On fractional-order maps and their synchronization
OOn fractional-order maps and theirsynchronization
Prashant M. Gade , ∗ and Sachin B. Bhalekar , Department of Physics,Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur, IndiaEmail: [email protected] (*Corresponding author) Department of Mathematics,Shivaji University, Kolhapur, India School of Mathematics and Statistics,University of Hyderabad, Hyderabad, IndiaEmail: [email protected] 10, 2020
Abstract
We study the stability of linear fractional order maps. We show thatin the stable region, the evolution is described by Mittag-Leffler func-tions and a well defined effective Lyapunov exponent can be obtained inthese cases. For one-dimensional systems, this exponent can be related tothe corresponding fractional differential equation. A fractional equivalentof map f ( x ) = ax is stable for a c ( α ) < a < where α is a fractionalorder parameter and a c ( α ) ≈ − α . For coupled linear fractional maps,we can obtain ‘normal modes’ and reduce the evolution to effectively one-dimensional system. If the eigenvalues are real the stability of the coupledsystem is dictated by the stability of effectively one-dimensional normalmodes. For complex eigenvalues, we obtain a much richer picture. How-ever, in the stable region, the evolution of modulus is dictated by Mittag-Leffler function and the effective Lyapunov exponent is determined bymodulus of eigenvalues. We extend these studies to synchronized fixedpoints of fractional nonlinear maps. After May’s influential paper [1], the difference equations gained wide popular-ity in all walks of science. They were studied from the viewpoint of nonlineardynamics and chaos which was ubiquitous in disciplines ranging from ecologyto economics. Mathematicians studied it as an interesting object in its right [2].The studies on maps are computationally less intensive. In certain cases, it waseasier to track them analytically. Feigenbaum’s study of period-doubling bifur-cations in logistic maps is an example [3,4]. Lower order difference equations canshow the same phenomena as higher-order differential equations. For example,we can observe chaos in a logistic map that has a single variable while we need1 a r X i v : . [ n li n . C D ] J u l t least three first-order autonomous differential equations to observe chaos. In-sights gained from studies in difference equations were often (though not always)useful in studies in differential equations and vice versa. From control problemsto synchronization schemes, many theoretical ideas have found applications inboth difference and differential equations [5]. In this work, we study fractionallinear difference equations and coupled fractional linear difference equations. Weshow that the analysis of linear fractional difference equations has remarkablesimilarities with corresponding fractional differential equations. For coupled dif-ference equations, the Jacobian, its eigenvalues and eigenvectors play a centralrole. We show that similar concepts can be very useful for coupled fractionaldifference equations.Synchronization of dynamical systems has been extensively studied in thepast three decades both theoretically as well as experimentally. Exact synchro-nization for master-slave type systems as well as synchronization for mutuallycoupled systems has been investigated extensively. Apart from exact synchro-nization, several other types such as anti-synchronization, anticipated synchro-nization, lag synchronization, phase synchronization, generalized synchroniza-tion, etc. have been studied. For spatially extended systems, the connectivitymatrix of underlying topology plays an important role in synchronization. Forexact synchronization, we have a clear mathematical formalism for finding nec-essary and sufficient conditions for synchronization.Ordinary differential equations have successfully described a variety of phys-ical system and found countless applications since Newton and Leibniz in-troduced them. However, memory plays an important role in many physi-cal systems. Apart from a mathematical curiosity, such systems are modeledby fractional-order differential equations from viewpoint of applications. Forfractional-order systems, fractional order maps have been introduced recently.These systems are not very well investigated yet. In a previous work [6], a cou-pled map lattice model of fractional order maps was studied. It was found thatit is possible to have synchronization even in the thermodynamic limit in thesesystems. However, the error reduced as a power-law. The