General Fractional Dynamics

• General Fractional Dynamics is a framework defined by nonlocal operators with memory kernels, using fractional integrals and derivatives to capture hereditary effects.
• It encompasses kernel-based formulations, variable order systems, and geometric methods to model scale dependence and noninteger evolution.
• The approach supports exact solvability and finds applications in anomalous diffusion, fractional gravity, viscoelasticity, and optimization.

General Fractional Dynamics (GFDynamics) denotes a family of frameworks for dynamical systems with nonlocality, hereditary effects, and noninteger-order evolution. In one formulation, it is the study of dynamical systems whose evolution depends on their entire past history, implemented through general fractional integrals (GFIs), general fractional derivatives (GFDs), and exact discrete-time nonlocal maps derived from them (Tarasov, 22 Sep 2025). In another, it is a geometric formalism based on fractional Caputo differential–integral calculus adapted to nonholonomic distributions, yielding fractional spacetime geometry, generalized tensor calculus, and fractional gravitational field equations (Vacaru, 2010). The same label is also used for dynamic-order systems, generator-based dynamic-memory operators, local fractional dynamics on fractal supports, and Hadamard-type non-local scaling constructions [(Sun et al., 2011); (Alzabut, 25 May 2026); (Datta et al., 2019); (Tarasov, 3 Sep 2025)]. Across these usages, the common theme is the replacement of local, integer-order evolution by operators whose kernels encode memory, scale dependence, or nonlocal geometry.

1. Kernel-based foundations of general fractional calculus

A central kernel-based formulation begins with a Sonin pair $(M,K)$ satisfying the convolution identity

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$

More generally, a Luchko pair of order $n\in\mathbb N$ satisfies

$(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$

On this basis one defines the general fractional integral

$$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$

the Riemann–Liouville-type derivative

$$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$

and the Caputo-type derivative

$$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$

For arbitrary order $n$, one similarly defines $I^{t,n}_{(M)}$, $D^{t,n}_{(K)}$, and $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$0 by replacing the first derivative with $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$1 or $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$2 (Tarasov, 22 Sep 2025).

The corresponding fundamental theorems generalize the ordinary inverse relations between differentiation and integration. For suitable $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$3, one has

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$4

together with

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$5

These identities are the structural basis for exact solution formulas and for the derivation of discrete-time nonlocal maps from continuously forced systems (Tarasov, 22 Sep 2025).

A related unification is the 1st-level GFD, defined from a triple $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$6 of 1st-level Sonin kernels satisfying $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$7. The operator

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$8

contains as special cases the Riemann–Liouville-type operator $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$9 and the Caputo-type operator $n\in\mathbb N$0. A Mikusiński-type operational calculus can then be built in a field of convolution quotients, yielding algebraic representations of these operators and closed-form solution formulas for linear initial-value problems (Alkandari et al., 2024).

A different proposal, also termed generalized fractional derivative, defines

$n\in\mathbb N$1

For functions expandable by Taylor series, that construction states the semigroup property $n\in\mathbb N$2, reproduces Caputo and Riemann–Liouville values on monomials, and gives a closed-form solution of a fractional Riccati equation (Abu-Shady et al., 2021). The literature therefore uses “general” both for kernel-based nonlocal operators and for specific generalized derivative proposals.

2. Memory generators, variable order, and parametrization with respect to another function

A recent extension replaces prescribed kernels by a Laplace-domain memory generator $n\in\mathbb N$3. The generated kernel is

$n\in\mathbb N$4

from which the dynamic-memory fractional integral

$n\in\mathbb N$5

and the Riemann–Liouville and Caputo dynamic-memory derivatives

$n\in\mathbb N$6

are defined. Under admissibility conditions $n\in\mathbb N$7, the kernels satisfy the semigroup law $n\in\mathbb N$8, giving

$n\in\mathbb N$9

By appropriate choices of $(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$0, the framework recovers Riemann–Liouville/Caputo, tempered Caputo, Hadamard-type, Katugampola, Prabhakar-type, and distributed-order operators, while also generating singular, nonsingular, tempered, logarithmic, oscillatory, crossover, and multiscale memory kernels within one analytical setting (Alzabut, 25 May 2026).

