been shown that these orbits can be some closed conics section (ellipse or circle) if the (local) equivalent system representation ( ˆf (X, X 2 ) and ˆf 2 (, X 2 )) can be designed such that condition (8) is fulfilled (see Corollary and its preliminary discussion). We think that this control method can be extended to deal with the problem of synchronization of chaotic systems. This is the objective of our future investigations. References Bhajekar, S., Jonckheere, E. A. & Hammad, A. [994] H control of chaos, Proc. 33rd Conf. Decision Control, Lake Buena Vista, FL, pp Chen, G. & Dong, X. [998] From Chaos to Order: Perspectives, Methodologies, and Applications (World Scientific, Singapore). Chen, G. [999] Controlling Chaos and Bifurcations in Engineering Systems (CRC Press). De Carlo, R. A., Zak, S. H. & Drakunov, S. V. [996] Variable structure, sliding-mode controller design, The Control Handbook, Elec. Eng. Handbook Series (CRC Press Inc.), pp Ditto, W. L., Spano, M. L. & Lindner, J. F. [995] Techniques for the control of chaos, Physica D86(82), González, G. A. [994] Controlling chaos of an uncertain Lozi system via adaptive techniques, Int. J. Bifurcation and Chaos 5, Hayes, S., Grebogi, C. & Ott, E. [993] Communicating with chaos, Phys. Rev. Lett. 7, Hübler, A., Georgii, R., Kuckler, M., Stelzl, W. & Lüscher, E. [988] Resonant stimulation of nonlinear damped oscillators by Poincaré maps, Helv. Phys. Acta 6, Hübler, A. W. [989] Adaptive control of chaotic systems, Helvetica Phys. Acta 62, Kim, J. H. & Stringer, J. (eds.) [992] Applied Chaos (John Wiley). Laval, L. & M Sirdi, N. K. [22] Stabilization of global invariant sets for chaotic systems: An energy based control approach, Int. J. Bifurcation and Chaos 2(6), Lenz, H. & Berstecher, R. [997] Sliding-mode control of chaotic pendulum: Stabilization and targeting of an instable periodic orbit, Proc. Int. Conf. Control of Oscillations and Chaos (COC 97, St. Petersburg, Russia), Vol. 3, pp Madan, R. N. [993] Chua s Circuit: A Paradigm for Chaos, World Scientific Series on Nonlinear Science. Serie B, Special Theme Issues and Proceedings, Vol. (World Scientific, Singapore). Ott, E., Grebogi, C. & Yorke, J. A. [99] Controlling chaos, Phys. Rev. Lett. 64, Peillon, X. [998] A compilation of references on chaotic circuits, Int. J. Chaos Th. Appl. 3(4). Petrov, V., Crowley, M. F. & Showalter, K. [994] An adaptive control algorithm for tracking unstable periodic orbits, Int. J. Bifurcation and Chaos 4, Pyragas, K. [992] Continuous control of chaos by selfcontrolling feedback, Phys. Lett. A7, Schroeder, M. [99] Fractals, Chaos, Power Laws (W. H. Freeman and Company, NY). Slotine, J. J. E. & Li, W. [99] Applied Nonlinear Control (Prentice Hall, Englewood Cliffs, NJ). Strogatz, S. H. [994] Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering (Addison-Wesley). Teodorescu, H. N. [998] An overview of the technical applications of chaos theory: Patents on chaos and its applications, Int. J. Chaos Th. Appl. 3(4).

9 On Stabilizing N-Dimensional Chaotic Systems 48 Utkin, V. I. [977] Variable structure systems with sliding mode: A survey, IEEE Trans. Autom. Contr. ACC-22(2), Yang, T. & Chua, L. O. [998] Control of chaos using sampled-data feedback control, Int. J. Bifurcation and Chaos 8, Yau, H. T., Chen, C. K. & Chen, C. L. [2] Sliding mode control of chaotic systems with uncertainties, Int. J. Bifurcation and Chaos (5), Yu, X. [996] Controlling Lorenz chaos, Int. J. Syst. Sci. 27(4),