Differential Equations with Fractional Derivative and Universal Map with Memory
aa r X i v : . [ n li n . C D ] J u l Journal of Physics A 42 (2009) 465102
Differential Equations with Fractional Derivativeand Universal Map with Memory
Vasily E. Tarasov , Courant Institute of Mathematical Sciences, New York University251 Mercer Street, New York, NY 10012, USA Skobeltsyn Institute of Nuclear Physics,Moscow State University, Moscow 119991, Russia
E-mail: [email protected]
Abstract
Discrete maps with long-term memory are obtained from nonlinear differential equa-tions with Riemann-Liouville and Caputo fractional derivatives. These maps are general-izations of the well-known universal map. The memory means that their present state isdetermined by all past states with special forms of weights. To obtain discrete map fromfractional differential equations, we use the equivalence of the Cauchy-type problems andto the nonlinear Volterra integral equations of second kind. General forms of the universalmaps with memory, which take into account general initial conditions, for the cases of theRiemann-Liouville and Caputo fractional derivatives, are suggested.
A dynamical system consists of a set of possible states, together with a rule that determines thepresent state in terms of past states. If we require that the rule be deterministic, then we candefine the present state uniquely from the past states. A discrete-time system without memorytakes the current state as input and updates the situation by producing a new state as output.All physical classical models are described by differential or integro-differential equations, and1he discrete-time systems can be considered as a simplified version of these equations. Adiscrete form of the time evolution equation is called the map. Maps are important becausethey encode the behavior of deterministic systems. The assumption of determinism is that theoutput of the map can be uniquely determined from the input. In general, the present stateis uniquely determined by all past states, and we have a discrete map with memory. Discretemaps are used for the study of evolution problems, possibly as a substitute of differentialequations [1, 2, 3]. They lead to a much simpler formalism, which is particularly useful insimulations. The universal discrete map is one of the most widely studied maps. In thispaper, we consider discrete maps with memory that can be used to study solutions of fractionaldifferential equations [4, 5, 6, 7].The nonlinear dynamics can be considered in terms of discrete maps. It is a very importantstep in understanding the qualitative behavior of systems described by differential equations.The derivatives of non-integer orders are a generalization of the ordinary differentiation ofinteger order. Fractional differentiation with respect to time is characterized by long-termmemory effects. The discrete maps with memory are considered in [8, 9, 10, 11, 12, 13, 14].The interesting question is a connection of fractional differential equations and discrete mapswith memory. It is important to derive discrete maps with memory from the equation of motion.In Ref. [14], we prove that the discrete maps with memory can be derived from differentialequations with fractional derivatives. The fractional generalization of the universal map wasobtained [14] from a differential equation with Riemann-Liouville fractional derivatives. TheRiemann-Liouville derivative has some notable disadvantages such as the hyper-singular im-proper integral, where the order of singularity is higher than the dimension, and nonzero of thefractional derivative of constants, which would entail that dissipation does not vanish for a sys-tem in equilibrium. The desire to formulate initial value problems for mechanical systems leadsto the use of Caputo fractional derivatives rather than Riemann-Liouville fractional derivative.It is possible to state that the Caputo fractional derivatives allows us to give more clearmechanical interpretation. At the same time, we cannot state that the Riemann-Liouville frac-tional derivative does not have a physical interpretation and that it shows unphysical behavior.Physical interpretations of the Riemann-Liouville fractional derivatives are more complicated2han Caputo fractional derivatives. But the Riemann-Liouville fractional derivatives naturallyappear for real physical systems in electrodynamics. We note that the dielectric susceptibil-ity of a wide class of dielectric materials follows, over extended frequency ranges, a fractionalpower-law frequency dependence that is called the ”universal” response [15, 16]. As was provedin [17, 18], the electromagnetic fields in such dielectric media are described by differentialequations with Riemann-Liouville fractional time derivatives. These fractional equations