The Effect of Explosive Divergence in a Coupled Map Lattice of Matrices

∗ Guangqing Lu a, Rasa Smidtaite b,c, Zenonas Navickas b, Minvydas Ragulskis a,b, a School of Electrical and Information Engineering of Jinan University, 206 Qianshan Road, Zhuhai, Guangdong 519070, P.R. China b Centre of Nonlinear Systems, Kaunas University of Technology, Studentu 50-146, Kaunas LT-51368, Lithuania c Department of Applied Mathematics, Kaunas University of Technology, Studentu 50-318, Kaunas LT-51368, Lithuania

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Article history: The extension of the Coupled map lattice (CML) model by replacing scalar nodal variables by matrix Received 16 April 2018 variables is investigated in this paper. The dynamics of the extended CML is investigated using formal

Revised 11 June 2018 analytical and computational techniques. Necessary conditions for the occurrence of the effect of explo- Accepted 12 June 2018 sive divergence in the extended CML are derived. It is demonstrated that the extended CML can generate complex fractal patterns representing spatiotemporal divergence which can be controlled by the coupling Keywords: parameter between the nodes. The logistic map Coupled map lattices Kaneko model

1. Introduction plications in physics, cryptography, steganography, economics, bi- ology, etc. CMLs play an important role in image encryption The Logistic map is a paradigmatic model exhibiting chaotic be- algorithms [13–16] . Chaotic CML based stream ciphers are pro- havior [1]: posed by Li et al. [17] and Wang et al. [18] . CMLs are introduced to the market maker framework in order to investigate price changes ( + ) ( ) ( ) ( ) x t 1 = ax t (1 − x t ) ; t = 0 , 1 , 2 , . . . ; 0 ≤ a ≤ 4 ; 0 ≤ x 0 ≤ 1 . in a multi-market system [19] . A multiple phenotype predator– (1) prey model with mutation modelled via a CML [20] is proposed by Abernethy et al. CMLs are used to study neuronal dynamics [21] , A variety of extensions of the discrete logistic map has been coherence resonance phenomena [22] , and even stochastic reso- proposed during the last decades. The phase-modulated logistic nance on excitable small-world networks via a pacemaker [23] . map [2] is introduced by Nandi et al. Logistic map with a squared CMLs are also used as models for chaotic variations in spatiotem- sine perturbation [3] is presented by de Carvalho et al. Alternat- poral systems [24] . ing sign generalized logistic map with three parameters [4] is dis- CMLs exhibit a variety of spatiotemporal phenomena [25–32] . cussed by Sayed et al. Three bi- and tri-parameter generalized lo- Frozen random pattern, defect chaotic diffusion pattern, pattern gistic maps with arbitrary powers [5] are presented by Radwan. competition intermittency and fully developed turbulence in logis- Generalized fractional order logistic map suitable for pseudoran- tic map lattice with mixed linear-nonlinear couplings and its possi- dom number key generators [6] is presented by Ismail et al. Logis- ble applications in cryptography [25] are reviewed by Zhang. Two tic map in two dimensions is used to encrypt digital images [7– qualitative classes spatiotemporal intermittency and spatial inter- 9] . Chaotic encryption algorithm based on 3D logistic map [10] is mittency [26,27] are analyzed in the sine circle map lattice by proposed by Khade et al. Modified three-dimensional logistic map Jabeen et al. Sufficient conditions for synchronization of coupled is applied in steganography [11] . Symmetric image encryption maps on certain graphs [28] are derived Medvedev et al. An an- method based on intertwining logistic map [12] is proposed by Ye alytical study of traveling waves in CML [29,30] is developed by et al. Herrera et al. Properties of rotating pulse waves and oscillations Coupled map lattices are widely used to study the dynam- in a unidirectionally CML of ring structure [31] are discussed by ics of spatially extended logistic map systems. CMLs have ap- Horikawa. Method of adaptive control (based on the quasi sliding mode control design) of spatiotemporal chaos in CML [32] is pro- posed by Rahmani. ∗ Corresponding author at: Centre of Nonlinear Systems, Kaunas University of The main objective of this manuscript as well as necessary pre- Technology, Studentu 50-146, Kaunas LT-51368, Lithuania. liminary information is depicted in the following section. E-mail address: [email protected] (M. Ragulskis). 2. Preliminaries and motivation 2.4. Motivation

