Chaotifying 2-D piecewise linear maps via a piecewise linear controller function Zeraoulia Elhadj 1,J.C.Sprott 2 1 Department of Mathematics, University of Tébéssa, (12000), Algeria. E-mail: zeraoulia@mail.univ-tebessa.dz and zelhadj12@yahoo.fr. 2 Department of Physics, University of Wisconsin, Madison, WI 53706, USA. E-mail: sprott@physics.wisc.edu. April 13, 2009 Abstract A simple method for chaotifying piecewise linear maps of the plane using a piecewise linear controller function is given. A domain of chaos in the resulting controlled map was determined exactly and rigorously. Keywords: Chaotification, piecewise linear map, piecewise linear controller, Lyapunov exponents. 1 Introduction A large number of physical and engineering systems have been found to be governed by a class of continuous or discontinuous maps [1-4-5-7-8-9-10-13- 14] where the discrete-time state space is divided into two or more compartments with different functional forms of the map separated by borderlines [14-15-16-17]. The theory of discontinuous maps is in its infancy, with some progress reported for 1-D and n-d discontinuous maps in [1-8-9-11-12-17], but these results are restrictive and cannot be obtained in the general n- dimensional context [12]. On the other hand, there are many works that focus on the chaotic behavior of discrete mappings. For example, they have 1
been studied as control and anti-control (chaotification) schemes using Lyapunov exponents [2-3-18-20] to prove the existence of chaos in n-dimensional discrete dynamical systems with the goal of making in some way an originally non-chaotic dynamical system chaotic or enhancing the chaos already existing in such a system. The goal of this work is to present a simple method for chaotifying an arbitrary 2-D piecewise linear map (continuous or not) using a simple piecewise linear controller function, which allows one to determine exactly and rigorously which portion of the bifurcation parameter space is characterized by the occurrence of chaos in the resulting controlled map using the standard definition of the Lyapunov exponents as the test for chaos. Consider an arbitrary piecewise-linear map f : D D, where D = D 1 D 2 R 2, defined by: AX k + b if X k D 1 X k+1 = f (X k )= (1) BX k + c if X k D 2, µ b11 b 12 b 21 b 22 µ a11 a where A = 12 and B = are 2 2 real matrices, and µ a 21 a 22 µ µ b1 c1 xk b = and c = are 2 1 real vectors, and X b 2 c k = R 2 y 2 k is the state variable. The Lyapunov exponents of a 2-D dynamical system are defined as follow: Let us consider the system X k+1 = f(x k ),X k R 2,k =0, 1, 2,... (2) where the function f : R 2 R 2 is the vector field associated with system (2). Let J (X k ) be its Jacobian evaluated at X k R 2, k =0, 1, 2,..., and define the matrix: T n (X 0 )=J(X n 1 ) J (X n 2 )...J (X 1 ) J (X 0 ). (3) Moreover, let J i (X 0,n) be the modulus of the i th eigenvalue of the n th matrix T n (X 0 ), where i =1, 2 and n =0, 1, 2,... Now, the Lyapunov exponents for a two-dimensional discrete-time system are defined by: ³ l i (X 0 ) = lim ln J i (X 0,n) 1 n,i=1, 2. (4) n + 2
Roughly speaking, chaotic behavior implies sensitive dependence on initial conditions, with at least one positive Lyapunov exponent. Based on this definition, we give in the next section a rigorous proof of chaos in the resulting controlled map (7) obtained below via a simple piecewise linear controller function applied to the map (1). While many algorithms for calculating the Lyapunov exponents would give spurious results for piecewise-linear discontinuous maps, the algorithm used here and given in [21] works for such cases. It essentially takes a numerical derivative and gives the correct result provided that is taken to ensure that the perturbed and unperturbed orbits lie on the same side of the discontinuity. This may require an occasional small perturbation into a region that is not strictly accessible to the orbit. 2 The chaotification method using a piecewise linear controller function The main idea of the chaotification method presented in this work is to introduce a piecewise linear controller function in such a way, such that the two (in both partitions) system matrices of the resulting controlled piecewise linear system have the same trace and determinant (invariants) and hence the same eigenvalues. Indeed, the controlled map is given by: AX k + b if X k D 1 g (X k )= + U (X k ), (5) BX k + c if X k D 2 where the controller function U (X k ) is defined by: U (X k )= Ã (b 11 + b 22 a 11 a 22 ) x k ³ a12 a 21 +b 11 a 22 b 11 b 22 +b 12 b 21 +a 22 b 22 a 2 22 a 12 x k µ 0 0 The controlled system (5) is now given by: if X k D 2.! if X k D 1 (6) 3