for”universal” electromagnetic waves in dielectric media are common to a wide class of materi-als, regardless of the type of physical structure, chemical composition, or of the nature of thepolarizing species. Therefore, we cannot state that Riemann-Liouville fractional time deriva-tives do not have a physical interpretation. The physical interpretation of these derivatives inelectrodynamics is connected with the frequency dependence of the dielectric susceptibility. Asa result, the discrete maps with memory that are connected with differential equations withRiemann-Liouville fractional derivatives are very important, and these derivatives naturallyappear for real physical systems.For computer simulation and physical application, it is very important to take into accountthe initial conditions for discrete maps with memory that are obtained from differential equa-tions with Riemann-Liouville fractional time derivatives. In Ref. [14], these conditions are notobtained. In this paper, to take into account the initial condition, we use the equivalence of thedifferential equation with Riemann-Liouville and Caputo fractional derivatives and the Volterraintegral equation. This approach is more general than the auxiliary variable method that isused in Ref. [14]. The proof of the result for Riemann-Liouville fractional derivatives is morecomplicated in comparison with the results for the Caputo fractional derivative. In this paper,we prove that the discrete maps with memory can be obtained from differential equations withthe Caputo fractional derivative. The fractional generalization of the universal map is derivedfrom a fractional differential equation with Caputo derivatives.The universal maps with memory are obtained by using the equivalence of the fractionaldifferential equation and the Volterra integral equation. We reduce the Cauchy-type problemfor the differential equations with the Caputo and Riemann-Liouville fractional derivativesto nonlinear Volterra integral equations of second kind. The equivalence of this Cauchy type3roblem for the fractional equations with the Caputo derivative and the correspondent Volterraintegral equation was proved by Kilbas and Marzan in [19, 20]. We also use that the Cauchy-type problem for the differential equations with the Riemann-Liouville fractional derivative canbe reduced to a Volterra integral equation. The equivalence of this Cauchy-type problem andthe correspondent Volterra equation was proved by Kilbas, Bonilla, and Trujillo in [21, 22].In Section 2, differential equations with integer derivative and universal maps without mem-ory are considered to fix notations and provide convenient references. In Section 3, fractionaldifferential equations with the Riemann-Liouville derivative and universal maps with memoryare discussed. In Section 4, the difference between the Caputo and Riemann-Liouville fractionalderivatives are discussed. In Section 5, fractional differential equations with the Caputo deriva-tive and correspondent discrete maps with memory are considered. A fractional generalizationof the universal map is obtained from kicked differential equations with the Caputo fractionalderivative of order 1 < α ≤
2. The usual universal map is a special case of the universal mapwith memory. Finally, a short conclusion is given in section 6.
In this section, differential equations with derivative of integer order and the universal mapwithout memory are considered to fix notations and provide convenient references.Let us consider the equation of motion D t x ( t ) + KG [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) = 0 (1)in which perturbation is a periodic sequence of delta-function-type pulses (kicks) following withperiod T = 2 π/ν , K is an amplitude of the pulses, D t = d /dt , and G [ x ] is some real-valuedfunction. It is well-known that this differential equation can be represented in the form of thediscrete map x n +1 − x n = p n +1 T, p n +1 − p n = − KT G [ x n ] . (2)Equations (2) are called the universal map. For details, see for example [1, 2, 3].4he traditional method of derivation of the universal map equations from the differentialequations is considered in Section 5.1 of [2]. We use another method of derivation of theseequations to fix notations and provide convenient references. We obtain the universal map byusing the equivalence of the differential equation and the Volterra integral equation. Proposition 1.
The Cauchy-type problem for the differential equations D t x ( t ) = p ( t ) , (3) D t p ( t ) = − K G [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) , (4) with the initial conditions x (0) = x , p (0) = p (5) is equivalent to the universal map equations of the form x n +1 = x + p ( n + 1) T − KT n X k =1 G [ x k ] ( n + 1 − k ) , (6) p n +1 = p − KT n X k =1 G [ x k ] . (7) Proof.