exponent is the sameas the fractional order of maps. We give certain pointers to understand thesefindings analytically in this work.As in differential equations, we start by studying the stability of a fixed pointin linear systems. We study two coupled fractional maps and derive conditionsfor synchronization analytically. For simple linear maps the bounds of synchro-nization and its relation to Mittag-Leffler function can be shown analytically.Asymptotically, Mittag-Leffler function behaves as a power-law and this couldbe the reason for the observation in previous work. For a linear function withconstant slope, this relation can be shown very clearly. We will be studyingsymmetric coupling. However, most of the studies can be easily extended toasymmetric coupling. We give conditions for the stability of the synchronizedstate and demonstrate it with certain examples. We propose that the usual def-inition of Lyapunov exponent using logarithm (which is inverse of exponential)should be appropriately modified to obtain an accurate quantifier that describesthe convergence of trajectories in a stable regime. (Mittag-Leffler function isa power-law asymptotically which is slower decay than exponential. Thus theLyapunov exponent will always be zero if we fit it with an exponential.) How-ever, the divergence of trajectories in the unstable region is still exponential andthe usual definition of Lyapunov exponent may hold in this case.2here are several definitions of fractional differential equations and the sameis true for fractional difference equations. We will study the definition obtainedby Gejji and Deshpande [7]. The evolution depends on the value at all previoustime-steps. The weight of previous values decays as a power-law and thereis a long term memory built-in in the system. It is not surprising that thefluctuations also decay extremely slowly and decay can be approximated bypower-law with power related to the order of fractional difference equation.This is in turn related to the properties of Mittag-Leffler function which playsan important role in fractional differential equations and plays a similar rolehere. In difference equations, the concept of coupled maps was introduced byKapral, Kuznetsov, and Kaneko. Kaneko can be credited for making it popular.The fractional difference equations have been studied only recently. In 1989,Miller and Ross began this investigations [8]. Some of the studies in fractionaldifference equations are due to Atici and coworkers [9,10], Holm [11], and others[12,13]. We extend this definition to the coupled fractional difference equations.While exploratory works on fractional equivalents of known nonlinear maps canyield useful insights, we focus on linear systems in this work. Linearizations ofnonlinear systems are a standard tool in difference equtions as well as coupleddifference equations. Most of analytic work in these systems is dependent onlinearization. Thus understanding linear systems and coupled linear systems isextremely important in nonlinear dynamics. We believe that studies in linearsystems can be equally useful in fractional difference equations. In this section, we introduce an alternative viz. Effective Lyapunov Exponent(ELE) for the classical Lyapunov exponent. We will follow the notation anddefinition used by Deshpande and Gejji [7]. They define the fractional equiv-alent for the x ( n + 1) = f ( x ( n )) in the following manner. They constructan discrete Caputo-type fractional difference operator and define u ( t ) = u + α ) (cid:80) tj =1 Γ( t − j + α )Γ( t − j +1) ( f ( j − , u ( j − − u ( j − in general. We assume thatthe function f does not depend on time. We define g α ( k ) = Γ( k + α )Γ( k +1) and alterna-tively write above expression as u ( t ) = u + α ) (cid:80) tj =1 g α ( t − j ) ( f ( u ( j − − u ( j − .Few one-dimensional maps such as the Gaussian map and Bernoulli maphave been studied in this work. Liu has numerically investigated coupled frac-tional Henon map [14]. Henon map is a two-dimensional map and Liu introducesmemory in only one of the variables. We will call such systems ‘fractional differ-ence equations of inhomogeneous order’. On the other hand, we will investigatemaps of homogeneous order.Consider f ( x ) = rx where r ∈ R u ( t ) = u + 1Γ( α ) t (cid:88) j =1 g α ( t − j ) ( r − u ( j − (1)can be identified with the continuous-time system D α u ( t ) = ( r − u ( t ) (2)for sufficiently small values of coefficient.3igure 1: Mittag-Leffler decay.The continuous-time system (2) has exact solution in terms of Mittag-Lefflerfunction