Discrete Maps with Memory: Fractional Dynamics

• Discrete maps with memory are discrete-time systems whose updates rely on a weighted history of past states, introducing non-Markovian behavior.
• Fractional calculus and exact memory kernels enable constructing diverse map families, including logistic, standard, and stochastic variants with power-law memory.
• Memory effects alter dynamics by suppressing chaos and reshaping attractors, with practical applications in economics, ecology, and physics.

Discrete maps with memory are discrete-time dynamical systems in which the update at step $n+1$ depends not only on the present state but also on earlier states, either through a finite remembered window or through the entire past trajectory. In the memoryless case one has $x_{n+1}=f(x_n)$, whereas in the memory-bearing case the evolution is temporally nonlocal and therefore non-Markovian in the current state alone. Across the literature, this notion appears in several mathematically distinct forms: weighted full-history recurrences, exact stroboscopic maps derived from fractional differential equations with kicks, fractional-difference maps with falling factorial-law memory, finite-step memory maps, variable-length stochastic chains represented by interval maps, and stochastic input-output maps with finite memory (Stanislavsky, 2011, Tarasov, 2011, Edelman, 2014, Collet et al., 2012).

1. Definitions and main classes

The most elementary distinction is between ordinary one-step maps and hereditary maps. In the former, the present state is sufficient for prediction. In the latter, the next iterate depends on a set of past states $\{x_k\}_{k\le n}$, usually through explicit kernels or selection rules. This dependence can be full-history, finite-delay, variable-length, or stochastic. A recurrent theme is that memory changes not only quantitative behavior but the very state description: a scalar recurrence may require a higher-dimensional lift, an augmented history state, or an integral representation to become Markovian (Stanislavsky, 2011, Góra et al., 2016).

Memory is not synonymous with ordinary delay. In the direct logistic-map construction of long-term memory, dependence extends over all previous iterates with slowly varying weights, and the paper explicitly distinguishes this from maps depending on only a few past values such as $x_n,x_{n-1}$ (Stanislavsky, 2011). By contrast, the one-step memory tent-map model uses precisely the pair $(x_{n-1},x_n)$, lifting the process to a planar map $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$ (Góra et al., 2016).

Class	Representative form	Representative papers
Full-history deterministic map	$$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$	(Stanislavsky, 2011)
Exact fractional map from kicked FDE	$$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$	(Tarasov, 2011, Tarasov, 2011)
Fractional-difference map	$$x_{n+1}=\cdots-\frac{1}{\Gamma(\alpha)}\sum (n-s-m+\alpha)^{(\alpha-1)}G_K(\cdot)$$	(Edelman, 2014)
One-step memory map	$x_{n+1}=\tau(\alpha x_n+(1-\alpha)x_{n-1})$	(Góra et al., 2016)
Variable-length or stochastic memory	conditional law or output mean depends on selected past states	(Collet et al., 2012, Ahmadypour et al., 2020, Saha, 18 Jun 2025)

This diversity suggests that “discrete maps with memory” is best understood as a structural category rather than a single formalism: the common feature is temporal nonlocality in discrete time.

2. Fractional-calculus derivations and exact memory kernels

A major strand derives discrete maps with memory from kicked fractional differential equations by exploiting the equivalence between fractional Cauchy-type problems and nonlinear Volterra integral equations of the second kind. In Tarasov’s Caputo-based formulation, for $x_{n+1}=f(x_n)$0, the kicked equation

$x_{n+1}=f(x_n)$1

with standard initial conditions yields the exact discrete map

$x_{n+1}=f(x_n)$2

for $x_{n+1}=f(x_n)$3. The defining feature is the kernel $x_{n+1}=f(x_n)$4, which makes every past kick contribute to the present state with a power-law weight (Tarasov, 2011).