Consider the nonlinear differential equation of second order D t x ( t ) = G [ t, x ( t )] , (0 ≤ t ≤ t f ) (8)on a finite interval [0 , t f ] of the real axis, with the initial conditions x (0) = x , ( D t x )(0) = p . (9)The Cauchy-type problem of the form (8), (9) is equivalent to the Volterra integral equation ofsecond kind x ( t ) = x + p t + Z t dτ G [ τ, x ( τ )] ( t − τ ) . (10)Using the function G [ t, x ( t )] = − KG [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) , nT < t < ( n + 1) T , we obtain x ( t ) = x + p t − KT n X k =1 G [ x ( kT )] ( t − kT ) . (11)For the momentum p ( t ) = D t x ( t ), equation (11) gives p ( t ) = p − KT n X k =1 G [ x ( kT )] . (12)The solution of the left side of the ( n + 1)th kick x n +1 = x ( t n +1 −
0) = lim ε → x ( T ( n + 1) − ε ) , (13) p n +1 = p ( t n +1 −
0) = lim ε → p ( T ( n + 1) − ε ) , (14)where t n +1 = ( n + 1) T , has the form (6) and (7).This ends the proof. (cid:3) Remark 1 .We note that equations (6) and (7) can be rewritten in the form (2). Using equations (6) and(7), the differences x n +1 − x n and p n +1 − p n give equations (2) of the universal map. Remark 2 .If G [ x ] = − x , then equations (2) give the Anosov-type system x n +1 − x n = p n +1 T, p n +1 − p n = KT x n . (15)For G [ x ] = sin( x ), equations (2) are x n +1 − x n = p n +1 T, p n +1 − p n = − KT sin( x n ) . (16)This map is known as the standard or Chirikov-Taylor map [1].6 Riemann-Liouville fractional derivative and universalmap with memory
In this section, we discuss nonlinear differential equations with the left-sided Riemann-Liouvillefractional derivative D αt defined for α > D αt x ( t ) = D nt I n − αt x ( t ) = 1Γ( n − α ) d n dt n Z t x ( τ ) dτ ( t − τ ) α − n +1 , ( n − < α ≤ n ) , (17)where D nt = d n /dt n , and I αt is a fractional integration [4, 7, 6].We consider the fractional differential equation D αt x ( t ) = G [ t, x ( t )] , (18)where G [ t, x ( t )] is a real-valued function, 0 ≤ n − < α ≤ n , and t >
0, with the intialconditions ( D α − kt x )(0+) = c k , k = 1 , ..., n. (19)The notation ( D α − kt x )(0+) means that the limit is taken at almost all points of the right-sidedneighborhood (0 , ε ), ε >
0, of zero as follows( D α − kt x )(0+) = lim t →
0+ 0 D α − kt x ( t ) , ( k = 1 , ..., n − , ( D α − nt x )(0+) = lim t →
0+ 0 I n − αt x ( t ) . The Cauchy-type problem (18) and (19) can be reduced to the nonlinear Volterra integralequation of second kind x ( t ) = n X k =1 c k Γ( α − k + 1) t α − k + 1Γ( α ) Z t G [ τ, x ( τ )] dτ ( t − τ ) − α , (20)where t >
0. The result was obtained by Kilbas, Bonilla, and Trujillo in [21, 22]. For α = n = 2,equation (20) gives (10).The Cauchy-type problem (18) and (19) and the nonlinear Volterra integral equation (20)are equivalent in the sense that, if x ( t ) ∈ L (0 , t f ) satisfies one of these relations, then it also7atisfies the other. In [21, 22] (see also Theorem 3.1. in Section 3.2.1 of [7]), this result isproved by assuming that the function G [ t, x ] belongs to L (0 , t f ) for any x ∈ W ⊂ R .Let us give the basic theorem regarding the nonlinear differential equation involving theRiemann-Liouville fractional derivative. Kilbas-Bonilla-Trujillo Theorem.
Let W be an open set in R and let G [ t, x ] , where t ∈ (0 , t f ] and x ∈ W , be a real-valued function such that G [ t, x ] ∈ L (0 , t f ) for any x ∈ W . Let x ( t ) be a Lebesgue measurable function on (0 , t f ) . If x ( t ) ∈ L (0 , t f ) , then x ( t ) satisfies almosteverywhere equation (18) and conditions (19) if, and only if, x ( t ) satisfies almost everywherethe integral equation (20). Proof.
