Discrete Maps with Memory (DMM)

• Discrete Maps with Memory (DMM) are discrete-time systems where each update depends on a weighted sum of all past states, embodying non-Markovian dynamics derived from fractional calculus.
• They are constructed exactly from fractional differential equations with periodic kicks, yielding recurrence relations for benchmark nonlinear models such as logistic and predator–prey systems.
• DMM frameworks utilize power-law and falling-factorial memory kernels to modulate stability and attractor behavior, offering insights into long-term dynamical phenomena across various applications.

Searching arXiv for core papers on discrete maps with memory, fractional maps, and related exact derivations. Discrete Maps with Memory (DMM) are discrete-time dynamical systems in which the state at step $n+1$ depends not only on the current state but on the entire preceding trajectory, typically through weighted sums over past states. In the literature synthesized here, DMM arise most systematically as exact discrete-time counterparts of fractional differential equations with periodic kicks, where the memory kernel is inherited from fractional calculus and usually has power-law form. This framework links non-Markovian discrete dynamics to Caputo, Riemann–Liouville, and Hilfer fractional derivatives, and yields explicit recurrence relations for benchmark nonlinear systems such as universal, standard, logistic, dissipative, macroeconomic, and predator–prey models (Tarasov, 2011, Tarasov, 2011, Edelman, 2013, Tarasova et al., 2016).

1. Definition and formal structure

In this context, a DMM is a recurrence relation in which the present state is determined by all past states with a specified memory kernel rather than by the most recent state alone. A generic formulation given in the literature is

$$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$

or, more explicitly, as a weighted history sum

$x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$

where $W$ is the discrete memory kernel (Tarasov, 2011, Tarasova et al., 2016). This is the basic distinction from ordinary Markovian maps such as $x_{n+1}=f(x_n)$.

A central subclass consists of maps with power-law memory. In these systems, the weight of a past state at lag $n-k$ behaves like

$(n-k)^{\alpha-1},$

or through a discrete increment kernel such as

$V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$

For $0<\alpha<1$, these weights decay slowly and algebraically, so distant states remain dynamically relevant (Tarasov, 2011, Tarasova et al., 2016, Edelman, 2013).

Several papers distinguish between “full memory” and long-term memory. Full memory assigns comparable effective weight to all past states, as in normalized sums of the form

$x_{n+1}=\frac{1}{n}\sum_{i=1}^{n} f(x_i),$

whereas long-term memory uses non-uniform weights that fade with lag, often in a fractional or asymptotically fractional manner (Stanislavsky, 2011). This suggests that DMM form a broad category encompassing both idealized averaging schemes and exact nonlocal maps derived from continuous-time memory equations.

2. Fractional-calculus origin

The main constructive route to DMM in this literature starts from fractional differential equations (FDEs) with respect to time. Fractional derivatives encode memory because they are integral operators with history-dependent kernels. For the Caputo derivative of order $$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$0, one has, for $$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$1,

$$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$2

so the derivative at time $$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$3 depends on the whole interval $$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$4 with a power-law kernel (Tarasov, 2011, Tarasov, 2011). In macroeconomic applications, the Caputo derivative is written as

$$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$5

generalizing standard accelerator and multiplier relations to power-law memory (Tarasova et al., 2016).

The same principle appears in logistic growth models,

$$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$6

and in predator–prey models,

$$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$7

where memory is combined with impulsive interactions (Tarasova et al., 2017, Tarasov, 22 Sep 2025). In all such cases, the nonlocality of the fractional operator is the continuous-time source of the discrete memory kernel.

A parallel line of work derives DMM from fractional difference equations rather than from continuous-time FDEs. There the relevant operator is a Caputo-like fractional difference, and the corresponding kernels are falling-factorial rather than exact power-law kernels, though they are asymptotically power-law: $$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$8 This yields discrete systems with falling factorial-law memory, asymptotically equivalent to power-law memory (Edelman, 2014).

