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Hamiltonian Chaos and Fractional Dynamics
... This fractional calculus extension of kinetic theory proceeds through the introduction of fractional time-and space-derivatives (of order β and α, respectively) into the governing equations. Specifically, the derivatives are expressed as convolutions of inverse powerlaw kernels with Φ β (t/τ) and X α (x/λ), respectively, with the well-behaved functions, f (t), t > 0 and g(x), x > 0 (for the definitions, see Ref. [28] Appendix C, p. 396, and Ref. [20,29], p. 236): ...
... Extending the Langevin equation, the Fokker-Planck equation [64] provides a more complete stochastic model of the time evolution of the particle distribution, P(x, t), one that includes both particle velocity, v(x, t), and diffusion, D(x, t) (see Ref. [28], pp. 271-220): ...
... Hence, it could be placed on the floor of the anomalous phase cube to replace the CTRW model. In a similar manner, the fractional/fractal extension of Hamiltonian dynamics on a fractal described by Zaslavsky [28] has the same general form as the CTRW model, but for his model the order of the fractional time and space derivatives are directly connected with a fractal structure in time and distance in the underlying phase space (p, q) of the Hamiltonian system. Another example is the generalized diffusion coefficient D(x, t) = D|x| a t b discussed by Evangelista and Lenzi which gives rise to anomalous diffusion depicted by stretched Bessel and Mittag-Leffler functions [29]. ...
The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases.
... (A) Fractional kinetics has been described in many reviews and books [13,17,[25][26][27], and [28][29][30][31][32]. Regarding anomalous and fractional diffusion, there are many works [13,17,[33][34][35][36][37]. These works use fractional calculus to describe nonlocality in space and time. ...
... Substitution of expression (25) into expression (26) gives the identity ...
... Substitution of expression (26) into expression (25) gives the identity ...
Nonlocal generalization of the standard (classical) probability theory of a continuous distribution on a positive semi-axis is proposed. An approach to the formulation of a nonlocal generalization of the standard probability theory based on the use of the general fractional calculus in the Luchko form is proposed. Some basic concepts of the nonlocal probability theory are proposed, including nonlocal (general fractional) generalizations of probability density, cumulative distribution functions, probability, average values, and characteristic functions. Nonlocality is described by the pairs of Sonin kernels that belong to the Luchko set. Properties of the general fractional probability density function and the general fractional cumulative distribution function are described. The truncated GF probability density function, truncated GF cumulative distribution function, and truncated GF average values are defined. Examples of the general fractional (GF) probability distributions, the corresponding probability density functions, and cumulative distribution functions are described. Nonlocal (general fractional) distributions are described, including generalizations of uniform, degenerate, and exponential type distributions; distributions with the Mittag-Leffler, power law, Prabhakar, Kilbas–Saigo functions; and distributions that are described as convolutions of the operator kernels and standard probability density.
... and 2 k denote, respectively, the square of the corresponding mass and wavenumber parameters ((156) in[1]).A reasonable hypothesis is that the primordial Universe evolves in far-fromequilibrium conditions and is unavoidably affected by the signature of stochastic fluctuations. The most straightforward way to account for the effect of these fluctuations on either curvature perturbations (7) or scalar field(10)is to use the Hamiltonian model of periodically kicked oscillator, extensively discussed in the nonlinear science literature.The equations describing the effect of periodic perturbations on the harmonic oscillator (14) are embodied in the Hamiltonian[2] ...
... of stability), along within regions of chaotic motion. As reported in[2][3][4], this phase-space configuration is typical for the transition to Hamiltonian chaos and the emergence of a fractal spacetime endowed with continuous dimensions. These findings carry significant implications for foundational physics, as reported in many contributions posted at[5]. ...
The goal of this paper is to analyze the likely transition from integrability to Hamiltonian chaos in the primordial Universe. The transition is driven by curvature fluctuations and favors the onset of a spacetime endowed with continuous dimensions.
... Physically, this infinite support can be approximated to be a finite one for the well-localized system/phenomenon. See the later discussion about Equation (31). This distribution obeys the Fokker-Planck equation [24] associated with the above-mentioned stochastic differential equation in the following general form: ...