2.1. Idempotents and nilpotents The replacement of the scalar variable in the Logistic map by a second order square matrix yields unexpected and non-trivial phe- Let us consider such second order real matrices X ∈ R 2 × 2 that nomena (explosive divergence, temporary divergence, etc.) [33,34] . λ λ both eigenvalues of X (denoted as 1 , 2 ) are real. Note that X is A natural question is: what happens if scalar nodal variables in the not necessarily a symmetric matrix. CML nodes are replaced by matrix variables? In other words, this

λ  = λ = 1 − λ , = , ;  = If 1 2 , then matrices Dk : λ −λ ( X l I); k l 1 2 k replacement increases the complexity of the spatially extended Lo- k l = gistic map at the nodes. How this increased complexity is repre- l are conjugate idempotents (I is the identity matrix): det Dk ; + = · = δ δ sented in the patterns generated by the lattice? The main objective 0 D1 D2 I; Dk Dl kl Dk ( kl is the Dirac delta function). Then X can be decomposed into the idempotent matrix form: of this paper is to seek answers to these questions.

X = λ1 D 1 + λ2 D 2 (2) 3. The Coupled map lattice of matrices

λ = λ = λ , = − λ = , If 1 2 0 then N : X 0 I is a nilpotent: det N 0 As stated previously, the main objective of this paper is to study 2 =   N ( is a zero matrix). Then X can be expressed in the form a CML where scalar nodal variables x (t)( i ) are replaced by matrix of a nilpotent matrix: variables X ( t ) ( i ) ∈ R 2 × 2 : X = λ I + N . (3) ( + ) ( ) ε ( ) ( ) 0 X t 1 ( i ) = ( 1 − ε ) f (X t ( i ) ) + f (X t ( i + 1 ) ) + f (X t ( i − 1 ) ) 2 (7) 2.2. The Logistic map of matrices Further on, this model is entitled as the Coupled map lattice of The extension of the Logistic map [33] by replacing the scalar matrices (CMLM). variable x ( t ) by a second order square matrix variable X ( t ) is dis- cussed by Navickas et al. It is shown [33] that if the matrix of ini- 3.1. Idempotent matrices of initial conditions with identical tial conditions X (0) is an idempotent matrix, then the evolution of idempotents the Logistic map of matrices is described by two separate scalar Logistic maps of eigenvalues: Let us assume that all matrices of initial conditions are idem-