3. Derivation from fractional equations with kicks

The mathematically characteristic construction of DMM in this corpus begins with a kicked fractional equation such as

$$x_{n+1} = F\Bigl(n, x_n, \sum_{k=0}^{n} W(n-k) G(x_k)\Bigr),$$9

or analogous equations with Riemann–Liouville or Hilfer derivatives (Tarasov, 2011, Tarasov, 2011, Tarasov, 3 Sep 2025). The periodic Dirac delta sequence enforces impulsive updates at times $x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$0, while the fractional derivative provides memory.

The key technical bridge is the equivalence between the Cauchy problem for a fractional differential equation and a nonlinear Volterra integral equation of the second kind. In Caputo form, one obtains

$x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$1

and with delta kicks the integral collapses into a discrete sum over past kick times (Tarasov, 2011, Tarasov, 2011, Tarasova et al., 2016). Evaluating this representation at $x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$2 yields exact discrete recurrence relations. The construction is analytic rather than numerical; the papers repeatedly emphasize that no approximation of the fractional derivative is used (Tarasov, 2011, Tarasova et al., 2016, Tarasov, 22 Sep 2025, Tarasov, 3 Sep 2025).

For $x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$3, the resulting Caputo universal map takes the form

$x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$4

$x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$5

which is the archetypal fractional universal map with memory (Tarasov, 2011). For $x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$6, only the coordinate-like variable remains, and the DMM reduces to a single history sum (Tarasova et al., 2016, Tarasova et al., 2017).

Hilfer-derivative constructions generalize this further. With order $x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$7 and type parameter $x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$8, the Hilfer derivative interpolates between the Riemann–Liouville case at $x_{n+1} = \sum_{k=0}^{n} W(n-k)\,G(x_k),$9 and the Caputo case at $W$0. The resulting DMM depend on kernels

$W$1

and provide a continuous family connecting RL-based and Caputo-based maps (Tarasov, 3 Sep 2025). This suggests that the space of exact DMM derivable from fractional dynamics is broader than the Caputo/RL dichotomy alone.

4. Memory kernels and orders

The most prominent kernel class is power-law memory. In the exact maps derived from Caputo or Riemann–Liouville equations, the history terms are weighted by factors such as

$W$2

or by the increment kernel

$W$3

For $W$4, the exponent $W$5, so the decay is slow and hyperbolic rather than exponential (Tarasova et al., 2016, Tarasov, 22 Sep 2025).

In fractional-difference formulations, the kernel has falling-factorial form,

$W$6

which is asymptotically equivalent to $W$7 for large $W$8. This establishes a discrete analog of the continuous-time fractional kernel while retaining purely discrete calculus throughout (Edelman, 2014).

A distinct but related construction appears in history-weighted logistic and tent maps where the weights are

$W$9

with normalization

$x_{n+1}=f(x_n)$0

These weights were motivated by robust algorithms for numerical fractional integration and define a normalized long-term memory map

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

Here $x_{n+1}=f(x_n)$2 yields no memory, $x_{n+1}=f(x_n)$3 yields full memory, and $x_{n+1}=f(x_n)$4 yields fractional memory (Stanislavsky, 2011).

These distinct kernel constructions should not be conflated. Exact maps derived from kicked FDEs use kernels fixed by fractional integral identities and Volterra equivalence; heuristic or quadrature-inspired maps use normalized weight families motivated by fractional integration algorithms (Tarasov, 2011, Stanislavsky, 2011). Both are DMM, but they occupy different methodological strata.

5. Canonical families and application domains

A large fraction of the DMM literature is organized around “$x_{n+1}=f(x_n)$5-families of maps,” in which the parameter $x_{n+1}=f(x_n)$6 is the order of the underlying fractional derivative and continuously interpolates between familiar integer-order maps and genuinely non-Markovian ones (Edelman, 2013, Edelman, 2014).

