Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo Families of Maps
Abstract
In this paper we extend the notion of an family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factoriallaw memory which is asymptotically powerlaw memory. We introduce the fractional difference Universal, Standard, and Logistic Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically powerlaw memory.
footnoteinfo]The author acknowledges support from the Joseph Alexander Foundation, Yeshiva University.
yu,ci]Edelman M.
fractinal derivative \sepfractional difference \sepattractor \sepdiscrete map \seppowerlaw \sepmemory
1 Introduction
Systems with memory are common in biology, social sciences, physics, and engineering (see review Edelman (2014c)). The most frequently encountered type of memory in natural and engineering systems is powerlaw memory. This leads to the possibility of describing them by fractional differential equations, which have powerlaw kernels. Nonlinear integrodifferential fractional equations are difficult to simulate numerically  this is why in Tarasov and Zaslavsky (2008) the authors introduced fractional maps, which are equivalent to fractional differential equations of nonlinear systems experiencing periodic delta functionkicks, and proposed to use them for the investigation of general properties of nonlinear fractional dynamical systems. Bifurcation diagrams in the fractional Logistic Map related to a scheme of numerical integration of fractional differential equations were considered by Stanislavsky (2006).
An adequate description of discrete natural systems with memory can be obtained by using fractional difference equations (see Miller and Ross (1989); Gray and Zhang (1988); Agarwal (2000); Atici and Eloe (2009); Anastassiou (2009); Chen et al. (2011); Wu et al. (2014); Wu and Baleanu (2014)). In Chen et al. (2011); Wu et al. (2014); Wu and Baleanu (2014) the authors demonstrated that in some cases fractional difference equations are equivalent to maps (which we will call fractional difference maps) with falling factoriallaw memory, where falling factorial function is defined as
(1) 
Falling factoriallaw memory is asymptotically powerlaw memory 
(2) 
and we may expect that fractional difference maps have properties similar to the properties of fractional maps.
The goal of the present paper is to introduce fractional difference families of maps depending on memory and nonlinearity parameters consistent with the previous research of fractional maps (see Edelman (2014c); Tarasov and Zaslavsky (2008); Edelman and Tarasov (2009); Tarasov (2009a, b, 2011); Edelman (2011); Edelman and Taieb (2013); Edelman (2013a, b)) in order to prepare a background for an investigation of general properties of systems with asymptotically powerlaw memory.
In the next section (Sec. 2) we will remind the reader how the regular Universal, Standard, and Logistic Maps (see Chirikov (1979.); Lichtenberg and Lieberman (1992); Zaslavsky (2008); May (1976)) are generalized to obtain fractional Caputo Families of Maps (FM). In Sec. 3 we’ll present some basics on fractional difference/sum operators, which will be used in Sec. 4 to derive the fractional difference Caputo Universal, Standard, and Logistic FMs. In Sec. 5 we’ll present some results on properties of fractional difference Caputo Standard FM.
2 Fractional Families of Maps
Fractional FM were introduced in Edelman (2013a), further investigated in Edelman (2013b), and reviewed in Edelman (2014c). The Universal FM was obtained by integrating the following equation:
(3) 
where , , , , with the initial conditions corresponding to the type of fractional derivative to be used. is a nonlinear function which depends on the nonlinearity parameter . It is called Universal because integration of Eq. (3) in the case and produces the regular Universal Map (see Zaslavsky (2008)). In what follows the author considers Eq. (3) with the leftsided Caputo fractional derivative (see Samko et al. (1993); Kilbas et al. (2006); Podlubny (1999))
(4) 
where , , is a fractional integral, is the gamma function, and the initial conditions are
(5) 
There are two reasons to restrict the consideration in this paper to the Caputo case (the RiemannLiouville case won’t be considered): a) as in the case of fractional differential equations, in the case of fractional difference equations it is much easier to define initial conditions for Caputo difference equations than for RiemannLiouville difference equations; b) the main goal of this work is to compare fractional and fractional difference maps, and the case of Caputo maps serves the purpose. Comparison of the RiemannLiouville and Caputo Standard Maps was considered in Edelman (2011).
The problem Eqs. (3)–(5) is equivalent to the Volterra integral equation of the second kind () Kilbas et al. (2006)
(6) 
After the introduction the Caputo Universal FM can be written as (see Tarasov (2011))
(7) 
where and .
In the case and with Eq. (7) produces the well–known Standard Map (see Chirikov (1979.)), which on a torus can be written as
(8) 
(9) 
This is why the Caputo Universal FM Eq. (7) with
(10) 
is called the Caputo Standard FM:
(11) 
where .
In the case and Eq. (7) produces the well–known Logistic Map (see May (1976))
(12) 
This is why the Caputo Universal FM Eq. (7) with
(13) 
is called the Caputo Logistic FM:
(14) 
where .
The Caputo Standard and Logistic FMs were investigated in detail in Edelman (2014c, 2013a, 2013b) for the case which is important in applications.