as below u ( t ) = u (0) E α (( r − t α ) . (3)Fig. 1 shows the Mittag-Leffler decay for various values of α .This can be alternatively written as u ( t ) = u (0) E α ( λ e t α ) . (4)where λ e is effective Lyapunov exponent. For a linear first order ordinary differ-ential equation x (cid:48) ( t ) = λx ( t ) , the solution would be x ( t ) = x (0) exp( λt ) where λ is Lyapunov exponent. When it is negative, the system goes to absorbing state.Our formulation could be considered as generalization λ e = lim t −→∞ t − α E − α (cid:18) u ( t ) u (0) (cid:19) . (5)This formulation is very similar to standard definition of Lyapunov exponentfor α = 1 where E α ( x ) = exp( x ) . We will demonstrate that this quantity is awell-defined quantity which indeed converges in the stable regime. On the otherhand, if we insist on using the definition of Lyapunov exponent used for ordinarydifferential equations, it will lead to zero value. The reason is that Mittag-Leffler function is a power-law asymptotically which is slower than exponential.Like Lyapunov exponent, the effective Lyapunov exponent is negative in theabsorbing state. In Fig. 2, we sketch the numerically computed λ e for variousvalues of r . It is clear that λ e = r − .The system is unstable for r > for any α . However, lower bound a c ( α ) depends on α . Lower bound a c ( α ) → − as α → . This is an expected limitfor integer order difference equation. We define two coupled maps in this setting and define. x ( t ) = x + 1Γ( α ) t (cid:88) j =1 g α ( t − j ) G ( x ( j − , y ( t − j )) ,y ( t ) = y + 1Γ( α ) t (cid:88) j =1 g α ( t − j ) G ( y ( j − , x ( t − j )) (6)4igure 2: Computation of λ e for few values of r . The value obtained is r − .where G ( a, b ) = δf ( a ) + qf ( b ) − a . Case 1: Real ‘Normal Modes’
First we consider the case f ( x ) = x for which the coefficient matrix of thesystem has real eigenvalues viz. δ + q and δ − q . Now we consider two newvariables u ( t ) = x ( t ) + y ( t ) and v ( t ) = x ( t ) − y ( t ) and obtain u ( t ) = u + 1Γ( α ) t (cid:88) j =1 g α ( t − j )(( δ + q − u ( j − , (7) v ( t ) = v + 1Γ( α ) t (cid:88) j =1 g α ( t − j )(( δ − q − v ( j − . (8)This is a simple decoupled system of linear difference equations.Again, the discrete-time equations above can be identified with the continuous-time system D α u ( t ) = ( δ + q − u ( t ) (9) D α v ( t ) = ( δ − q − v ( t ) (10)for sufficiently small values of coefficient and exact solution in terms of Mittag-Leffler function is given as u ( t ) = u (0) E α (( δ + q − t α ) . (11) v ( t ) = v (0) E α (( δ − q − t α ) . (12)Thus the effective Lyapunov exponents are given by δ + q − and δ − q − for the above decoupled system. We compute the system for T = 8 × − time-steps. The decay is extremely slow for smaller α . After discarding first2000 time-steps, we check if the distance of ( x ( t ) , y ( t )) from origin every 100time-steps and the trajectory is stable if the distance does not increase till time T . The stability regime for the above system is plotted as a function of δ and q in Fig. 3. The stability region is a rhombus bounded by lines parallel to δ = q δ = − q . For all values of α , the system is unstable for δ + q − > and δ − q − > . The other two bounding lines change with α . For q = 0 , wehave an effectively one-dimensional system. The bounds for q = 0 are δ = 1 and δ = a c ( α ) . As α → , a c ( α ) → − . The lines enclosing stability region aregiven by δ ± q = 1 and δ ± q = a c ( α ) .‘ It is clear from a c ( α ) are close to − α .This explains the rhombus structure in Fig. 3. (For f ( x ) = rx , the phase di-agram will be similar except that the values of δ and q will be scaled to δr and qr .) Synchronization and antisynchronization
We consider the linear system (6) with real normal modes. As shown in (7)–(8),the sum and difference variables u ( t ) = x ( t ) + y ( t ) and v ( t ) = x ( t ) − y ( t ) areeffectively decoupled. We can also say the u corresponds to (1 , mode and v corresonds to (1 , − mode. If the effective Lyapunov exponent correspondingto variable u is in the unstable region while the one corresponding to v is instable region, we will find that v ( t ) → . Thus x ( t ) → y ( t ) implying synchro-nization. On the other hand if λ e corresponding to v ( t ) is in unstable regionand corresponding to u ( t ) is in the stable region we will observe that v ( t ) → .This phenomenon is termed as antisynchronization. Fig. 4 shows both thesephenomena for different values of parameters. We also find that the decayingmode decays slower than exponential making it necessary to give a new defini-tion for Lyapunov exponent. However, growing mode increases exponentially.The reason may be that approximating fractional difference equation by frac-tional differential equation may not be valid for large values. In coupled maplattices, it is known that the condition for chaotic synchronization is that allmodes except one corresponding to the mean, i.e. eigenmode (1 , , . . . shoulddecay [15]. We can obtain normal modes in fractional system in a similar mannerand find conditions for chaotic synchronization. Case 2: Complex ‘Normal Modes’