Closely related constructions were given for Riemann–Liouville and Caputo derivatives in the universal-map setting. For $x_{n+1}=f(x_n)$5, the Caputo universal map takes the form

$x_{n+1}=f(x_n)$6

$x_{n+1}=f(x_n)$7

In the Riemann–Liouville formulation, analogous maps contain fractional-power initial terms inherited from fractional initial conditions. These constructions reduce to the ordinary universal map at $x_{n+1}=f(x_n)$8, which is the memoryless integer-order limit (Tarasov, 2011).

Tarasov and Zaslavsky extended the same logic to fractional universal, standard, dissipative, and kicked damped rotator maps. In the basic $x_{n+1}=f(x_n)$9 universal-map case,

$\{x_k\}_{k\le n}$0

with

$\{x_k\}_{k\le n}$1

The asymptotic decay $\{x_k\}_{k\le n}$2 gives long-term power-law memory (Tarasov et al., 2011).

A further unification is provided by Hilfer-derivative maps. For the kicked equation

$\{x_k\}_{k\le n}$3

the exact piecewise solution produces discrete maps with memory for arbitrary positive order $\{x_k\}_{k\le n}$4. The endpoint values $\{x_k\}_{k\le n}$5 and $\{x_k\}_{k\le n}$6 recover the Riemann–Liouville and Caputo cases, so the Hilfer framework interpolates between the two established classes (Tarasov, 3 Sep 2025).

Discrete fractional calculus leads to a related but distinct kernel. In the fractional difference Caputo framework, the map

$\{x_k\}_{k\le n}$7

has falling factorial-law memory, with asymptotic kernel

$\{x_k\}_{k\le n}$8

This is asymptotically power-law memory rather than exactly power-law memory at finite lag (Edelman, 2014).

3. Canonical families and representative constructions

A direct hereditary construction appears in the study of logistic maps under long-term memory. The starting point is the replacement of $\{x_k\}_{k\le n}$9 by a normalized weighted history sum,

$x_n,x_{n-1}$0

with explicit coefficients $x_n,x_{n-1}$1 chosen in analogy with numerical fractional integration. The special cases are central: $x_n,x_{n-1}$2 gives the ordinary memoryless map, $x_n,x_{n-1}$3 gives long-term memory, and $x_n,x_{n-1}$4 yields a trapezoid-like full-memory average over the entire history (Stanislavsky, 2011).

This direct construction was applied to the triangular map

$x_n,x_{n-1}$5

and the quadratic logistic map

$x_n,x_{n-1}$6

The weighting scheme preserves fixed points and induces nonlocal temporal coupling across the whole trajectory (Stanislavsky, 2011).

Economic models provide another route to logistic maps with memory. Starting from

$x_n,x_{n-1}$7

and adding periodic crisis terms modeled by delta functions, the exact discrete analog for $x_n,x_{n-1}$8 becomes

$x_n,x_{n-1}$9

with

$(x_{n-1},x_n)$0

After rescaling, this yields an explicit logistic map with power-law memory (Tarasova et al., 2017). Closely related accelerator models give exact discrete accelerators with memory, such as

$(x_{n-1},x_n)$1

showing that capital adjustment depends on the entire output history rather than only on the current period (Tarasova et al., 2016).

The broader $(x_{n-1},x_n)$2-family program collects universal, standard, and logistic maps into continuous families indexed by fractional order. For $(x_{n-1},x_n)$3, one obtains the fractional standard family; for $(x_{n-1},x_n)$4, the fractional logistic family. Integer $(x_{n-1},x_n)$5 recovers the classical finite-dimensional maps, while noninteger $(x_{n-1},x_n)$6 gives full-history memory (Edelman, 2013).

Multidimensional generalized fractional maps extend these ideas to dissipative maps such as Hénon and Lozi. In their common form,

$(x_{n-1},x_n)$7

the kernel $(x_{n-1},x_n)$8 is either exact power-law or asymptotically power-law-like. This includes both fractional and fractional-difference Hénon and Lozi maps (Edelman, 2024). Fractional generalizations of Zaslavsky and Hénon maps were also derived directly from kicked damped equations with non-integer-order derivatives (Tarasov, 2011).