This theorem is proved in [21, 22] (see also Theorem 3.1. in Section 3.2.1 of [7]). (cid:3)
In Ref. [14] we consider a fractional generalization of equation (1) of the form D αt x ( t ) + K G [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) = 0 , (1 < α ≤ , (21)where t >
0, and D αt is the Riemann-Liouville fractional derivative defined by (17). Let usgive the following theorem for equation (21). Proposition 2.
The Cauchy-type problem for the fractional differential equation of theform (21) with the initial conditions ( D α − t x )(0+) = c , ( D α − t x )(0+) = ( I − αt x )(0+) = c (22) is equivalent to the equation x ( t ) = c Γ( α ) t α − + c Γ( α − t α − − KT Γ( α ) n X k =1 G [ x ( kT )] ( t − kT ) α − , (23) where nT < t < ( n + 1) T . roof. Using the function G [ t, x ( t )] = − K G [ x ] ∞ X k =1 δ (cid:16) tT − k (cid:17) , (24)equation (21) has the form of (18) with the Riemann-Liouville fractional derivative of order α ,where 1 < α ≤
2. It allows us to use the Kilbas-Bonilla-Trujillo theorem. As a result, equation(21) with initial conditions (19) of the form (22) is equivalent to the nonlinear Volterra integralequation x ( t ) = c Γ( α ) t α − + c Γ( α − t α − − K Γ( α ) ∞ X k =1 Z t dτ G [ x ( τ )] ( t − τ ) α − δ (cid:16) τT − k (cid:17) , (25)where t >
0. If nT < t < ( n + 1) T , then the integration in (25) with respect to τ gives (23).This ends the proof. (cid:3) To obtain equations of discrete map a momentum must be defined. There are two possiblitiesof defining the momentum: p ( t ) = D α − t x ( t ) , p ( t ) = D t x ( t ) . (26)Let us use the first definition. Then the momentum is defined by the fractional derivativeof order α −
1. Using the definition of the Riemann-Liouville fractional derivative (17) in theform D αt x ( t ) = D t I − αt x ( t ) , (1 < α ≤ , (27)we define the momentum p ( t ) = D α − t x ( t ) = 1Γ(2 − α ) ddt Z t x ( τ ) dτ ( t − τ ) α − , (1 < α ≤ , (28)where x ( τ ) is defined for τ ∈ (0 , t ). Then D αt x ( t ) = D t p ( t ) , (1 < α ≤ . (29)Using momentum p ( t ) and coordinate x ( t ), equation (21) can be represented in the Hamiltonianform D α − t x ( t ) = p ( t ) , (30)9 t p ( t ) = − KG [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) . (31) Proposition 3.
The Cauchy type problem for the fractional differential equations of theform (30) and (31) with the initial conditions ( D α − t x )(0+) = c , ( D α − t x )(0+) = ( I − αt x )(0+) = c (32) is equivalent to the discrete map equations x n +1 = c T α − Γ( α ) ( n + 1) α − + c T α − Γ( α −
1) ( n + 1) α − − KT α Γ( α ) n X k =1 G [ x k ] ( n + 1 − k ) α − , (33) p n +1 = c − KT n X k =1 G [ x k ] . (34) Proof.