... This equation still does not explicitly contain the (generalized) diffusion coefficient and therefore is invariant under the rescaling transformations of ξ and t. It is however known that, in order to obtain the Lévy distribution as a solution of the Fokker-Planck equation, the operator ∂ 2 /∂ ξ 2 should be fractionalized [30,31] and replaced by e.g. Riesz's fractional Laplacian. ...
The Omori-Utsu law shows the temporal power-law-like decrease of the frequency of earthquake aftershocks and, interestingly, is found in a variety of complex systems/phenomena exhibiting catastrophes. Now, it may be interpreted as a characteristic response of such systems to large events. Here, hierarchical dynamics with the fast and slow degrees of freedom is studied on the basis of the Fokker-Planck theory for the load-state distribution to formulate the law as a relaxation process, in which diffusion coefficient in the space of the load state is treated as a fluctuating slow variable. The evolution equation reduced from the full Fokker-Planck equation and its Green's function are analyzed for the subdynamics governing the load state as the fast degree of freedom. It is shown that the subsystem has the temporal translational invariance in the logarithmic time, not in the conventional time, and consequently the aging phenomenon appears.
... Fractional calculus (FC) has gained considerable importance in many fields of applied sciences and engineering for solving various differential equations and investigating behaviors of mathematical models simulating real-world problems. Its amazing presence is evident in the modeling of several natural phenomena such as Hamiltonian chaos and fractional dynamics [1], bio-engineering [2], viscoelasticity [3], vibrations and diffusion [4], physics [5,6], financial economics [7] and references cited therein. For more detailed theoretical aspect of FC, see [8][9][10][11][12] and references therein. ...
... Theorem 4.2. Assume that α k ∈ (1,2], β k ∈ [0, 1], ρ k ∈ R + , γ k = (β k (2ρ k − α k ) + α k )/ρ k , ψ k ∈ C(J, R) where ψ ′ k > 0, k = 1, 2, . . . , m and f ∈ C(J × R 3 , R). ...
This paper investigates a class of nonlinear impulsive fractional integro-differential equations with mixed nonlocal boundary conditions (multi-point and multi-term) that involves (ρ k , ψ k)-Hilfer fractional derivative. The main objective is to prove the existence and uniqueness of the solution for the considered problem by means of fixed point theory of Banach's and O'Regan's types, respectively. In this contribution, the transformation of the considered problem into an equivalent integral equation is necessary for our main results. Furthermore, the nonlinear functional analysis technique is used to investigate various types of Ulam's stability results. The applications of main results are guaranteed with three numerical examples.
... The topic of coupled fractional-order systems, complemented with different kinds of boundary conditions, constitute an interesting area of research, because such systems appear in mathematical models of real-world problems, such as ecology [1], chaos and fractional dynamics [2], financial economics [3], bio-engineering [4], etc. Nonlocal boundary conditions are found to be more plausible and practical in contrast to the classical boundary conditions in view of their applicability to describe the changes happening within the given domain. In the literature, there are many fractional derivative operators, such as Riemann-Liouville, Caputo, Hadamard, Hilfer, Katugampola, etc., see the monographs [5][6][7][8][9][10]. ...
... where all constants and notations are as in the problem (2). The following lemma is not difficult to derive and, therefore, we omit the proof. ...
In this paper, we introduce and study a new class of coupled and uncoupled systems, consisting of mixed-type ψ1-Hilfer and ψ2-Caputo fractional differential equations supplemented with asymmetric and symmetric integro-differential nonlocal boundary conditions (systems (2) and (13), respectively). As far as we know, this combination of ψ1-Hilfer and ψ2-Caputo fractional derivatives in coupled systems is new in the literature. The uniqueness result is achieved via the Banach contraction mapping principle, while the existence result is established by applying the Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.
... For instance, in the diffusion process, the operator was used to derive heat kernel estimates for many symmetric jump-type processes (see Ref. [2]) and to study the acoustic wave equation. In astrophysics, it is used to describe the dynamics in the Hamiltonian chaos (see Ref. [3]). It also has various applications in probability and finance, in which this operator is defined as the generator of α-stable Lévy processes that represent random motions, such as the Brownian motion and the Poisson process (see Ref. [4]), anomalous diffusion and quasi-geostrophic flows, turbulence and water waves, molecular dynamics, and relativistic quantum mechanics of stars, see Refs. ...