Another axis of generalization defines nonlocal operators of a function $(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$1 with respect to another function $(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$2. For $(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$3, the left-sided parametric GFI is

$(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$4

the Riemann–Liouville-type parametric GFD is

$(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$5

and the Caputo-type parametric GFD is

$(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$6

Through the substitution operator $(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$7, these operators are conjugate to ordinary GF operators on $(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$8, which yields semigroup and fundamental-theorem results. Special cases include the ordinary Riemann–Liouville and Caputo operators for $(M*K)(t)=h_n(t)=\frac{t^{n-1}}{(n-1)!}.$9, Hadamard-type operators for $$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$0, and Erdélyi–Kober operators for $$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$1 with suitable prefactors (Tarasov, 2 Sep 2025).

Variable order enters through dynamic-order systems, in which the order itself is generated by an auxiliary dynamics. In the Caputo sense,

$$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$2

with $$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$3 determined by another system rather than prescribed a priori (Sun et al., 2011). This construction links variable memory directly to multi-system interaction rather than to a fixed external schedule.

3. Fractional geometry, spacetime, and vector calculus

In fractional gravity, GFDynamics is built upon two key ingredients: a nonholonomic manifold endowed with a nonlinear connection splitting and a fractional differential–integral calculus of Caputo type. A fractional nonholonomic manifold $$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$4 carries a Whitney sum

$$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$5

with local coefficients $$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$6. The formalism introduces fractional frame and coframe fields adapted to this splitting, a distinguished metric

$$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$7

and a metric-compatible fractional d-connection $$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$8 with induced torsion and curvature. The Einstein d-tensor is

$$I^{t}_{(M)}[\tau]\,X(\tau)=(M*X)(t)=\int_{0}^{t}M(t-\tau)\,X(\tau)\,d\tau,$$9

and the fractional Einstein equations are

$$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$0

Fractional Levi–Civita configurations are recovered by the additional nonholonomic constraints

$$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$1

For a $$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$2 coordinate splitting and a general off-diagonal ansatz, the field equations decouple into a 2D fractional Laplace equation for $$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$3, evolution-type equations for $$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$4, and algebraic or integrable equations for the N-connection coefficients $$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$5. This enables exact solutions including pure vacuum off-diagonal configurations, fractional black ellipsoids obtained by deforming the Schwarzschild horizon with a small eccentricity parameter $$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$6, and black holes embedded in fractional KdV or sine-Gordon backgrounds (Vacaru, 2010).

The same geometric machinery extends beyond gravity to fractional Lagrange–Finsler dynamics, fractional nonholonomic Ricci flows, fractional gauge and Dirac fields on N-anholonomic spinor bundles, and fractional kinetic and diffusion models on nonholonomic fiber bundles (Vacaru, 2010). This suggests a geometric branch of GFDynamics in which memory and nonlocality are expressed by adapted connections, torsion, and nonholonomic splitting rather than solely by scalar convolution kernels.

General fractional vector calculus provides an analogous extension in space. Starting from one-dimensional GFI/GFD operators associated with Sonine-pair kernels, it defines regional general fractional gradient, divergence, and curl, together with line general fractional circulation, surface general fractional flux, and general fractional volume integrals. In Cartesian coordinates,

$$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$7

$$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$8

and $$D^{t}_{(K)}[\tau]\,X(\tau)=\frac{d}{dt}\bigl(K*X\bigr)(t),$$9 is defined componentwise in the usual antisymmetric pattern. The generalized gradient, Green, Stokes, and Gauss theorems are proved for suitable kernels and regularity classes, including orthogonal curvilinear coordinates such as spherical and cylindrical systems. The same work explicitly notes difficulties caused by the violation of the standard product rule, chain rule, and semigroup property in the nonlocal case (Tarasov, 2021).