We use Proposition 2 to prove this statement. To obtain an equation for themomentum (28), we use the following fractional derivatives of power functions (see Section 2.1in [7]), a D αt ( t − a ) β − = Γ( β )Γ( β − α ) ( t − a ) β − − α , α ≥ , β > , t > a, (35) D αt t α − k = 0 , k = 1 , ..., n, n − < α ≤ n. (36)These equations give D αt t α − = Γ( α ) , D αt t α − = 0 , and a D αt ( t − a ) α − = Γ( α ) . We note that equation (23) for x ( τ ) can be used only if τ ∈ ( nT, t ), where nT < t < ( n +1) T .The function x ( τ ) in the fractional derivative D αt of the form (28) must be defined for all τ ∈ (0 , t ). We cannot take the derivative D αt of the functions ( τ − kT ) α − that are defined10or τ ∈ ( kT, t ). In order to use equation (23) on the interval (0 , t ), we must modify the sum inequation (23) by using the Heaviside step function. Then equation (23) has the form x ( τ ) = c Γ( α ) τ α − + c Γ( α − τ α − − KT Γ( α ) n X k =1 G [ x ( kT )] ( τ − kT ) α − θ ( τ − kT ) , (37)where τ ∈ (0 , t ). Using the relation D αt (cid:16) ( t − a ) α − θ ( t − a ) (cid:17) = a D αt ( t − a ) α − = Γ( α ) , (38)equations (28) and (37) give p ( t ) = c − KT n X k =1 G [ x ( kT )] , (39)where nT < t < ( n + 1) T . Then the solution of the left side of the ( n + 1)-th kick p n +1 = c − KT n X k =1 G [ x k ] . (40)As a result, we obtain a universal map with memory in the form of equations (33) and (34).This ends the proof. (cid:3) Remark 3 .For α = n = 2 equations (33) and (34) give the usual universal map (6) and (7). Remark 4 .We note that the map (33) and (34) with c = p , c = 0was obtained in [14] in the form x n +1 = T α − Γ( α ) n X k =1 p k +1 V α ( n − k + 1) , (41) p n +1 = p n − KT G ( x n ) , (1 < α ≤ , (42)11here p = c , and the function V α ( z ) is defined by V α ( z ) = z α − − ( z − α − , ( z ≥ . (43)In Ref. [14], we obtain these map equations by using an auxiliary variable ξ ( t ) such that C D − αt ξ ( t ) = x ( t ) . The nonlinear Volterra integral equations and the general initial conditions (32) are not usedin [14]. In the general case, the fractional differential equation of the kicked system (21) isequivalent to the discrete map equations x n +1 = T α − Γ( α ) n X k =1 p k +1 V α ( n − k + 1) + c T α − Γ( α −
1) ( n + 1) α − , (44) p n +1 = p n − KT G ( x n ) , (1 < α ≤ , (45)where p = c . Here we take into account the initial conditions (32). The second term of theright-hand side of equation (44) is not considered in [14]. Using − < α − <
0, we havelim n →∞ ( n + 1) α − = 0 . Therefore, the case of large values of n is equivalent to c = 0.Let us give the proposition regarding the second definition of the momentum p ( t ) = D t x ( t ). Proposition 4.
The Cauchy-type problem for the fractional differential equations D t x ( t ) = p ( t ) , (46) D αt x ( t ) = − K G [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) , (1 < α < , (47) with the initial conditions ( D α − t x )(0+) = c , ( D α − t x )(0+) = ( I − αt x )(0+) = c (48)12 s equivalent to the discrete map equations x n +1 = c T α − Γ( α ) ( n + 1) α − + c T α − Γ( α −
1) ( n + 1) α − − KT α Γ( α ) n X k =1 G [ x k ]( n + 1 − k ) α − . (49) p n +1 = c T α − Γ( α −
1) ( n +1) α − + c ( α − T α − Γ( α −
1) ( n +1) α − − KT α − Γ( α − n X k =1 G [ x k ]( n +1 − k ) α − . (50) Proof.