... The concerning parabolic system includes global existence and global existence numbers, blow-up rates, and blow-up sets, uniqueness or non-uniqueness and so on. For system (3) there are problems such as existence or non-existence, uniqueness or non-uniqueness and so on. Most works were restricted to special cases, such as, n 1 = m 2 = 0 (single equation), m 1 = n 2 = 0 (variational system) or m 1 = p, m 2 = p + 1, n 1 = q + 1, n 2 = q (see [12]). ...
In this paper, we are concerned with the anti-symmetric solutions to the following elliptic system involving fractional Laplacian {(−Δ)su(x)=um1(x)vn1(x),u(x)≥0,x∈R+n,(−Δ)sv(x)=um2(x)vn2(x),v(x)≥0,x∈R+n,u(x′,−xn)=−u(x′,xn),x=(x′,xn)∈Rn,v(x′,−xn)=−v(x′,xn),x=(x′,xn)∈Rn, where 0<s<1, mi,ni>0(i=1,2),n>2s,R+n={(x′,xn)|xn>0}. We first show that the solutions only depend on xn variable by the method of moving planes. Moreover, we can obtain the monotonicity of solutions with respect to xn variable (for the critical and subcritical cases mi+ni≤n+2sn−2s(i=1,2) in the L2s space). Furthermore, when m1=n2=p,n1=m2=q, in the cases p+q+2s≥1, we obtain a Liouville theorem for the cases p+q≤n+2sn−2s in the L2s space. Then, through the doubling lemma, we obtain the singularity estimates of the positive solutions on a bounded domain Ω. Using the anti-symmetric property of the solutions, one can extend the space from L2s to L2s+1, we can still prove the Liouville theorem in the extended space. With the extension, we prove the existence of nontrivial solutions.
... As traditional prototype of Hamiltonian chaos, the conservative Standard Map models a "kicked rotor" system whose evolution equations are given by [4,7] ' sin ...
It is known that both classical and Quantum Field Theory (QFT) are built on the fundamental principle of stationary action. The goal of this introductory work is to analyze the breakdown of stationary action under nonadiabatic conditions. These conditions are presumed to develop far above the Standard Model scale and favor the onset of Hamiltonian chaos and fractal spacetime. The nearly universal transition to nonadiabatic behavior is illustrated using a handful of representative examples. If true, these findings are likely to have far-reaching implications for phenomena unfolding beyond the Standard Model scale and in early Universe cosmology.
... , m. Banach contraction mapping principle, Krasnosel'skiȋ's fixedpoint theorem, and Laray-Schauder nonlinear alternative are used to establish the existence and uniqueness results. Nonlocal fractional order coupled systems are also significant, as such systems often occur in applications, for example in fractional dynamics [32], bio-engineering [33], financial economics [34], etc. In a series of papers [35][36][37], a variety of coupled systems for (k, ψ)-Hilfer differential equations of fractional order were investigated. ...
This paper deals with a nonlocal fractional coupled system of (k,ψ)-Hilfer fractional differential equations, which involve, in boundary conditions, (k,ψ)-Hilfer fractional derivatives and (k,ψ)-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples.
... These systems often exhibit long-range interactions and memory e¤ects, which are challenging to model using classical calculus. Fractional calculus provides a more suitable mathematical framework to accurately capture the underlying physics [4]. A signi…cant application of fractional calculus in particle physics is in the modeling of transport phenomena in plasmas. ...
Incorporating a topological defect and anisotropic plasma, this work used the generalized fractional of the Nikiforov–Uvarov technique to solve the fractional-radial Schrödinger equation in the longitudinal-transverse plane. The study produced wave functions and energy eigenvalues in their fractional forms. The results showed that the presence of an anisotropic plasma and a topological defect increases the dissociation energy of bottomonium. Furthermore, regardless of whether the fractional or classical models are taken into account, it was shown that the effect of temperature on the dissociation energy is stronger than the effect of baryonic chemical potential. In addition, the dissociation energy of bottomonium is significantly larger at lower chemical potential levels. Last but not least, the energy of bottomonium is only little influenced by magnetic auxiliaries.