4. Mechanisms producing fractional dynamics

One mechanism derives fractional dynamics from ordinary Markovian dynamics by subordination. In the ordinary Langevin framework, an internal time $$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$0 is replaced by the inverse $$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$1-stable hitting-time process

$$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$2

where $$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$3 is a strictly increasing $$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$4-stable subordinator. If $$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$5 solves the ordinary Fokker–Planck equation, then the subordinated density is

$$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$6

and the resulting master equation becomes the time-fractional Fokker–Planck equation in Riemann–Liouville form. In this construction the real-time process is non-Markovian, all moments are finite, the stationary state remains Gibbs–Boltzmann, the Einstein relation remains valid, and the H-theorem holds (Stanislavsky, 2011).

A broader Langevin-based account expresses fractional dynamics through generalized Langevin equations with power-law memory kernels. For

$$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$7

choosing

$$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$8

yields a fractional-kernel regime with Green’s function

$$D^{t,*}_{(K)}[\tau]\,X(\tau)=\int_{0}^{t}K(t-\tau)\,\frac{dX}{d\tau}(\tau)\,d\tau.$$9

velocity autocorrelation $n$0, and mean-squared displacement asymptotics

$n$1

The same line of work relates sampled stationary GLE solutions to ARMA or VARMA models and studies ergodic versus non-ergodic behavior, including superstatistical random-parameter mixtures (Ślęzak, 2018).

Dynamic-order coupling is a different mechanism. A prototypical two-system model is

$n$2

with $n$3. Under Lipschitz and bounded-order hypotheses, the system can be recast as a Volterra integral system, and a fixed-point argument yields local existence and uniqueness. In this interpretation, larger $n$4 corresponds to shorter memory, while smaller $n$5 corresponds to longer memory. Canonical examples include anomalous relaxation and anomalous diffusion, with the diffusion example coupled through $n$6 to an auxiliary thermal dynamics (Sun et al., 2011).

A third mechanism is the deformation of ordinary calculus into local fractional dynamics on fractal supports through asymptotic duality and a renormalization-group map $n$7. In that setting, the ordinary derivative is invariant under the duality-enabled RG transformation,

$n$8

and a local fractional derivative on a fractal curve $n$9 is defined by

$I^{t,n}_{(M)}$0

The approach yields local-fractional versions of one- and two-dimensional wave equations on fractal strings and membranes. The same work also identifies limitations: a concrete dynamics requires choosing a seed fractal $I^{t,n}_{(M)}$1, nonlinear dynamics on fractals remain largely unexplored, and a classification of weakly-dual arithmetic asymptotics is still under development (Datta et al., 2019).

5. Exact solvability, operational calculi, and nonlocal maps

Operational methods are a recurrent feature of GFDynamics. In the 1st-level GFD theory, the convolution ring $I^{t,n}_{(M)}$2 is embedded into a field of convolution quotients, and the algebraic inverse of the GFI kernel $I^{t,n}_{(M)}$3 is denoted $I^{t,n}_{(M)}$4. The 1st-level algebraic GFD takes the form

$I^{t,n}_{(M)}$5

and the resolvent expansion

$I^{t,n}_{(M)}$6

produces explicit solution formulas. For the initial-value problem

$I^{t,n}_{(M)}$7

the solution is

$I^{t,n}_{(M)}$8

which reduces in the power-law case to the standard Mittag–Leffler expressions for Riemann–Liouville- and Caputo-type equations (Alkandari et al., 2024).

Exact discrete-time memory maps arise when general fractional equations are driven by periodic kicks. For the Caputo-type equation

$I^{t,n}_{(M)}$9

application of the GFI and the fundamental theorem yields, for $D^{t,n}_{(K)}$0,

$D^{t,n}_{(K)}$1

Sampling at $D^{t,n}_{(K)}$2 and subtracting successive steps gives the exact nonlocal map

$D^{t,n}_{(K)}$3

Analogous constructions exist for Riemann–Liouville-type equations, for GFI-driven equations, and for arbitrary order via generalized momenta (Tarasov, 22 Sep 2025).