Now let us consider another coupled system x ( t ) = x + 1Γ( α ) t (cid:88) j =1 g α ( t − j )( δx ( j −
1) + qy ( t − j ) − x ( j − , (13) y ( t ) = y + 1Γ( α ) t (cid:88) j =1 g α ( t − j )( δy ( j − − qx ( t − j ) − y ( j − . (14)6igure 4: Chaotic synchronization and antisynchronization for symmetricallycoupled maps. α = 0 . ,a) δ =0.85, q = − . and b) δ = 0 . , q = 0 . The eigenvalues of coefficient matrix of the system (13)–(14) are δ ± ιq .We have plotted stability region in δ − q space for various values of α inFig. 5. This structure is far richer than one obtained in Fig. 3. As α → , thestability region tends to the unit circle in the complex plane which is a stabilityregion for integer-order difference maps in two dimensions.As α decreases, the stability region gradually deforms from an unit circle toa non-convex shape.Consider z ( t ) = x ( t ) + ιy ( t ) . z ( t ) = 1Γ( α ) t (cid:88) j =1 g α ( t − j ) ((( δ − − ιq ) x ( j −
1) + ι ( δ − − ιq ) y ( j − . (15)Thus z ( t ) = 1Γ( α ) t (cid:88) j =1 g α ( t − j )( δ − − ιq ) z ( j − . (16)Similarly, if we define ¯ z ( t ) = x ( t ) − ιy ( t ) , we get ¯ z ( t ) = 1Γ( α ) t (cid:88) j =1 g α ( t − j )( δ −
1) + ιq )¯ z ( j − . (17)The variables z and ¯ z can be associated to ordered pairs ( x, y ) and ( x, − y ) describing a complex number and its conjugate in complex plane.The equivalent continuous-time system of (13)–(14) is given by D α x = ( δ − x + qy (18) D α y = − qx + ( δ − y. (19)The general solution of system (18)–(19) is x ( t ) + ιy ( t ) = E α (( δ − − ιq ) t α ) ( x (0) + ιy (0)) . (20)If we define z ( t ) = x ( t ) + ιy ( t ) and λ (cid:48) e = ( δ − − ιq ) , we have7igure 5: Stability region for coupled maps with antisymmetric coupling. Itis clear that the stability region approaches unit circle as α → while it issignificantly different for small α . The stability region is shown for differentvalues of α in ascending order in a) and in descending order in b). z ( t ) = E α ( λ (cid:48) e t α ) z (0) . (21)This motivates us to define the effective Lyapunov exponent λ (cid:48) e of the abovesystem as λ (cid:48) e = lim t −→∞ t − α E − α (cid:18) z ( t ) z (0) (cid:19) . (22)This is a well-defined quantity and we always find λ (cid:48) e = δ − − ιq numer-ically. This is a complex number. On the other hand, Lyapunov exponentsfor dynamical systems have real Lyapunov exponents. We believe that a singlereal number can determine stability for difference equation because the stabilitycondition for integer-order differential equation is that the real part of eigen-values is less than zero. This is a single condition. On the other, for fractionalorder differential equations, stability condition is given by two lines. Thus asingle number may not be enough to determine stability of fractional equations.However, if we insist that the effective Lyapunov exponents should be real, wecan take −| λ (cid:48) e | as an effective Lyapunov exponent. Alternatively, we may definethe effective Lyapunov exponent λ e of system (18)–(19) as λ e = lim t −→∞ (cid:12)(cid:12)(cid:12)(cid:12) t − α E − α (cid:18) x ( t ) + ιy ( t ) x (0) + ιy (0) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:112) ( δ − + q . (23)Note that, √ x ( t ) + y ( t ) √ x (0) + y (0) = | E α (( δ − − ιq ) t α ) | . In general, it is not possible8igure 6: Effective Lyapunov exponent λ e is plotted as function of expression (cid:112) ( δ − + q .to find the inverse of composite function on right side. However, for α ∈ [0 . , and ( δ, q ) in the stability region we observed that the quantity √ x ( t ) + y ( t ) √ x (0) + y (0) decays with λ e and we have λ e ≈ lim t −→∞ t − α E − α (cid:18) √ x ( t ) + y ( t ) √ x (0) + y (0) (cid:19) .In Fig. 6 we have plotted effective Lyapunov found numerically for variousvalues of δ , q and α as a function of above quantity and it is clear that there isan excellent match. Let us try to extend the analysis to nonlinear systems. Let us