Representative classes

Family	Memory source	Illustrative form	arXiv id
Universal / Standard maps	Fractional kicked equations	History sums with $x_{n+1}=f(x_n)$7 or $x_{n+1}=f(x_n)$8	(Tarasov, 2011)
Logistic and triangular maps	Normalized long-term memory weights $x_{n+1}=f(x_n)$9	$n-k$0	(Stanislavsky, 2011)
Fractional difference $n-k$1-families	Caputo fractional difference	Falling-factorial memory kernels	(Edelman, 2014)
Economic accelerator / logistic models	Caputo derivatives + kicks	Exact maps for capital/output with memory	(Tarasova et al., 2016, Tarasova et al., 2017)
Predator–prey maps	Fractional Lotka–Volterra / Kolmogorov with kicks	Exact maps for $n-k$2 with memory	(Tarasov, 22 Sep 2025)

In the universal-map setting, choosing $n-k$3 yields fractional standard maps, while $n-k$4 yields Anosov-type systems with memory (Tarasov, 2011, Tarasov, 2011). Fractional Zaslavsky and Henon maps extend this logic to dissipative and generalized dissipative settings, with position updates determined by memory sums over past momenta (Tarasov, 2011).

In macroeconomics, DMM arise from accelerator and multiplier equations with power-law memory and periodic crises. For $n-k$5, the capital stock map becomes

$n-k$6

and the increment form is

$n-k$7

up to the paper’s normalization conventions (Tarasova et al., 2016). The same method yields logistic maps with memory from economic growth models with competition and crises (Tarasova et al., 2017).

Predator–prey models provide a more recent biological instantiation. There the exact fractional Lotka–Volterra maps in logarithmic variables take forms such as

$n-k$8

with an analogous formula for $n-k$9 (Tarasov, 22 Sep 2025). This is a direct example of a two-population DMM with power-law fading memory.

6. Dynamical behavior, regimes, and interpretation

The defining dynamical effect of DMM is temporal nonlocality. Every update depends on the entire past, so these maps are non-Markovian unless embedded into an augmented state space that explicitly stores history variables (Tarasov, 2011, Edelman, 2013). This suggests an effectively infinite-dimensional phase-space structure for non-integer orders.

Several recurring qualitative effects are reported across the literature. One is slow, algebraic relaxation rather than exponential relaxation. In fractional maps, convergence to fixed points or cycles often follows power-law rates, and bifurcation diagrams can depend on iteration length because long transients remain dynamically relevant (Edelman, 2013, Edelman, 2014).

Another is the emergence of attractor classes not present in ordinary maps. Fractional maps with power-law memory have been reported to exhibit periodic sinks, chaotic attractors, attracting slow diverging trajectories, attracting accelerator mode trajectories, and “cascade of bifurcations type trajectories” (Edelman, 2013). In the standard-map family for $(n-k)^{\alpha-1},$0, analytical thresholds for stability loss of the fixed point and birth of period-two sinks are derived in terms of

$(n-k)^{\alpha-1},$1

with

$(n-k)^{\alpha-1},$2

and further parameter curves govern higher-order structures (Edelman, 2013). These are concrete signatures of memory-modified stability.

In the normalized long-term memory logistic and triangular maps of Safonov, increasing the memory parameter suppresses chaos. Numerical bifurcation studies over $(n-k)^{\alpha-1},$3 with $(n-k)^{\alpha-1},$4 iterations show shrinking chaotic bands as $(n-k)^{\alpha-1},$5 increases, and the paper states that after $(n-k)^{\alpha-1},$6 the dynamics tends to a regular stable fixed point (Stanislavsky, 2011). For the triangular map, the Lyapunov exponent satisfies

$(n-k)^{\alpha-1},$7

with equality at $(n-k)^{\alpha-1},$8, supporting the interpretation that long-term memory decreases sensitivity to initial conditions (Stanislavsky, 2011).

By contrast, in the broader fractional-map literature, memory can also enrich rather than suppress complex dynamics. Fractional universal maps are reported to possess “new regular and strange attractors” and “new type of attractors” for some parameter ranges (Tarasov, 2011, Tarasov et al., 2011). This is not a contradiction so much as a model-dependent consequence: memory can stabilize, destabilize, or restructure the phase portrait depending on the underlying nonlinearity, order interval, and parameterization. A plausible implication is that “memory” is not itself equivalent to “damping,” even though in some constructions it has a smoothing effect.