... Fractional coupled systems are also important, as such systems appear in the mathematical models associated with fractional dynamics [26], bio-engineering [27], financial economics [28], etc. In [29], the authors studied the existence and Ulam-Hyers stability results of a coupled system of ψ-Hilfer sequential fractional differential equations. ...
In this paper, a new class of coupled hybrid systems of proportional sequential ψ-Hilfer fractional differential equations, subjected to nonlocal boundary conditions were investigated. Based on a generalization of the Krasnosel’skii˘’s fixed point theorem due to Burton, sufficient conditions were established for the existence of solutions. A numerical example was constructed illustrating the main theoretical result. For special cases of the parameters involved in the system many new results were covered. The obtained result is new and significantly contributes to existing results in the literature on coupled systems of proportional sequential ψ-Hilfer fractional differential equations.
... Following this exceptional theory, the subject of FC caught great attention in mathematics and other areas of scientific research. Since then, FC has been extensively used to examine various physical facts, such as electromagnetic, viscoelastic, and damping theories, artificial intelligence, fluid dynamics, wave propagation theories, chaotic dynamics, heat transfer analysis, appliances, control systems, and various inherent algorithms [28][29][30][31][32][33][34]. ...
This article aims to examine the nonlinear excitations in a coupled Hirota system described by the fractal fractional order derivative. By using the Laplace transform with Adomian decomposition (LADM), the numerical solution for the considered system is derived. It has been shown that the suggested technique offers a systematic and effective method to solve complex nonlinear systems. Employing the Banach contraction theorem, it is confirmed that the LADM leads to a convergent solution. The numerical analysis of the solutions demonstrates the confinement of the carrier wave and the presence of confined wave packets. The dispersion nonlinear parameter reduction equally influences the wave amplitude and spatial width. The localized internal oscillations in the solitary waves decreased the wave collapsing effect at comparatively small dispersion. Furthermore, it is also shown that the amplitude of the solitary wave solution increases by reducing the fractal derivative. It is evident that decreasing the order α modifies the nature of the solitary wave solutions and marginally decreases the amplitude. The numerical and approximation solutions correspond effectively for specific values of time (t). However, when the fractal or fractional derivative is set to one by increasing time, the wave amplitude increases. The absolute error analysis between the obtained series solutions and the accurate solutions are also presented.
... For stochastic variables, such scaling rules are typically unknown and valid only statistically. Nevertheless, concepts of fractality continue to arise in physics [4]. In particular, fractal dimensions often emerge in the fundamental phenomenon of diffusion [5]. ...
The Kardar–Parisi–Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation–dissipation theorem (FDT) for spatial dimension d>1. Notably, these questions were answered exactly only for 1+1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d+1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension dn. We present relations between the KPZ exponents and two emergent fractal dimensions, namely df, of the rough interface, and dn. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α, the surface fractal dimension df and, through our relations, the noise fractal dimension dn. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.
... There has also been shown a significant interest in studying the fractional-order coupled systems, as such systems appear in the mathematical models associated with bioengineering [13], financial economics [14], fractional dynamics [15], etc. In [16], the authors studied a coupled system of ψ-Hilfer fractional differential equations. ...
In this paper, we investigate a nonlinear coupled integro-differential system involving generalized Hilfer fractional derivative operators ((k,ψ)-Hilfer type) of different orders and equipped with non-local multi-point ordinary and fractional integral boundary conditions. The uniqueness results for the given problem are obtained by applying Banach’s contraction mapping principle and the Boyd–Wong fixed point theorem for nonlinear contractions. Based on the Laray–Schauder alternative and the well-known fixed-point theorem due to Krasnosel’skiĭ, the existence of solutions for the problem at hand is established under different criteria. Illustrative examples for the main results are constructed.
... Fractal dimensions provide a way to characterize the irregular and self-repeating patterns that arise in the trajectories of chaotic systems. By quantifying the fractal dimension of attractors or basin boundaries, researchers gain insights into the complexity of classical chaotic behavior [55]. ...