For the Caputo Standard and Logistic FMs are identically zeros: .

For the Caputo Standard FM is
(15) and the Caputo Logistic FM is
(16) 
For the 1D Standard Map is the Circle Map with zero driving phase
(17) and the 1D Logistic FM is the Logistic Map Eq. (12).

For the Caputo Standard FM is
(18) (19) where and the Caputo Logistic FM is
(20) (21)
3 Fractional Difference/Sum Operators
In this paper we will adopt the definition of the fractional sum ()/difference () operator introduced in Miller and Ross (1989) as
(24) 
Here is defined on and on , where , and falling factorial is defined by Eq. (1). As Miller and Ross noticed, their way to introduce the discrete fractional sum operator based on the Green’s function approach is not the only way to do so. In Gray and Zhang (1988) the authors defined the discrete fractional sum operator generalizing the fold summation formula in a way similar to the way in which the fractional Riemann–Liouville integral is defined in fractional calculus by extending the Cauchy fold integral formula to the real variables. They mentioned the following theorem but didn’t present a proof. {thm} For
(25) 
where , are the summation variables. {pf} Indeed, this formula is obviously true for . Let’s assume that Eq. (25) is true for :
(26) 
where is the number of combinations from a given set of elements. Then, for Eq. (26) gives
(27) 
Now Eq. (25) can be obtained from
(28) 
Here we used the identity
(29) 
which is true for and can be proven by induction for any
(30) 
This ends the proof.
As we see, two different approaches are consistent with the definition of the fractioanal sum operator given by Miller and Ross (see also Atici and Eloe (2009)). For and Anastassiou (2009) defined the fractional (left) Caputolike difference operator as
(31) 
where is the th power of the forward difference operator defined as . The proof (see Miller and Ross (1989) p.146) that in the limit approaches the identity operator can be easily extended to the operator. In this case the definition Eq. (31) can be extended to all real with for . Then, the Anastassiou’s fractional Taylor difference formula Anastassiou (2009)
(32) 
where is defined on , , and for is valid for any real and for integer is identical to the integer discrete Taylor’s formula (see p.28 in Agarwal (2000))
(33) 
As it was noticed in Wu et al. (2014) and Wu and Baleanu (2014), Lemma 2.4 from Chen et al. (2011) on the equivalency of the fractional Caputolike difference and sum equations can be extended to all real and formulated as follows: {thm} The Caputolike difference equation
(34) 
with the initial conditions
(35) 
is equivalent to the fractional sum equation
(36)  
where . Here we should notice that the authors of Wu et al. (2014) and Wu and Baleanu (2014) didn’t consider the Caputo difference operator with integer . As a result, Theorem 3 is not valid for integer values of with their definition .
4 Fractional Difference Families of Maps
In the following we assume that is a nonlinear function with the nonlinearity parameter and adopt the Miller and Ross proposition to let . Now, with , Theorem 3 can be formulated as {thm} For , the Caputolike difference equation
(38) 
where , with the initial conditions
(39) 
is equivalent to the map with falling factoriallaw memory
(40) 
where which we will call the fractional difference Caputo Universal Family of Maps. The fractional difference Caputo Universal FM is similar to the general form of the Caputo Universal FM Eq. (7). Both of them can be written as
(41) 
where are the initial value of momenta defined as for fractional maps and as for fractional difference maps; for fractional maps and for fractional difference maps; for fractional maps and for the fractional difference maps. for fractional maps and for fractional difference maps. Asymptotically, both expressions for coincide because of Eq. (2).
4.1 Fractional Difference Universal Fm
Let’s consider the case . Then the difference Eq. (34) produces
(42) 
and the equivalent sum equation is
(43) 
After the introduction with the assumption the map equations indeed can be written as the well–known 2D Universal Map
(44) 
(45) 
which for produces the Standard Map Eqs. (8) and (9). In the rest of this paper we’ll call Eq. (40) with the fractional difference Caputo Standard Family of Maps.
In the case the fractional difference Caputo Universal FM is
(46) 
which produces the Logistic Map if . In the rest of this paper we’ll call Eq. (40) with the fractional difference Caputo Logistic Family of Maps.
4.2 Difference Caputo Standard and Logistic FMs