4. Dynamical effects of memory

Long-term memory can suppress, reorganize, or qualitatively replace the standard route to chaos. In the direct logistic-map construction, numerical bifurcation diagrams with $(x_{n-1},x_n)$9 iterations show that increasing $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$0 weakens period-doubling, washes out fine structure, and shrinks chaotic bands. The reported threshold is $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$1, with the abstract stating that for $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$2 the memory effects dominate chaos. The same paper remarks that for the triangular map the Lyapunov exponent decreases for $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$3, and interprets memory as an intrinsic dissipative mechanism (Stanislavsky, 2011).

Fractional maps exhibit a broader range of behaviors. The fractional-map survey reports periodic sinks, attracting slow diverging trajectories, attracting accelerator mode trajectories, chaotic attractors, and cascade of bifurcations type trajectories. It also emphasizes power-law convergence to attractors, non-uniqueness of solutions, intersection of trajectories and overlapping of attractors, and intermittent cascade of bifurcations type behaviors (Edelman, 2013). Fractional-difference maps display qualitatively similar properties, but the finite-lag discrepancy between falling factorial-law memory and exact power-law memory can matter, especially for $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$4 and particularly as $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$5 (Edelman, 2014).

For multidimensional generalized fractional maps, asymptotic $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$6-cycles satisfy explicit algebraic conditions derived from the memory kernel. In the Hénon case, the fixed points remain the same as in the classical map, but the period-two branches and bifurcation thresholds become memory-dependent through coefficients such as $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$7 (Edelman, 2024).

A different memory mechanism appears in the one-step memory tent-map model

$$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$8

For $$G_\alpha(x,y)=\bigl(y,\tau(\alpha y+(1-\alpha)x)\bigr)$$9, the orbits are described statistically by an absolutely continuous invariant measure in two dimensions. As $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$0 approaches $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$1 from below, the support becomes thinner. At $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$2, all points have period $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$3 or are eventually period $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$4. For $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$5, all starting points in $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$6 except $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$7 are attracted to the fixed point $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$8. At $$x_{n+1}=\frac{1}{(1+\alpha)n^\alpha}\sum_{i=0}^n c_i^{(n)}f(x_i)$$9, every point except $$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$0 is periodic, eventually periodic, or attracted to the line $$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$1, whose non-fixed points are period $$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$2 (Góra et al., 2016).

5. Stochastic, symbolic, and finite-memory variants

Not all discrete maps with memory are fractional. Collet and Galves study chains of infinite order and chains with memory of variable length, then construct topological Markov interval maps whose invariant probability measures coincide with the stationary laws of the given chains. The central formula,

$$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$3

shows that the derivative of the deterministic interval map is the reciprocal of the transition probability conditioned on the entire symbolic past. In the variable-length case, derivative constancy on context-determined cylinders characterizes maps associated with probabilistic context trees (Collet et al., 2012).

Memory can also be finite and explicitly engineered for optimization. For symmetric linear maps

$$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$4

the robust worst-case convergence problem under only eigenvalue bounds $$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$5 has a sharp solution: one memory slot is already optimal, and adding more memory slots cannot improve the guaranteed convergence rate. The optimal one-memory parameters are

$$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$6

with $$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$7 (Sarlette, 2014).

In communication theory, a discrete Poisson channel with memory is a stochastic input-output map whose output means are

$$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$8

The channel is finite-memory and causal. Under only a total-power constraint and $$x_{n+1}^{(s)}=\cdots-\frac{K T^{\alpha-s}}{\Gamma(\alpha-s)}\sum_{k=1}^{n}(n+1-k)^{\alpha-1-s}G[x_k]$$9, the optimal binary codebook places all power in one slot for one codeword and in a slot shifted by the memory length for the other, so that the two convolved mean trajectories do not overlap (Ahmadypour et al., 2020).