potent matrices with the same idempotents D1 and D2 : (t +1 ) (t ) (t ) λ = aλ 1 − λ ; (0) ( 0 ) ( 0 ) 1 1 1 ( ) = λ ( ) + λ ( ) X i 1 i D1 2 i D2 (8) ( + ) ( ) ( ) (4) λ t 1 = λ t − λ t , 2 a 2 1 2 Note that ( ) ( ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) 0 ( ) + 0 ( ) = λ ( ) + λ ( ) + λ ( ) + λ ( ) λ λ (0) X k X l 1 k 1 l D1 2 k 2 l D2 where 1 and 2 are two different distinct eigenvalues of X . In other words, the dynamics of the map is fully determined by (9) the initial eigenvalues and the parameter a . (0) ( ) ( ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) However, if the matrix of initial conditions X is a nilpotent 0 ( ) 0 ( ) = λ ( )λ ( ) + λ ( )λ ( ) X k X l 1 k 1 l D1 2 k 2 l D2 (10) matrix, then the Logistic map of matrices can be divided into two Then, different scalar maps. The first one is a scalar Logistic map of the ( 0 ) ( ) ( ) ( ) λ (0) 0 = 0 − 0 eigenvalue ( 0 is the eigenvalue of X ). The other scalar map is f X ( i) aX ( i) I X ( i) μ( t ) μ(0) = the map incorporating the supplementary parameter ( ( 0 ) ( 0 ) ( 0 ) ( 0 ) = aλ ( i ) 1 − λ ( i ) D + aλ ( i ) 1 − λ ( i ) D , 1 ) [33] : 1 1 1 2 2 2 (11) ( + ) ( ) ( ) λ t 1 = λ t − λ t ; 0 a 0 1 0 (5) where I is the second order identity matrix. ( + ) ( ) (t ) μ t 1 = μ t − λ . Therefore, a 1 2 0 (1) ( 1 ) ( 1 ) X ( i ) = L ( i )D + L ( i )D (12) It is demonstrated [34] that the Logistic map of matrices can 1 1 2 2 exhibit the effect of explosive divergence if and only if the ma- where trix of initial conditions is a nilpotent matrix and the Lyapunov ex- ( 1 ) ( 1 ) ε ( 1 ) ( 1 ) ( ) = ( − ε )λ ( ) + λ ( + ) + λ ( − ) ; L1 i 1 1 i 2 1 i 1 1 i 1 ponent of the complementary scalar iterative map is greater than (13) ( 1 ) ( 1 ) ε ( 1 ) ( 1 ) zero. ( ) = ( − ε )λ ( ) + λ ( + ) + λ ( − ) , L2 i 1 2 i 2 2 i 1 2 i 1 ( ) ( ) λ 1 (s ) = f (λ 0 (s )) ; k = 1 , 2 ; s = 1 , . . . , n ; n is the number of k k 2.3. Coupled map lattices nodes in the CMLM with periodic boundary conditions ( (i − 1) = n when i = 1 ; (i + 1) = 1 when i = n ). The paradigmatic model of coupled map lattices is the Kaneko Thus, finally, model with periodic boundary conditions [35] : (t+1) (t +1 ) (t +1 ) X ( i ) = L ( i )D + L ( i )D (14) 1 1 2 2 ( + ) ( ) ε ( ) ( ) x t 1 ( i ) = ( 1 − ε ) f (x t ( i ) ) + f (x t ( i + 1 ) ) + f (x t ( i − 1 ) ) where

2 (t +1 ) (t ) ε (t ) (t ) (6) L ( i ) = ( 1 − ε ) f L ( i ) + f L ( i + 1 ) + f L ( i − 1 ) ; 1 1 2 1 1 (t +1 ) (t ) ε (t ) (t ) = , , . . . , , ( ) = ( − ε ) ( ) + ( + ) + ( − ) , where t is a discrete time step, i is a lattice point (i 1 2 N N L2 i 1 f L2 i 2 f L2 i 1 f L2 i 1 is the system size), ε is a coupling parameter, f ( x ) is the mapping (15) function. Further on f ( x ) is set as the Logistic mapping function: ( ) ( ) f ( x ) = ax ( 1 − x ). L 0 ( s ) = λ 0 ( s ); k = 1 , 2 ; s = 1 , . . . , n. k k Fig. 1. The dynamics of the CMLM with identical idempotents does not reveal new features compared to the standard CML because the evolution of each nodal matrix is 1 . 5 4 −0 . 5 − 4 described by two separate idempotents D and D ( Eq. (14) ). Computational experiments are performed at D = and D = at a = 3 . 8284 . 1 2 1 −0 . 1875 − 0 . 5 2 0 . 1875 1 . 5

Parts A, B, C and D show the evolution of matrix elements x 11 , x 12 , x 21 and x 22 accordingly. Parts E and F show the evolution of lattices of eigenvalues λ1 and λ2 ( Eq. (15) ). The coupling parameter ε = 0 . 3 . The x -axis denotes the number of the node; the y -axis –the iteration number.