This paper aims to explore the intricate relationships between the Schrödinger equations that govern the evolution of quantum systems, the phenomenon of quantum measurements, and the underlying fractal geometry apparent in various quantum systems. Our objective is to establish a theoretical framework that uses the mathematics of fractals as a bridge to unify our understanding of quantum dynamics and measurements. Methodology: We start with a comprehensive review of the basic principles of quantum mechanics, focusing on the Schrödinger equations and the postulates of quantum measurements. Subsequently, we delve into fractal geometry, outlining key concepts and properties of fractals. Employing mathematical and computational models, we analyze the fractal-like structures in the energy spectra of certain quantum systems and examine how these fractal properties may relate to the wave functions governed by the Schrödinger equations. We further investigate the implications of these fractal properties on the phenomena of quantum measurements. Main Findings: Our analysis reveals that the fractal nature inherent in certain quantum states, described mathematically, offers a fresh perspective on understanding the relationship between the Schrödinger equations and quantum measurements. We demonstrate through specific models that the fractal-like structures in the energy spectra of quantum systems provide insights into the process of wavefunction collapse during quantum measurements. These fractal structures act as a mathematical conduit that helps to elucidate the probabilistic nature of quantum measurements and potentially solves some of the paradoxes associated with wave function collapse. Significance: This study provides a novel interdisciplinary approach to understanding quantum mechanics, bringing the rich mathematics of fractal geometry into the foundational questions of quantum theory. The proposed framework not only sheds light on the intricate nature of quantum dynamics and measurements but also suggests practical applications, particularly in the realm of quantum computing and quantum information theory. By unifying concepts that have traditionally been treated separately, this work may pave the way for new theoretical developments and technologies in quantum physics.
... Other invariant curves form "islands" characterized by regular motion inside the core and mixing domains outside. Islands of first-order nonlinear resonances are surrounded by smaller, higher-order resonance islands, forming chains with narrow stochastic layers in between (see, e.g., [16]). The vortex core is a robust structure. ...
In the Lagrangian approach, the transport processes in the ocean and atmosphere are studied by tracking water or air parcels, each of which may carry different tracers. In the ocean, they are salt, nutrients, heat, and particulate matter, such as plankters, oil, radionuclides, and microplastics. In the atmosphere, the tracers are water vapor, ozone, and various chemicals. The observation and simulation reveal highly complex patterns of advection of tracers in turbulent-like geophysical flows. Transport barriers are material surfaces across which the transport is minimal. They can be classified into elliptic, parabolic, and hyperbolic barriers. Different diagnostics in detecting transport barriers and the analysis of their role in the dynamics of oceanic and atmospheric flows are reviewed. We discuss the mathematical tools, borrowed from dynamical systems theory, for detecting transport barriers in simple kinematic and dynamic models of vortical and jet-like flows. We show how the ideas and methods, developed for simple model flows, can be successfully applied for studying the role of barriers in oceanic and atmospheric flows. Special attention is placed on the significance of transport barriers in important practical issues: anthropogenic and natural pollution, advection of plankton, cross-shelf exchange, and propagation of upwelling fronts in coastal zones.
... This is a key distinction between the two types of operators. This indicates that the next state of the system is dependent not only on the state in which the system is presently running but also on all of the states in which the system has operated in the past (for further information, see [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and the references therein). ...
... We use a variety of real-world examples to assess the performance of the new method on our problem, see [10,14,16,29,30,36]. The integration is carried out using the three-step fractional Adams-Bashforth methods for Caputo, Caputo Fabrizio, and Atangana-Baleanu. ...
In the present paper, we introduce a two-order nonlinear fractional sequential Langevin equation using the derivatives of Atangana-Baleanu and Caputo-Fabrizio. The existence of solutions is proven using a fixed point theorem under a weak topology, and an illustrative example is then given. Furthermore, we present new fractional versions of the Adams-Bashforth three-step approach for the Atangana-Baleanu and Caputo derivatives. New nonlinear chaotic dynamics are performed by numerical simulations.