QX k + b if X k D 1 g (X k )= (7) BX k + c if X k D 2, where the matrix Q is given by: Ã! b 11 + b 22 a 22 a 12 Q = b 11 a 22 b 11 b 22 +b 12 b 21 +a 22 b 22 a 2. (8) 22 a 12 a 22 The Jacobian matrix of the controlled map (7) is given by: Q if X k D 1 J (X k )= (9) B if X k D 2. It is shown in [22] that if we consider a system x k+1 = f (x k ),x k Ω R n, such that: kf 0 (x)k = p λ max (f(x) T f(x)) N<+, (10) with a smallest eigenvalue of f(x) T f(x) that satisfies: λ min f 0 (x) T f 0 (x) θ>0, (11) where N 2 θ, then, for any x 0 Ω, all the Lyapunov exponents at x 0 are located inside ln θ, ln N,thatis, 2 ln θ 2 l i (x 0 ) ln N,i =1, 2,..., n, (12) where l i (x 0 ) are the Lyapunov exponents for the map f and k.k is the Euclidian norm in R n. We remark that J (X k ) given in (9) is not welldefined due to the discontinuity, but, since B and Q havethesameeigenvalues, then one has that kqk = kbk = p µ λ max (B T B). Because B T B = b 2 11 + b 2 21 b 11 b 12 + b 21 b 22 b 11 b 12 + b 21 b 22 b 2 12 + b 2 is at least a positive semi-definite matrix, 22 then all its eigenvalues are real and positive, i.e. λ max B T B λ min B T B 0. Hence the eigenvalues of B T B are given by: ½ λmax B T B = 1 2 b2 11 + 1 2 b2 12 + 1 2 b2 21 + 1 2 b2 22 + 1 2 d λ min B T B = 1 2 b2 11 + 1 2 b2 12 + 1 2 b2 21 + 1 2 b2 22 1 (13) 2 d, 4
where d = (b 11 + b 22 ) 2 +(b 12 b 21 ) 2 (b 12 + b 21 ) 2 +(b 11 b 22 ) 2 > 0 (14) pfor all b 11,b 12,b 21, and b 22. Condition (10) gives kf 0 (x)k = kbk = kqk = λmax (B T B)=N < +, because B and Q have the same eigenvalues. Condition (11) gives the inequality: θ 2 b 2 11 + b 2 12 + b 2 21 + b22 2 θ +(b11 b 22 b 12 b 21 ) 2 0, (15) with the condition θ< b2 11 + b 2 12 + b 2 21 + b 2 22. (16) 2 Since the discriminant of (15) is equal to d>0, then (11) holds when: θ λ max B T B, or θ λ min B T B. (17) The condition θ λ max B T B is impossible because of condition (16), so that θ must satisfy the condition: Now, if θ<λ min B T B = b2 11 + b 2 12 + b 2 21 + b 2 22 d. (18) 2 ½ 2 <b 2 11 + b 2 12 + b 2 21 + b 2 22 < (b 11 b 22 b 12 b 21 ) 2 +1 (19) b 11 b 22 b 12 b 21 > 1, then λ min B T B > 1, i.e. θ =1, and one has: 0 <l i (x 0 ) ln N,i =1, 2, (20) i.e. the controlled map (7) converges to a hyperchaotic attractor for all parameters b 11,b 12,b 21, and b 22 satisfying (19). Finally, the following theorem is proved: Theorem 1 The piecewise linear controller function (6) makes the map (1) chaotic in the following case: ½ 2 <b 2 11 + b 2 12 + b 2 21 + b 2 22 < (b 11 b 22 b 12 b 21 ) 2 +1 (21) b 11 b 22 b 12 b 21 > 1. Finally, we note that Theorem 1 does not garanty the boundness of the resulting controlled map (7). This problem is still open for general piecewiselinear map and flows. 5
3 Example In this section, we make chaotic the following piecewise linear map using the above method: µ xk αy k +1 if y γx k 0 k f (x k,y k )= µ (22) xk + αy k +1 if y βx k < 0, k where α, β, and γ are bifurcation parameters. The map (22) is a special case AX k + b, if y k 0 ofthemap(1)sinceonecanrewriteitasf (X k )=, µ µ BX µ k + c, if y k < 0 1 α 1 α 1 where A =,B =, and b = c =, and the two γ 0 β 0 0 sub regions are: D 1 = {(x k,y k ) R 2 /y k 0} and D 2 = {(x k,y k ) R 2 /y k < 0}. Thus, the resulting controlled map is given by: µ µ µ 1 α xk 1 + if y β 0 y k 0 k 0 µ 1 α yk + x g (x k,y k )= µ µ µ = k βx 1 α xk 1 k sgn(y k ) + if y β 0 y k 0 k < 0, (23) which is the so called discrete butterfly presentedin[19],wheresgn(.) is the standard signum function that gives ±1 depending on the sign of its argument. For the controlled map (23), condition (21) becomes: Ã! β α > max p β 2 1, 1, β > 1. (24) β For illustration, assume that β < 1. Then we remark that β β 2 1 < 1 = 1 β β, and thus conditions (24) become β β 2 1 α > 1,β < 1. (25) β 6 =,
Figure 1: Regions of dynamical behaviors in the αβ-plane for the controlled map (23) with 2 β< 1 and 0.1 1 1 <α< +0.4. β β Using the obtained analytical results, Figure 1 shows that for 2 β< 1, and 0.1 1 1 <α< +0.4, the controlled map (23) converges to bounded β β hyperchaotic attractors or unbounded orbits for α > 1. In this figure, β white indicates unbounded solutions, blue indicates periodic solutions, and red indicates chaotic solutions in the αβ-plane for the controlled map (23), where we use 5000 different initial conditions and 10 6 iterations for each point. A chaotic attractor for the case with α =0.6 and β = 2 is shown in Fig 2. On the other hand, it is necessary to verify the hyperchaoticity of the attractors by calculating both Lyapunov exponents using the formula: l 1 (x 0 )+ l 2 (x 0 )=ln det(j) =ln αβ averaged along the orbit where det(j) is the determinant of the Jacobian matrix. The result is shown in Fig. 3 for 0.4 α 0.9, with β = 2. 7
Figure 2: The new piecewise linear chaotic attractor obtained from the controlled map (23) with its basin of attraction in white for α =0.6, β= 2, and the initial condition x 0 = y 0 =0.01. 8
Figure 3: Variation of the Lyapunov exponents of the controlled map (23) versus the parameter 0.4 α 0.9, with β = 2, where the two vertical lines indicate the border between the montiened dynamical behaviors. 9
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