Thus, the dynamics of the CMLM is described by two sepa- where ⎧ rate recurrences of eigenvalues. In other words, the model will ( 1 ) ( 1 ) ε ( 1 ) ( 1 ) ⎨ L ( i ) = ( 1 − ε )λ ( i ) + λ ( i + 1 ) + λ ( i − 1 ) ; not exhibit new features compared to the standard CML model. 0 0 2 0 0 ( ) ( 1 ) 1 ( ) = ( − ε ) − λ ( ) (21) The following computational experiment illustrates the dynamics M i 1 a 1 2 0 i ⎩ ε ( 1 ) ( 1 ) + − λ ( + ) − λ ( − ) . of CMLM (Fig. 1). 2 a 2 2 0 i 1 2 0 i 1 Thus, finally, 3.2. Nilpotent matrices of initial conditions with identical nilpotents ( + ) (t +1 ) ( + ) t 1 ( ) = ( ) + t 1 ( ) X i L0 i I M i N (22) Let us assume that all matrices of initial conditions are nilpo- tent matrices with the same nilpotent N : where ⎧ (t +1 ) (t ) (0) ( 0 ) ⎪ L ( i ) = ( 1 − ε ) f L ( i ) X ( i ) = λ ( i )I + N . (16) ⎪ 0 0 0 ⎨ ε (t ) (t ) + f L ( i + 1 ) + f L ( i − 1 ) ; 2 0 0 Then, ( + ) (t ) (23) ⎪ M t 1 ( i ) = ( 1 − ε )a 1 − 2 L ( i ) ⎩⎪ 0 (0) ( 0 ) ( 0 ) ( 0 ) ε (t ) (t ) ( ) f X ( i ) = aλ ( i ) 1 − λ ( i ) I + a 1 − 2 λ ( i ) N + − ( + ) − ( − ) t ( ), 0 0 0 2 a 2 2L0 i 1 2L0 i 1 M i ( 1 ) ( ) = λ ( ) + μ 1 ( ) ; ( ) ( ) 0 i I i N (17) 0 ( ) = λ 0 ( ) ; (0) ( ) = μ(0) ( ) = ; = , . . . , . L0 i 0 i M i i 1 i 1 n Such a CMLM can exhibit different features compared to the ( + ) standard CML because the iterative variation of M t 1 (i ) can yield ( ) ( 1 ) ( 1 ) ( ) ( 1 ) 2 0 ( ) = λ ( ) − λ ( ) + μ 1 ( ) − λ ( ) f X i a 0 i 1 0 i I a i 1 2 0 i N the effect of explosive divergence. This effect can occur when the ( 2 ) (2) matrix of initial conditions is a nilpotent matrix and the Lyapunov = λ ( i )I + μ ( i )N ; (18) ( ) 0 μ(t+1) = μ(t) ( − λ t ) exponent of the scalar iterative map a 1 2 0 is greater than zero. These effects are demonstrated in Fig. 2 . It is ( ) ( m ) ( ) interesting to observe that numerical values of the elements of the f m X 0 ( i ) = λ ( i )I + μ m ( i )N , (19) 0 matrices and parameters M ( t ) ( i ) diverge rapidly (all values higher ( m ) than 5 are truncated to 5 in Fig. 2 ). However, the evolution of where μ ( i ) are defined by Eq. (5) . ( ) L t (i ) ( Fig. 2 part F) remains bounded in the interval [0; 1] and The first iteration of the CMLM yields: 0 forms a pattern similar to the pattern generated by the idempo- ( ) ( 1 ) ( ) 1 ( ) = ( ) + 1 ( ) X i L0 i I M i N (20) tent matrices (Fig. 1). This effect can be explained by the structure 1 4 Fig. 2. The effect of explosive divergence in the CMLM with identical nilpotents ( Eq. (22) ). Computational experiments are performed at N = and a = 3 . 8284 . −0 . 25 − 1 ) ( t ) (t ( ) The dynamics of matrix elements x11 , x12 , x21 and x22 is shown in parts A, B, C and D. The evolution of M (i) is show in part E; the evolution of L0 i –inpart F. The number of nodes is 200, the coupling parameter ε = 0 . 3 . All values higher than 5 are truncated to 5 in parts A –E for the clarity of presentation. The effect of explosive

(t +1) (t ) divergence is not observed in part F because the evolution of L0 is affected only by L0 (Eq. (23)).