A different stochastic example is the random walk with $$x_{n+1}=\cdots-\frac{1}{\Gamma(\alpha)}\sum (n-s-m+\alpha)^{(\alpha-1)}G_K(\cdot)$$0 memory channels. In the two-channel case,

$$x_{n+1}=\cdots-\frac{1}{\Gamma(\alpha)}\sum (n-s-m+\alpha)^{(\alpha-1)}G_K(\cdot)$$1

followed by saturation to keep $$x_{n+1}=\cdots-\frac{1}{\Gamma(\alpha)}\sum (n-s-m+\alpha)^{(\alpha-1)}G_K(\cdot)$$2. Each $$x_{n+1}=\cdots-\frac{1}{\Gamma(\alpha)}\sum (n-s-m+\alpha)^{(\alpha-1)}G_K(\cdot)$$3 is a uniformly sampled past increment from the entire history, so the map has complete-history memory. For RW2MC, the exact mean and second moment show diffusive, superdiffusive, and ballistic-second-moment regimes, and the model admits a detailed correspondence with a three-color Pólya-type urn evolving by two drawings (Saha, 18 Jun 2025).

6. Exactness, applications, and limitations

Several papers emphasize that the maps are exact discrete analogs rather than finite-difference approximations. This is explicit in the Caputo-based universal maps (Tarasov, 2011), the economic logistic and accelerator maps (Tarasova et al., 2017, Tarasova et al., 2016), the predator-prey maps with memory and kicks (Tarasov, 22 Sep 2025), and the Hilfer-derivative discrete maps with memory (Tarasov, 3 Sep 2025). In these constructions, exactness comes from solving the kicked fractional equation via a Volterra or fractional-integral representation and then evaluating the solution at kick times.

Applications span economics, ecology, and physical systems. The economic literature interprets memory as power-law fading in investment-output dynamics and crises as periodic delta-like price splashes (Tarasova et al., 2017, Tarasova et al., 2016). Predator-prey models use Caputo derivatives and impulsive interactions to produce exact Lotka–Volterra and Kolmogorov maps with memory (Tarasov, 22 Sep 2025). Fractional-map papers point to viscoelastic media, dielectric materials, Hamiltonian systems, adaptation in biological systems, and human memory as settings where power-law memory is natural (Edelman, 2013).

The mathematical costs of memory are substantial. Full-history maps require summation over all previous states, and the direct logistic-memory paper explicitly notes that the duration of calculations grows notably with $$x_{n+1}=\cdots-\frac{1}{\Gamma(\alpha)}\sum (n-s-m+\alpha)^{(\alpha-1)}G_K(\cdot)$$4 (Stanislavsky, 2011). Several studies also delimit their scope carefully. The direct logistic-memory work reports $$x_{n+1}=\cdots-\frac{1}{\Gamma(\alpha)}\sum (n-s-m+\alpha)^{(\alpha-1)}G_K(\cdot)$$5 empirically rather than deriving it rigorously (Stanislavsky, 2011). The accelerator and predator-prey papers focus on derivation and exact correspondence rather than full stability or bifurcation theory (Tarasova et al., 2016, Tarasov, 22 Sep 2025). The fractional-map survey explicitly excludes short-memory truncations, concentrating instead on exact power-law memory over the full past (Edelman, 2013).

Taken together, these results define discrete maps with memory as a broad family of nonlocal discrete-time systems in which hereditary dependence is the organizing principle. Whether introduced by normalized full-history averages, by fractional differential equations with kicks, by discrete fractional calculus, by symbolic context trees, or by stochastic finite-memory channels, memory changes the effective state space, the admissible invariant measures, and the route from regular dynamics to instability or chaos. In many cases, this suggests that memory should be treated not as a perturbation of a standard map but as a structural modification of the update law itself.