... Generalized Yang-Fourier transforms which is obtained by authors by generalization of Yang-Fourier transforms is a technique of fractional calculus for solving mathematical, physical and engineering problems. The fractional calculus is continuously growing in last five decades [1][2][3][4][5][6][7]. Most of the fractional ordinary differential equations have exact analytic solutions, while others required either analytical approximations or numerical techniques to be applied, among them: fractional Fourier and Laplace transforms [8,41], heat-balance integral method [9][10][11], variation iteration method (VIM) [12][13][14], decomposition method [15,41], homotopy perturbation method [16,41] etc. ...
... In particular, the ergodic theorem, the fluctuation-dissipation theorem, analyticity, unitarity, locality, finiteness in all orders of perturbation theory and renormalizability are either violated or lose their conventional meaning [4][5][6][7][8][9][10][11][12][13]. ...
This report expands on the preprint "Dimensional Regularization as Mass Generating Mechanism", posted at the following sites: https://doi.org/10.32388/DW6ZZS , https://www.researchgate.net/publication/370670747
... The works [121,122] deals with the fractional generalization of the diffusion equation, fractality of the chaotic dynamics and kinetics, and also includes material on non-ergodic and non-well-mixing Hamiltonian dynamics. ...
In this article we would like to consider some approaches to non-integer integro-differentiations and its implementation in computer algebra system Wolfram Mathematics.
... Thereby, it evolved in many theoretical and applications area. For application details in ecology, chaos and fractional dynamics, medical sciences, financial economics bio-engineering, immune system, etc., we refer the reader to the works [2][3][4][5][6][7][8][9]. For more theoretical aspects of fractional calculus, we refer the reader to the monographs [10][11][12][13][14][15][16][17][18]. ...
... For various geometrical considerations related to the emergence question see also [14, 15, 25, 30–35, 51, 62] and in connection with chaos we cite e.g. [1, 25, 38, 41, 50, 51, 62, 63].Volume 2 PROGRESS IN PHYSICS April, 2009 ...
According to the experimental analysis conducted by P.-M. Robitaille, the 2.7 K microwave background, first detected by Penzias and Wilson, is not of cosmic origin, but originates from the Earth, and is generated by oceanic water. In examining this problem two fields must be considered: (1) the Earth Microwave Background, the EMB, presentwith the 2.7 K monopole and 3.35 mK dipole components; (2) the weak Intergalactic Microwave Background, the IMB, which is connected to the entire Metagalaxy. This conclusion meets our theoretical considerations. First, the field density of the EMB, being inversely proportional to the field volume, should decrease with the cube of thedistance from the Earth’s surface, while its dipole anisotropy, which is due to the motion of the entire field in common with the Earth, is independent from altitude. Therefore, the EMB monopole should not be found at the 2nd Lagrange point (1.5 mln km from the Earth), while the dipole anisotropy should remain the same as near the Earth. Second, according to General Relativity, the motion through the IMB in a referred direction manifests the three-dimensional rotation of the entire space of the Metagalaxy.
... Fractional calculus, as a branch of mathematics, is one of the modern mathematical tools to solve complex problems in science and engineering technology [1][2][3][4][5][6][7][8][9]. The significance of fractional calculus is to extend the integer order calculus in the general sense to any fractional order, which is an extension of integer order calculus operation. ...
Because of the nonlocal and nonsingular properties of fractional derivatives, they are more suitable for modelling complex processes than integer derivatives. In this paper, we use a fractional factor to investigate the fractional Hamilton’s canonical equations and fractional Poisson theorem of mechanical systems. Firstly, a fractional derivative and fractional integral with a fractional factor are presented, and a multivariable differential calculus with fractional factor is given. Secondly, the Hamilton’s canonical equations with fractional derivative are obtained under this new definition. Furthermore, the fractional Poisson theorem with fractional factor is presented based on the Hamilton’s canonical equations. Finally, two examples are given to show the application of the results.
... The Hamiltonian of the model is given by [11,18] Fig. 6 shows a color-coded representation of the map at 0.128 E = . As chaos sets in above a critical value of the driving parameter(s), analysis shows that chaotic orbits repeatedly "stick" to the border of critical tori with a power-like distribution of sticking times [12][13][14][15]. This effect generates a long-time correlation of chaotic orbits and an anomalous diffusion of momentum in phase-space. ...