( + ) of the iterative model in Eq. (23) –the evolution of M t 1 is af- ⎧ ( ) ( + ) (t +1 ) (t ) (t ) ( t ) t , t 1 ⎪ λ ( ) = ( )λ ( ) − λ ( ) ; fected both by M and L but the evolution of L is affected ⎪ 0 i a i 0 i 1 0 i 0 0 ⎪ ( ) (t) ⎨ (t +1 ) (t ) t μ ( i ) = ( 1 − ε )a ( i )μ ( i ) 1 − 2 λ ( i ) only by L0 . 0 ε (t) (t ) (25) ⎪ + a ( i + 1 )μ ( i + 1 ) 1 − 2 λ ( i + 1 ) ⎪ 2 0 ⎩ ( ) (t ) + ( − )μ t ( − ) − λ ( − ) ; 3.3. The general case a i 1 i 1 1 2 0 i 1

(0) (0) Let the matrix of initial conditions at the i th node reads: μ (i ) = 1 ; 0 ≤ λ (i ) ≤ 1 ; 0 ≤ a (i ) ≤ 4 ; i = 1 , . . . , n . ( ) ( ) 0 0 ( ) 0 ( ) ( ) x11 i x12 i In other words, the simplified nilpotent model of CMLM mimics X 0 ( i ) = . Then, ( ) ( ) the evolution of an isolated Logistic map of matrices with a nilpo- 0 ( ) 0 ( ) x21 i x22 i ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) a x ( i ) 1 − x ( i ) − x ( i )x ( i ) ax ( i ) 1 − x ( i ) − x ( i ) ( ) 11 11 12 21 12 11 22 f X 0 ( i ) = . (24) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( 0 ) ( ) − ( ) − ( ) ( ) − ( ) − ( ) ( ) ax21 i 1 x11 i x22 i a x22 i 1 x22 i x12 i x21 i

Now, insertion of Eq. (24) in Eq. (23) yields an iterative algo- rithm for the evolution of CMLM. However, an explicit derivation ( + ) μ( t ) of X t 1 ( i ) becomes infeasible because no additional requirements tent initial matrix –exceptthat the nilpotent parameters (i) are are raised for the structure of initial matrices. linked by the Kaneko model. Moreover, all parameters of the Lo- gistic map a ( i ) can be different. Note that the simplified nilpotent model of CMLM comprises two scalar maps – therefore the lattice (t ) λ ( ) μ(t) 4. The simplified nilpotent model of CMLM parameters 0 i and (i) are computed directly instead of per- forming matrix computations in the lattice. As mentioned previously, a CMLM with idempotent matrices of initial conditions cannot exhibit the effect of explosive divergence. 4.1. The dynamics of the simplified nilpotent model of CMLM at the Therefore, the further study is focused on nilpotent matrices of ini- onset of chaos tial conditions. The evolution of an isolated Logistic map of matri- ces is governed by Eq. (5) . The relations in Eq. (5) serve as a mo- Let us consider the simplified nilpotent model of CMLM with tivation for the design of a simplified nilpotent model of CMLM: all parameters a ( i ) are set to 3.57; i = 1 , . . . , 30 . This value of a ( i ) Fig. 3. The coupling parameter between the nodes can suppress fractal patterns generated by the simplified nilpotent model of CMLM ( Eq. (25) ) at the onset of chaos. All ) = , . . . , λ(0 ( ) ε parameters a(i) are set to 3.57; i 1 30; 0 i are randomly distributed in the interval [0; 1]. The coupling parameter is set to 0.05 (part A), 0.15 (part B) and 0.5 (part C). All values higher than 5 are truncated to 5.