We define the momentum p ( t ) = D t x ( t ) . If nT < t < ( n + 1) T , then the differentiation of (23) with respect to t gives p ( t ) = c Γ( α − t α − + c ( α − α − t α − − KT Γ( α − n X k =1 G [ x ( kT )] ( t − kT ) α − . (51)Here we use the relation Γ( α ) = ( α − α − , (1 < α ≤ . Using equations (23) and (51), we can obtain the solution of the left side of the ( n + 1)-th kick(13) and (14). As a result, we have equations (49) and (50).This ends the proof. (cid:3) Remark 5 .Equations (49) and (50) describe a generalization of equations (6) and (7). If α = n = 2, and c = x , c = p , then equation (49) gives (6) and (7). Remark 6 .In equations (50) and (51), we can use c ( α − α −
1) = c Γ( α − < α <
2. 13 emark 7 .If we use the definition p ( t ) = D t x ( t ), then the Hamiltonian form of the equations of motionwill be more complicated than (30) and (31) since D t I − αt x ( t ) = I − αt D t x ( t ) . Remark 8 .Note that we use the usual momentum p ( t ) = D t x ( t ). In this case, the values c and c are notconnected with p (0) and x (0). If we use the momentum p ( t ) = D α − t x ( t ), then c = p (0). In Ref. [14] we consider nonlinear differential equations with Riemann-Liouville fractionalderivatives. The discrete maps with memory are obtained from these equations. The problemswith initial conditions for the Riemann-Liouville fractional derivative are not discussed.The Riemann-Liouville fractional derivative has some notable disadvantages in applicationsin mechanics such as the hyper-singular improper integral, where the order of singularity ishigher than the dimension, and nonzero of the fractional derivative of constants, which wouldentail that dissipation does not vanish for a system in equilibrium. The desire to use the usualinitial value problems for mechanical systems leads to the use of Caputo fractional derivatives[7, 6] rather than the Riemann-Liouville fractional derivatives.The left-sided Caputo fractional derivative [23, 24, 25, 7] of order α > C D αt f ( t ) = 1Γ( n − α ) Z t dτ D nτ f ( τ )( t − τ ) α − n +1 = I n − αt D nt f ( t ) , (52)where n − < α < n , and I αt is the left-sided Riemann-Liouville fractional integral of order α > I αt f ( t ) = 1Γ( α ) Z t f ( τ ) dτ ( t − τ ) − α , ( t > . (53)This definition is, of course, more restrictive than the Riemann-Liouville fractional derivative[4, 7] in that it requires the absolute integrability of the derivative of order n . The Caputo14ractional derivative first computes an ordinary derivative followed by a fractional integral toachieve the desire order of fractional derivative. The Riemann-Liouville fractional derivative iscomputed in the reverse order. Integration by part of (52) will lead to C D αt x ( t ) = D αt x ( t ) − n − X k =0 t k − α Γ( k − α + 1) x ( k ) (0) . (54)It is observed that the second term in equation (54) regularizes the Caputo fractional derivativeto avoid the potentially divergence from singular integration at t = 0. In addition, the Caputofractional differentiation of a constant results in zero C D αt C = 0 . Note that the Riemann-Liouville fractional derivative of a constant need not be zero, and wehave D αt C = t − α Γ(1 − α ) C. If the Caputo fractional derivative is used instead of the Riemann-Liouville fractional deriva-tive, then the initial conditions for fractional dynamical systems are the same as those for theusual dynamical systems. The Caputo formulation of fractional calculus can be more applicablein mechanics than the Riemann-Liouville formulation.
In this section, we study a generalization of differential equation (1) by the Caputo fractionalderivative. The universal map with memory is derived from this fractional equation.We consider the nonlinear differential equation of order α , where 0 ≤ n − < α ≤ n , C D α x ( t ) = G [ t, x ( t )] , (0 ≤ t ≤ t f ) , (55)involving the Caputo fractional derivative C D αt on a finite interval [0 , t f ] of the real axis, withthe initial conditions ( D kt x )(0) = c k , k = 0 , ..., n − . (56)15ilbas and Marzan [19, 20] proved the equivalence of the Cauchy-type problem of the form(55), (56) and the Volterra integral equation of second kind x ( t ) = n − X k =0 c k k ! t k + 1Γ( α ) Z t dτ G [ τ, x ( τ )] ( t − τ ) α − (57)in the space C n − [0 , t f ]. For α = n = 2 equation (57) gives (10).Let us give the basic theorem regarding the nonlinear differential equation involving theCaputo fractional derivative. Kilbas-Marzan Theorem.
The Cauchy-type problem (55) and (56) and the nonlinearVolterra integral equation (57) are equivalent in the sense that, if x ( t ) ∈ C [0 , t f ] satisfies oneof these relations, then it also satisfies the other. Proof.