In the case the 0D Standard Map turns into the Sine Map (see, e.g., Lalescu (2010))
(47) 
The 0D Logistic Map is
(48)
4.3 Fractional Difference Caputo Standard and Logistic FMs

For the fractional difference Standard Map is
(49) which after the shift of the independent variable coincides with the “fractional sine map” proposed in Wu et al. (2014).

The fractional difference Logistic Map can be writen as
(50) The fractional Logistic Map introduced in Wu and Baleanu (2014) does not converge to the Logistic map in the case .
4.4 Difference Caputo Standard and Logistic FMs
4.5 Fractional Difference Caputo Standard and Logistic FMs

For the fractional difference Standard Map is
(52) 
The fractional difference Logistic Map is
(53)
Let’s introduce ; then these maps can be written as 2D maps with memory:

The fractional difference Standard Map is
(54) (55) which in the case is identical to the ”fractional standard map” introduced in Wu et al. (2014) (Eq. (18) with there).

The fractional difference Logistic Map is
(56) (57)
4.6 Difference Caputo Standard and Logistic FMs
5 Conclusion
As we saw in Sec. 3, the fractional difference operator is a natural extension of the difference operator. The simplest fractional difference equations (of the Eq. (34) type), where the fractional difference on the left side is equal to a nonlinear function on the right side, are equivalent to maps with falling factoriallaw (asymptotically powerlaw) memory Eq. (36). Systems with powerlaw memory play an important role in nature (see Edelman (2014c)) and investigation of their general properties is important for understanding behavior of natural systems.
Properties of the fractional difference Caputo Standard FM were investigated in detail in Edelman (2014a) (see also Sec. 3 in Edelman (2014b). Qualitatively, properties of the fractional difference and fractional maps (maps with falling factorial and powerlaw memory) are similar. The similarity reveals itself in the dependence of systems’ properties on the memory () and nonlinearity () parameters (bifurcation diagrams, see Figs. 1, 2, and 3), powerlaw convergence to attractors, nonuniqueness of solutions (intersection of trajectories and overlapping of attractors), and cascade of bifurcations and intermittent cascade of bifurcations type behaviors (see Figs. 4 and 5).
The differences of the properties of the falling factoriallaw memory maps from the powerlaw memory maps are the results of the differences in weights of the recent (with ) values of the maps’ variables at the time instants in the definition of the present values at time and are significant when (especially when ), see Fig. 1.
The author acknowledges support from the Joseph Alexander Foundation, Yeshiva University. The author expresses his gratitude to E. Hameiri, H. Weitzner, and G. Ben Arous for the opportunity to complete this work at the Courant Institute and to V. Donnelly for technical help.
Footnotes
 thanks: [
References
 Agarwal, R. (2000). Difference equations and inequalities. Marcel Dekker, New York.
 Anastassiou, G. (2009). Discrete fractional calculus and inequalities. http://arxiv.org/abs/0911.3370.
 Atici, F. and Eloe, P. (2009). Initial value problems in discrete fractional calculus. Proc.Am.Math.Soc., 137, 981–989.