This is Part 2 of the brief tutorial "Hamiltonian Chaos and the Fractal Topology of Spacetime" posted at https://www.researchgate.net/publication/369584882.
... Concerning the importance of coupled fractional differential systems, it is well-known that such systems appear in the mathematical models of many physical phenomena related to bio-engineering [15], fractional dynamics [16], financial economics [17], etc. In [18,19], some interesting results forψ-Hilfer fractional differential coupled systems were obtained. ...
We investigate a nonlinear, nonlocal, and fully coupled boundary value problem containing mixed (k,ψ^)-Hilfer fractional derivative and (k,ψ^)-Riemann–Liouville fractional integral operators. Existence and uniqueness results for the given problem are proved with the aid of standard fixed point theorems. Examples illustrating the main results are presented. The paper concludes with some interesting findings.
... 2. The emergence of a spacetime equipped with continuous dimensionality above the Fermi scale follows from several premises, one of them being the onset of Hamiltonian chaos and fractional dynamics [9][10] We believe that, besides 1) -4), a scenario worthy of investigation is the fluidgravity correspondence inspired by the gauge-gravity duality of string theory [14]. A drawback of this duality is that it operates with a negative cosmological constant, clearly at odds with current astrophysical observations. ...
Complex Ginzburg-Landau equation (CGLE) is a universal model of nonequilibrium dynamical systems. Focusing on the primordial stages of cosmological evolution, this work points out that the connection between CGLE and the Navier-Stokes (NS) equation bridges the gap between fluid flows and the mathematics of General Relativity (GR).
... For the first time, nonlocal (fractional) generalizations of the concepts of the average value and nonlocal generalizations of the normalization conditions for probability density functions were suggested in 2004 [16,[29][30][31]. The fractional and nonlocal probability theories can be important for nonlocal statistical mechanics [37][38][39][40][41], fractional kinetics and anomalous transport [21,22,[42][43][44][45], and non-Markovian quantum dynamics [46,47]. ...
A generalization of probability theory is proposed by using the Riemann–Liouville fractional integrals and the Caputo and Riemann–Liouville fractional derivatives of arbitrary (non-integer and integer) orders. The definition of the fractional probability density function (fractional PDF) is proposed. The basic properties of the fractional PDF are proven. The definition of the fractional cumulative distribution function (fractional CDF) is also suggested, and the basic properties of these functions are also proven. It is proven that the proposed fractional cumulative distribution functions generate unique probability spaces that are interpreted as spaces of a fractional probability theory of arbitrary order. Various examples of the distributions of the fractional probability of arbitrary order, which are defined on finite intervals of the real line, are suggested.
... Due to the recent widespread interest in discrete fractional calculus, many difference operators, notably the Grünwald-Letnikov difference operator [3], have been proposed in the literature. Many efforts have been made to thoroughly analyze the dynamics of both classical systems and fractional systems since the discovery of chaos phenomena in dynamical systems [3,4]. Regarding the latter, various works have been conducted on the topic of studying chaotic behaviors in nonlinear maps defined by fractional order difference equations [5][6][7]. ...
Fractional order maps are a hot research topic; many new mathematical models are suitable for developing new applications in different areas of science and engineering. In this paper, a new class of a 2D fractional hyperchaotic map is introduced using the Caputo-like difference operator. The hyperchaotic map has no equilibrium and lines of equilibrium points, depending on the values of the system parameters. All of the chaotic attractors generated by the proposed fractional map are hidden. The system dynamics are analyzed via bifurcation diagrams, Lyapunov exponents, and phase portraits for different values of the fractional order. The results show that the fractional map has rich dynamical behavior, including hidden homogeneous multistability and offset boosting. The paper also illustrates a novel theorem, which assures that two hyperchaotic fractional discrete systems achieve synchronized dynamics using very simple linear control laws. Finally, the chaotic dynamics of the proposed system are stabilized at the origin via a suitable controller.
... As phase transitions abound in Nature, the control parameter can take on many forms, from temperature to pressure, density, chemical potential, 3) Emergence of fractal spacetime from the fragmented structure of phase-space in Hamiltonian chaos [16]. ...