Fig. 4. The coupling parameter between the nodes cannot suppress the effect of explosive divergence generated by the simplified nilpotent model of CMLM ( Eq. (25) ) in the ) = , . . . , λ(0 ( ) ε state of a fully developed chaos. All parameters a(i) are set to 3.8; i 1 30; 0 i are randomly distributed in the interval [0; 1]. The coupling parameter is set to 0.3 (part A), 0.4 (part B) and 0.5 (part C). All values higher than 5 are truncated to 5.

corresponds to the onset of chaos for an isolated scalar Logistic 4.2. The avalanche of divergence caused by one node ( ) λ 0 ( ) map [1]. Initial eigenvalues 0 i are randomly distributed in the interval [0; 1]; the coupling parameter ε is set to 0.05, 0.15 and 0.5 Let us reconsider Fig. 3 A –at the beginning numerical values of ( Fig. 3 ). Fig. 3 shows only the evolution of μ( t ) ( i ) - the dynamics of μ( t ) ( i ) exceed the threshold at some nodes –but the whole lattice ( ) λ t ( ) calms down after 120 time forward iterations. However, the situa- 0 i is described by a closed-form iterative equation in Eq. (23). The system experiences complex transitions at ε = 0 . 05 tion changes completely when the parameter a is changed in one ( Fig. 3 A). A complex fractal-type pattern is generated in the pro- of the nodes ( a ( 13 ) = 3 . 6 ). Initially, the evolution of the system is cess –however none of the nodes does experience the effect of almost identical to the non-perturbed system. But the perturbed the explosive divergence. The whole lattice calms down after 120 node experiences the explosive divergence after 35 time forward time forward iterations when the coupling parameter ε is set to steps ( Fig. 3 B). Moreover, the divergence of the 13th node affects 0.15 ( Fig. 3 B). Finally, the variability of the system is completely the dynamics of the adjacent nodes –and the avalanche of di- suppressed after 20 time forward iterations when ε is set to 0.5 vergence is observed in Fig. 5 A. Note that the initial distribution ( ) ( 0 ) λ 0 ( ) of λ ( i ) and the coupling parameter ε are kept the same as in (Fig. 3C). Note that the initial distribution of 0 i is kept identi- 0 cal in all three simulations. Fig. 3 A. However, the whole lattice experiences the effect of explosive Computational experiments are continued with the same set of divergence when all parameters a ( i ) are set to 3.8; i = 1 , . . . , 30 parameters except the coupling parameter ε. It appears that the ( Fig. 4 ). This value of a ( i ) corresponds to the fully developed chaos avalanche of divergence caused by a single node can be controlled for an isolated scalar Logistic map [1] . The increased coupling be- by increasing the coupling parameter ( ε = 0 . 17 in Fig. 5 B; ε = 0 . 19 tween the nodes cannot suppress the effect of explosive divergence in Fig. 5 C). The simplified nilpotent model of CMLM calms down ( ) λ 0 ( ) after 200 time forward steps in Fig. 5 C. (Fig. 4). Note that the initial distribution of 0 i is kept identical in all simulations. Fig. 5. The explosive divergence at one node excites neighboring nodes of the simplified nilpotent model of CMLM ( Eq. (25) ). The coupling parameter between the nodes ) = , . . . , λ(0 ( ) can control the complexity of the spatiotemporal fractal pattern of divergence. All parameters a(i) are set to 3.57; i 1 30 and 0 i are randomly distributed in the interval [0; 1]. A perturbation of a single node ( a (13) is set to 3.6) causes the avalanche of explosive divergences in the adjacent nodes (part A). The coupling parameter ε = 0 . 15 is set in part A; ε = 0 . 17 in part B; ε = 0 . 19 –in part C. All values higher than 5 are truncated to 5.

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