Hadamard-type non-local scaling yields a parallel exact mapping theory. With the scale-invariant integral

$D^{t,n}_{(K)}$4

and the associated Hadamard-type fractional differential operator

$D^{t,n}_{(K)}$5

one considers nonlinear equations with periodic kicks. Exact continuous-time solutions are obtained by converting the Cauchy problem to a Volterra integral equation and evaluating against the delta train. Sampling then gives mappings with non-local scaling in time whose kernels are

$D^{t,n}_{(K)}$6

For zero initial data, the resulting mappings are independent of the kick period $D^{t,n}_{(K)}$7 (Tarasov, 3 Sep 2025).

6. Representative domains, applications, and open questions

On networks, fractional dynamics is formulated through the fractional Laplacian matrix

$D^{t,n}_{(K)}$8

where $D^{t,n}_{(K)}$9. The associated fractional random-walk transition matrix is built from the modified fractional Laplacian $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$00, and the stationary distribution is

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$01

The average return probability is

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$02

and the global exploration time is

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$03

For a ring, the hopping probabilities have the asymptotic tail

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$04

and the return probability decays as

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$05

This produces long-range dynamics, Lévy-flight behavior on the ring, and a dynamically induced small-world property on any network (Riascos et al., 2015).

In viscoelastic structural dynamics, a distributed-order constitutive law

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$06

specializes to the fractional Kelvin–Voigt model

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$07

Applied to a geometrically nonlinear viscoelastic cantilever beam, Hamilton’s principle leads to a nonlinear time-fractional PDE, which is reduced by an assumed-mode decomposition to a nonlinear time-fractional ODE. The linear part is integrated numerically by a direct L1-difference scheme, and the nonlinear system is treated semi-analytically by a method of multiple scales. The reported anomalous dynamic qualities are far-from-equilibrium power-law amplitude decay rates, super-sensitivity of amplitude response at free vibration, and bifurcation in steady-state amplitude at primary resonance (Suzuki et al., 2020).

In optimization, the proposed fractional-order Nesterov flow is

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$08

with the fractional operators interpreted in the sense of Caputo, Riemann–Liouville, and Grünwald–Letnikov derivatives. The analysis focuses on the regime $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$09, especially $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$10, where the ordinary Nesterov flow fails to guarantee convergence. In the fractional setting, convergence can still be established, and the proof uses fractional Opial-type lemmas and Lyapunov memory functionals. For convex functions, the framework yields weak convergence of trajectories and asymptotic decay of functional values; for strongly convex functions, it gives explicit rates expressed through Mittag–Leffler decay (Ranoto, 15 Sep 2025).

In the superdiffusive regime $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$11, one studies the abstract Cauchy problem

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$12

where $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$13 is any positive self-adjoint operator on a Hilbert space. The Caputo derivative is

$$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$14

The theory establishes existence and regularity of weak and strong energy solutions without assuming that $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$15 has compact resolvent, and treats examples including Schrödinger operators and nonlocal operators (Alvarez et al., 2 Sep 2025).

Open questions are stated explicitly in several branches of the literature. Dynamic-order systems raise questions on rigorous stability and bifurcation theory, system identification of $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$16, efficient numerical schemes for variable-order kernels, extension to space–time variable orders $$(M*K)(t)=\int_{0}^{t}M(t-\tau)\,K(\tau)\,d\tau=1,\qquad t>0.$$17, and coupling with stochastic processes (Sun et al., 2011). The local-fractional fractal-support approach notes that nonlinear dynamics and interacting field theories on fractals remain largely unexplored (Datta et al., 2019). The optimization framework points to nonconvex objectives, adaptive order, and stochastic fractional flows as immediate extensions (Ranoto, 15 Sep 2025). Taken together, these strands suggest not a single closed formalism but a technical research program centered on nonlocal operators, exact memory representations, and generalized geometries for systems in which present evolution is shaped by prior states, scale transformations, or auxiliary processes.