--- title: General Fractional Dynamics url: https://www.emergentmind.com/topics/general-fractional-dynamics-gfdynamics type: topic --- # General Fractional Dynamics 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 [2510.17806]. 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 [1004.0628]. 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 [1103.0082; 2605.26326; 1904.00777; 2509.15228]. 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 \(D^{t,n,*}_{(K)}\) by replacing the first derivative with \(d^n/dt^n\) or \(d^nX/d\tau^n\) [2510.17806]. The corresponding fundamental theorems generalize the ordinary inverse relations between differentiation and integration. For suitable \(X\), one has \[ D^{t}_{(K)}\,I^{t}_{(M)}\,X=X,\qquad D^{t,*}_{(K)}\,I^{t}_{(M)}\,X=X, \] together with \[ I^{t}_{(M)}\,D^{t,*}_{(K)}\,X=X-X(0),\qquad I^{t}_{(M)}\,D^{t}_{(K)}\,X=X. \] These identities are the structural basis for exact solution formulas and for the derivation of discrete-time nonlocal maps from continuously forced systems [2510.17806]. A related unification is the 1st-level GFD, defined from a triple \((\kappa,k_1,k_2)\) of 1st-level Sonin kernels satisfying \((\kappa*k_1*k_2)(t)=1\). The operator \[ {}_{1L}D_{(k_1,k_2)}\,f(t):=\Bigl(I_{(k_1)}\;\frac{d}{dt}\;I_{(k_2)}\,f\Bigr)(t) \] contains as special cases the Riemann–Liouville-type operator \(D_{(k)}f=d/dt(k*f)\) and the Caputo-type operator \({}_*D_{(k)}f=I_{(k)}f'\). 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 [2406.08642]. A different proposal, also termed generalized fractional derivative, defines \[ D^{\alpha}_{\beta}\,f(t)=\lim_{\varepsilon\to0} \frac{f\!\bigl(t + \tfrac{\varepsilon\,\Gamma(\beta+1)}{\Gamma(\beta-\alpha+1)}\,t^{1-\alpha}\bigr)-f(t)}{\varepsilon} = \frac{\Gamma(\beta+1)}{\Gamma(\beta-\alpha+1)}\,t^{1-\alpha}\,\frac{d\,f}{dt}(t). \] For functions expandable by Taylor series, that construction states the semigroup property \(D^\alpha D^\beta f=D^{\alpha+\beta}f\), reproduces Caputo and Riemann–Liouville values on monomials, and gives a closed-form solution of a fractional Riccati equation [2108.06354]. 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 \(\Phi(p,\Theta)\). The generated kernel is \[ \Psi_\alpha(t;\Theta)=\mathcal L^{-1}\{\Phi(p,\Theta)^{-\alpha}\}(t),\qquad \alpha>0, \] from which the dynamic-memory fractional integral \[ I_\Phi^\mu x(t)=(\Psi_\mu*x)(t)=\int_0^t \Psi_\mu(t-s;\Theta)\,x(s)\,ds \] and the Riemann–Liouville and Caputo dynamic-memory derivatives \[ D_\Phi^{\mu,RL}x(t)=\frac{d}{dt}\,(\Psi_{1-\mu}*x)(t),\qquad D_\Phi^{\mu,C}x(t)=\Psi_{1-\mu}*x'(t) \] are defined. Under admissibility conditions \((A1)-(A4)\), the kernels satisfy the semigroup law \(\Psi_\alpha*\Psi_\beta=\Psi_{\alpha+\beta}\), giving \[ I_\Phi^\alpha I_\Phi^\beta=I_\Phi^{\alpha+\beta}, \qquad D_\Phi^{\mu,RL}I_\Phi^\mu x=x, \qquad I_\Phi^\mu D_\Phi^{\mu,C}x=x-x(0). \] By appropriate choices of \(\Phi\), 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 [2605.26326]. Another axis of generalization defines nonlocal operators of a function \(f(x)\) with respect to another function \(g(x)\). For \(g\in{\cal G}(a,b]\), the left-sided parametric GFI is \[ \bigl(I^{(M)}_{a+,g}f\bigr)(x) =\int_{a}^{x}M\!\bigl(g(x)-g(u)\bigr)\,f(u)\,g'(u)\,du, \] the Riemann–Liouville-type parametric GFD is \[ \bigl(D^{(K)}_{a+,g}f\bigr)(x) =\frac1{g'(x)}\frac{d}{dx} \int_{a}^{x}K\!\bigl(g(x)-g(u)\bigr)\,f(u)\,g'(u)\,du, \] and the Caputo-type parametric GFD is \[ \bigl(D^{(K),*}_{a+,g}f\bigr)(x) =\int_{a}^{x}K\!\bigl(g(x)-g(u)\bigr)\,f'(u)\,du. \] Through the substitution operator \(Q_g\), these operators are conjugate to ordinary GF operators on \((g(a),g(b))\), which yields semigroup and fundamental-theorem results. Special cases include the ordinary Riemann–Liouville and Caputo operators for \(g(x)=x\), Hadamard-type operators for \(g(x)=\ln(x/a)\), and Erdélyi–Kober operators for \(g(x)=x^\sigma\) with suitable prefactors [2509.12218]. Variable order enters through dynamic-order systems, in which the order itself is generated by an auxiliary dynamics. In the Caputo sense, \[ {}^{C}D_t^{\alpha(t)}\,x(t) = \frac{1}{\Gamma\bigl(1-\alpha(t)\bigr)} \int_{0}^{t}(t-\tau)^{-\alpha(t)}\,\frac{d}{d\tau}x(\tau)\,d\tau, \] with \(\alpha(t)\in(0,1)\) determined by another system rather than prescribed a priori [1103.0082]. 