In [19, 20] (see also [7], Theorem 3.24.) this theorem is proved by assuming that afunction G [ t, x ] for any x ∈ W ⊂ R belong to C γ (0 , t f ) with 0 ≤ γ < γ < α . Here C γ (0 , t f )is the weighted space of functions f [ t ] given on (0 , t f ], such that t γ f [ t ] ∈ C (0 , t f ). This endsthe proof. (cid:3) We consider the fractional differential equation of the form C D αt x ( t ) + K G [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) = 0 , (1 < α < , (58)where C D αt is the Caputo fractional derivative, with the initial conditions x (0) = x , ( D x )(0) = p . (59)Using p ( t ) = D t x ( t ), equation (58) can be rewritten in the Hamilton form. Proposition 5.
The Cauchy-type problem for the fractional differential equations D t x ( t ) = p ( t ) , (60)16 D α − t p ( t ) = − K G [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) , (1 < α < , (61) with the initial conditions x (0) = x , p (0) = p (62) is equivalent to the discrete map equations x n +1 = x + p ( n + 1) T − KT α Γ( α ) n X k =1 ( n + 1 − k ) α − G [ x k ] , (63) p n +1 = p − KT α − Γ( α − n X k =1 ( n + 1 − k ) α − G [ x k ] . (64) Proof.
We use the Kilbas-Marzan theorem with the function G [ t, x ( t )] = − KG [ x ( t )] ∞ X k =1 δ (cid:16) tT − k (cid:17) . The Cauchy-type problem (58) and (59) is equivalent to the Volterra integral equation of secondkind x ( t ) = x + p t − K Γ( α ) ∞ X k =1 Z t dτ ( t − τ ) α − G [ x ( τ )] δ (cid:16) tT − k (cid:17) , (65)in the space of continuously differentiable functions x ( t ) ∈ C [0 , t f ].If nT < t < ( n + 1) T , then equation (65) gives x ( t ) = x + p t − KT Γ( α ) n X k =1 ( t − kT ) α − G [ x ( kT )] . (66)We define the momenta p ( t ) = D t x ( t ) . (67)Then equations (66) and (67) give p ( t ) = p − KT Γ( α − n X k =1 ( t − kT ) α − G [ x ( kT )] , ( nT < t < ( n + 1) T ) , (68)where we use Γ( α ) = ( α − α − n + 1)-th kick (13) and (14) can be represented byequations (63) and (64), where we use the condition of continuity x ( t n + 0) = x ( t n − (cid:3) Remark 9 .Equations (63) and (64) define a generalization of the universal map. This map is derived froma fractional differential equation with Caputo derivatives without any approximations. Themain property of the suggested map is a long-term memory that means that their present statedepends on all past states with a power-law form of weights.
Remark 10 .If α = 2, then equations (63) and (64) give the universal map of the form (6) and (7) thatis equivalent to equations (2). As a result, the usual universal map is a special case of thisuniversal map with memory. Remark 11 .By analogy with Proposition 5, it is easy to obtain the universal map with memory fromfractional equation (58) with α > The suggested discrete maps with memory are generalizations of the universal map. These mapsdescribe fractional dynamics of complex physical systems. The suggested universal maps withmemory are equivalent to the correspondent fractional kicked differential equations. We obtaina discrete map from fractional differential equation by using the equivalence of the Cauchy-typeproblem and the nonlinear Volterra integral equation of second kind. An approximation forfractional derivatives of these equations is not used.It is important to obtain and to study discrete maps which correspond to the real physicalsystems described by the fractional differential equations. Media with memory in mechanics andelectrodynamics, we can consider viscoelastic and dielectric materials as a media with memory.We note that the dielectric susceptibility of a wide class of dielectric materials follows, over18xtended frequency ranges, a fractional power-law frequency dependence that is called the”universal” response [15, 16]. As was proved in [17, 18], the electromagnetic fields in suchdielectric media are described by differential equations with fractional time derivatives. Thesefractional equations for electromagnetic waves in dielectric media are common to a wide class ofmaterials, regardless of the type of physical structure, chemical composition, or of the nature ofthe polarizing species, whether dipoles, electrons or ions. We hope that it is possible to obtainthe discrere maps with memory which correspond to the real dielectric media described by thefractional differential equations.
Acknowledgments
This work was supported by the Office of Naval Research, Grant No. N00014-02-1-0056 andRosnauka No.02.740.11.0244.
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