consider stabilityof the fixed point of a nonlinear system. Nonlinear system can be linearized by f ( x ) = f ( x ∗ ) + f (cid:48) ( x ∗ )( x − x ∗ ) + . . . . We define y = x − x ∗ . Now x n +1 = f ( x n ) isequivalent to y n +1 = x n +1 − x ∗ = f ( x n ) − x ∗ = f (cid:48) ( x ∗ ) y n = ry n where r = f (cid:48) ( x ∗ ) .Thus we can conjecture that the stability regime for the fixed point is given by a c ( α ) < f (cid:48) ( x ∗ ) < . Interestingly, this conjecture works. Consider the cases ofBernoulli map and Gauss map considered by Deshpande and Gejji [7].For Bernouli map f ( x ) = rx | mod , r > , we can guess that for r < , thefixed point is stable for any value of α . This is precisely what Deshpande andGejji [7] find. In the case of Gauss map given by f ( x ) = exp( − . x ) + β , thefixed point can be found numerically using bisection method or other methods.The fixed point is close to β . The slope is negative and the stability regime isdependent on value of α . One expects this fixed point to be more stable if α increases. It can be checked that a c (0 .
4) = − . , a c (0 .
6) = − . and a c (0 .
8) = − . . The critical points corespond to values of β at which the f (cid:48) ( x ∗ ) matches with these values. Thus we expect the fixed point to be stablefor β > . for α = 0 . , for β > . for α = 0 . and β > . for α = 0 . (cf.Fig. 7).This can be confirmed from the above paper [7] as well as independentsimulations. We start with β as an initial condition.9igure 7: a)Bifurcation diagram for Gauss map as a function of β for variousvalues of α . b) Local slope of fixed point x ∗ is plotted as a function of β . Verticallines show the values of β at which f (cid:48) ( x ∗ ) = a c ( α ) for these values of α . Thefixed point is stable in the expected region even for a nonlinear function. Analytic studies in nonlinear dynamics or coupled nonlinear systems are oftenbased on local linearization of dynamics. Thus linear systems serve as a basiswhich helps undertanding dynamics in these systems. We have studied the frac-tional equivalent of linear maps, which are not studied before to our knowledge.Our studies indicate that the results obtained are useful in studies of nonlinearsystems as well. For coupled systems, we have studied the stability of the fixedpoint. We also find that the conditions for chaotic synchronization and antisyn-chronization and find that the conditions are very similar to those obtained forcoupled integer order maps.For one-dimensional f ( x ) = rx , we find that the stability regime is given by a c ( α ) < r < . In the stable regime the dynamics is governed by Mittag-Lefflerfunction. We also define the effective Lyapunov exponent and find that r − iseffective Lyapunov exponent in this case.For two coupled linear systems, the behavior is different for symmetric andantisymmetric coupling. The analysis is motivated by study of linear differenceequations which is essentially the theory of matrices. We can reduce dynamics to‘normal modes’ which could be real or complex. We can find effective Lyapunovexponents in these cases as well. When the normal modes are real, the stabilitycondition is the same as the condition for a single linear map for each mode.A much richer picture is observed for complex normal modes. The effectiveLyapunov exponents, in this case, are complex which is not entirely unexpected.The stability region of a continuous-time fractional-order dynamical systemis a superset of that of classical integer-order one. As we increase the fractional10rder to 1, the cone-like stability region of the fractional case gets contracted tothe left-half complex plane and we get the usual region of stability of classicalcase. In this article, we showed that the stability properties of discrete-time sys-tems are different. The cardioid-like stability region of fractional order systemgets deformed and converted to the unit disc as we increase the order to 1.We also extend this work to fixed points of nonlinear maps and confirm thata similar criterion holds. This work can be extended in many directions. We cantry to find the stability of periodic orbits of the higher period and find routesto chaos in low-dimensional fractional difference equations. Acknowledgements
PMG thanks DST-SERB for financial assistance (Ref. EMR/2016/006686).SBB acknowledges the Science and Engineering Research Board (SERB), NewDelhi, India for the Research Grant (Ref. MTR/2017/000068) under Mathe-matical Research Impact Centric Support (MATRICS) Scheme.
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