 Chen, F., Luo, X., and Zhou, Y. (2011). Existence results for nonlinear fractional difference equation. Adv.Differ.Eq., 2011, 713201.
 Chirikov, B. (1979.). A universal instability of many dimensional oscillator systems. Phys. Rep., 52, 263–379.
 Edelman, M. (2011). Fractional standard map: Riemannliouville vs. caputo. Commun. Nonlin. Sci. Numer. Simul., 16, 4573–4580.
 Edelman, M. (2013a). Fractional maps and fractional attractors. part i: families of maps. Discontinuity, Nonlinearity, and Complexity, 1, 305–324.
 Edelman, M. (2013b). Universal fractional map and cascade of bifurcations type attractors. Chaos, 23, 033127.
 Edelman, M. (2014a). Caputo standard family of maps: fractional difference vs. fractional. Phys. Lett. A (submitted).
 Edelman, M. (2014b). Fractional maps and fractional attractors. part ii: fractional difference families of maps. arXiv:1404.4906v2.
 Edelman, M. (2014c). Fractional maps as maps with powerlaw memory. In A. Afraimovich, A. Luo, and X. Fu (eds.), Nonlinear dynamics and complexity, 79–120. Springer, New York.
 Edelman, M. and Taieb, L. (2013). New types of solutions of nonlinear fractional differential equations. In A. Almeida, L. Castro, and F.O. Speck (eds.), Advances in Harmonic Analysis and Operator Theory; Series: Operator Theory: Advances and Applications, volume 229, 139–155. Springer, Basel.
 Edelman, M. and Tarasov, V.E. (2009). Fractional standard map. Phys. Lett. A, 374, 279–285.
 Gray, H. and Zhang, N.F. (1988). On a new definition of the fractional difference. Math. Comput., 50, 513–529.
 Kilbas, A., Srivastava, H., and Trujillo, J. (2006). Theory and application of fractional differential equations. Elsevier, Amsterdam.
 Lalescu, C. (2010). Patterns in the sine map bifurcation diagram. arXiv:1011.6552.
 Lichtenberg, A. and Lieberman, M. (1992). Regular and chaotic dynamics. Springer, Berlin.
 May, R. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467.
 Miller, K. and Ross, B. (1989). Fractional difference calculus. In H. Srivastava and S. Owa (eds.), Univalent functions, fractional calculus, and their applications, 139–151. Ellis Howard, Chichester, 1st edition.
 Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego.
 Samko, S., Kilbas, A., and Marichev, O. (1993). fractional integrals and derivatives theory and applications. Gordon and Breach, New York.
 Stanislavsky, A. (2006). Longterm memory contribution as applied to the motion of discrete dynamical system. Chaos, 16, 043105.
 Tarasov, V. (2009a). Differential equations with fractional derivative and universal map with memory. J. Phys. A, 42, 465102.
 Tarasov, V. (2009b). Discrete map with memory from fractional differential equation of arbitrary positive order. J. Math. Phys., 50, 122703.
 Tarasov, V. (2011). Fractional dynamics: application of fractional calculus to dynamics of particles, fields, and media. Springer, New York.
 Tarasov, V. and Zaslavsky, G. (2008). Fractional equations of kicked systems and discrete maps. J. Phys. A, 41, 435101.
 Wu, G.C. and Baleanu, D. (2014). Discrete fractional logistic map and its chaos. Nonlin. Dyn., 75, 283–287.
 Wu, G.C., Baleanu, D., and Zeng, S.D. (2014). Discrete chaos in fractional sine and standard maps. Phys. Lett. A, 378, 484–487.
 Zaslavsky, G. (2008). Hamiltonian chaos and fractional dynamics. Oxford University Press, Oxford.