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 \(\bigl(V,N,d,I\bigr)\) carries a Whitney sum \[ T^\alpha V=h^\alpha V\oplus v^\alpha V, \] with local coefficients \({}^\alpha N_i^a(x,y)\). The formalism introduces fractional frame and coframe fields adapted to this splitting, a distinguished metric \[ {}^\alpha\mathbf g = {}^\alpha g_{ij}\,{}^\alpha e^i\otimes{}^\alpha e^j + {}^\alpha h_{ab}\,{}^\alpha e^a\otimes{}^\alpha e^b, \] and a metric-compatible fractional d-connection \({}^\alpha\mathbf D\) with induced torsion and curvature. The Einstein d-tensor is \[ {}^\alpha E_{\mu\nu} = {}^\alpha R_{\mu\nu} -\tfrac12\,{}^\alpha g_{\mu\nu}\,{}^\alpha R, \] and the fractional Einstein equations are \[ {}^\alpha E_{\mu\nu}={}^{\alpha}\Upsilon_{\mu\nu}. \] Fractional Levi–Civita configurations are recovered by the additional nonholonomic constraints \[ {}^\alpha C^i_{ja}=0,\qquad {}^\alpha\Omega^a_{\ ij}=0. \] For a \(2+2\) coordinate splitting and a general off-diagonal ansatz, the field equations decouple into a 2D fractional Laplace equation for \({}^\alpha g_{ij}\), evolution-type equations for \({}^\alpha h_3,{}^\alpha h_4\), and algebraic or integrable equations for the N-connection coefficients \(w_i,n_i\). This enables exact solutions including pure vacuum off-diagonal configurations, fractional black ellipsoids obtained by deforming the Schwarzschild horizon with a small eccentricity parameter \(\varepsilon\), and black holes embedded in fractional KdV or sine-Gordon backgrounds [1004.0628]. 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 [1004.0628]. 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, \[ \mathrm{Grad}^{(K)}_W U = e_x D_{x,*}^{(K_1)}U + e_y D_{y,*}^{(K_2)}U + e_z D_{z,*}^{(K_3)}U, \] \[ \mathrm{Div}^{(K)}_W F = D_{x,*}^{(K_1)}F_x + D_{y,*}^{(K_2)}F_y + D_{z,*}^{(K_3)}F_z, \] and \(\mathrm{Curl}^{(K)}_W F\) 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 [2111.02716]. ## 4. Mechanisms producing fractional dynamics One mechanism derives fractional dynamics from ordinary Markovian dynamics by subordination. In the ordinary Langevin framework, an internal time \(t_s\) is replaced by the inverse \(\alpha\)-stable hitting-time process \[ S(t)=\inf\{\tau>0\mid T(\tau)>t\}, \] where \(T(\tau)\) is a strictly increasing \(\alpha\)-stable subordinator. If \(p_{\rm parent}(x,t_s)\) solves the ordinary Fokker–Planck equation, then the subordinated density is \[ p(x,t)=\int_0^\infty p_{\rm parent}(x,t_s)\,p_{\rm directing}(t_s,t)\,dt_s, \] 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 [1111.3283]. A broader Langevin-based account expresses fractional dynamics through generalized Langevin equations with power-law memory kernels. For \[ m\,\ddot X(t) = -\int_0^t K(t-s)\,\dot X(s)\,ds + F(t), \qquad \langle F(0)F(t)\rangle=k_BT\,K(t), \] choosing \[ K(t)=\frac{z}{\Gamma(2h-1)}\,t^{2h-2},\qquad 0}(t) =:l_{\kappa,\lambda}(t) \] produces explicit solution formulas. For the initial-value problem \[ {}_{1L}D_{(k_1,k_2)}y(t)-\lambda y(t)=f(t), \qquad \bigl(I_{(k_2)}y\bigr)(0)=y_0, \] the solution is \[ y(t)= (l_{\kappa,\lambda}*f)(t)+y_0\,(l_{\kappa,\lambda}*k_1)(t), \] which reduces in the power-law case to the standard Mittag–Leffler expressions for Riemann–Liouville- and Caputo-type equations [2406.08642]. Exact discrete-time memory maps arise when general fractional equations are driven by periodic kicks. For the Caputo-type equation \[ D^{t,*}_{(K)}[\tau]\,X(\tau) = \lambda\,\mathcal G\bigl(t,X(t)\bigr) \sum_{k=1}^{\infty}\delta\Bigl(\frac{t+\epsilon}{T}-k\Bigr), \] application of the GFI and the fundamental theorem yields, for \(Tn