7. Methodological variants, related meanings, and misconceptions

One potential source of confusion is terminological breadth. In the fractional-dynamics literature, DMM refers to discrete maps with memory derived from or modeled after nonlocal kernels in time (Tarasov, 2011, Tarasova et al., 2016). In a distinct machine-learning and program-semantics context, “Dataflow Matrix Machines” are also abbreviated DMM, but they refer to countable recurrent networks operating on streams of V-values and evolving via a two-stroke linear/nonlinear update. Those systems are discrete-time dynamical systems with state and self-referential matrix updates, but they are conceptually separate from fractional maps with power-law memory (Bukatin et al., 2017). The shared abbreviation does not indicate shared theory.

A second misconception is that DMM are merely numerical discretizations of fractional equations. In the core Tarasov line of work and its applications, the maps are derived exactly from Volterra-equivalent integral representations and periodic delta kicks; they are not finite-difference approximations of derivatives (Tarasov, 2011, Tarasov, 2011, Tarasova et al., 2016, Tarasov, 22 Sep 2025). By contrast, some history-weighted constructions inspired by numerical fractional integration are motivated differently and are better understood as model definitions rather than exact stroboscopic images of kicked FDEs (Stanislavsky, 2011).

A third issue concerns “memory” versus finite-dimensional state augmentation. For integer $(n-k)^{\alpha-1},$9, many $V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$0-families reduce to ordinary Markovian maps in finite-dimensional phase space: $V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$1 yields the one-dimensional universal or logistic-type map, $V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$2 yields the classical universal or standard map, and $V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$3 yields a three-dimensional volume-preserving universal map (Edelman, 2013). Genuine long-term memory appears for non-integer orders, where the history cannot be collapsed exactly to finitely many state coordinates without changing the model class.

More recent memcomputing literature uses “Digital Memcomputing Machines” for continuous-time dynamical systems with explicit memory variables that induce time non-locality. Under numerical integration these become discrete-time maps with internal memory variables, but the memory is represented by slow auxiliary coordinates rather than by an explicit power-law kernel over the full past (Sipling et al., 11 Jun 2025). This suggests a broader systems-theoretic notion of DMM as discrete-time nonlocal dynamics, extending beyond fractional kernels.

8. Computational considerations and broader significance

From a computational viewpoint, DMM with explicit history sums are intrinsically more expensive per step than memoryless maps, since naive evaluation of the recurrence at step $V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$4 requires summing over all $V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$5. This implies $V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$6 work per step in straightforward implementations (Tarasova et al., 2016). The same source notes that for translation-invariant power-law kernels, FFT-based convolution can plausibly accelerate these calculations, though this point is presented as an implicit computational aspect rather than a developed algorithmic result (Tarasova et al., 2016).

The fractional-difference framework adds a practically useful discrete alternative: instead of sampling a continuous-time FDE with kicks, one starts from a discrete fractional difference equation and obtains an equivalent map with falling-factorial memory, asymptotically power-law. This provides a numerically natural discrete setting while preserving the characteristic long-memory phenomenology (Edelman, 2014).

Across the literature, DMM are positioned as reduced or exact discrete-time analogs of systems with hereditary behavior in viscoelasticity, dielectric relaxation, anomalous diffusion, biological adaptation, human memory, ecology, and macroeconomics (Edelman, 2013, Tarasov, 2011, Tarasova et al., 2017, Tarasov, 22 Sep 2025). The unifying idea is that memory kernels with slow decay alter both local stability and global phase-space organization, often producing attractors and bifurcation structures not available to ordinary maps.

At the level of theory, the most important general conclusion is the exact correspondence

$V_{\alpha}(z)=(z+1)^{\alpha-1}-z^{\alpha-1}.$7

with variations depending on the derivative type, kick structure, and whether the setting is continuous-fractional or fractional-difference (Tarasov, 2011, Tarasova et al., 2016, Edelman, 2014). This correspondence makes DMM a canonical discrete language for studying long-term memory in nonlinear dynamics.