We argue here that the onset of classical chaos above the Fermi scale underlies the construction of Effective Field Theory (EFT). According to this view, particle physics and gravitational dynamics are low-energy manifestations of chaotic behavior and multifractal geometry.
... From Table I, it is seen that the naïve mean-field exponents are simply the usual mean-field ones whereas the corrected ones are distinctive. We note in passing that power-law interactions mathematically correspond to fractional calculus [86][87][88][89][90] and general fractional dynamic equations for both longrange spatial and temporal interactions have been proposed [91]. In epidemic spreading, long-range spatial infections [92] and long-range temporal infections with algebraically distributed waiting times for future infections [93][94][95][96] have also been analyzed. ...
We develop a systematic theory for the critical phenomena with memory in all spatial dimensions, including $d<d_c$, $d=d_c$, and $d>d_c$, the upper critical dimension. We show that the Hamiltonian plays a unique role in dynamics and the dimensional constant $\mathfrak{d}_t$ that embodies the intimate relationship between space and time is the fundamental ingredient of the theory. However, its value varies with the space dimension continuously and vanishes exactly at $d=4$, reflecting reasonably the variation of the amount of the temporal dimension that is transferred to the spatial one with the strength of fluctuations. Such variations of the temporal dimension save all scaling laws though the fluctuation-dissipation theorem is violated. Various new universality classes emerge.
... In the second half of the last century, the search for more serious numerical algorithms began to control many of the nonlinear problems that appeared with the emergence of fractional derivatives and their emerging applications. However, many studies came into existence, as in the following literature [9][10][11][12][13]. To date, effective analytical and numerical schemes have been developed and successfully applied to deal with different classes of FDMs [14][15][16][17][18][19][20]. ...
Studying and analyzing the random motion of a particle immersed in a liquid represented in the Langevin fractional model by Caputo’s independent derivative is one of the aims of applied physics. In this article, we will attend to a new, accurate, and comprehensive numerical solution to the aforementioned model using the reproducing kernel Hilbert approach. Basically, numerical and exact solutions of the fractional Langevin model are represented using an infinite/finite sum, simultaneously, in the Σ 2 Ξ space. The proof has been sketched for many mathematical theorems such as independence, convergence, error behavior, and completeness of the solution. A sufficient set of tabular results and two-dimensional graphs are shown, and absolute/relative error graphs that express the dynamic behavior of the fractional parameters α , β are utilized as well. From an analytical and practical point of view, we noticed that the simulation process and the iterative approach are appropriate, easy, and highly efficient tools for solving the studied model. In conclusion, what we have carried out is presented with a set of recommendations and an outlook on the most important literature used.
... Since the discovery of chaos, various works have been made to deeply analyze the dynamics of classical systems and fractional systems [4,5]. Referring to the latter, several papers have been published on the study of chaotic behaviors in nonlinear maps described by difference equations of fractional order [6,7]. ...
Recent works have focused the analysis of chaotic phenomena in fractional discrete memristor. However, most of the papers have been related to simulated results on the system dynamics rather
than on their hardware implementations. This work reports the implementation of a new chaotic
fractional memristor map with “hidden attractors”. The fractional memristor map is developed
based on a memristive map by using the Grunwald–Letnikov difference operator. The fractional
memristor map has flexible fixed points depending on a system’s parameters. We study system
dynamics for different values of the fractional orders by using bifurcation diagrams, phase portraits,
Lyapunov exponents, and the 0–1 test. We see that the fractional map generates rich dynamical
behavior, including coexisting hidden dynamics and initial offset boosting.
... A remarkable property of these systems is that they can be described as geodesic flows on Riemannian manifolds [3][4][5]. It is also known that the ergodic regime of Hamiltonian dynamics arises from the divergence of geodesics on manifolds with negative curvature (so-called hyperbolic manifolds), which drive the transition from integrability to chaos [3][4][5][6][7][8]. Research shows that Hamiltonian chaos is not restricted to hyperbolic manifolds, but that it extends to large N -body systems analogous to fluid flows ( → N ) and to systems displaying parametric instability [3-5]. ...
We argue here that the high-energy behavior of fundamental interactions can be interpreted as manifestation of Kolmogorov (-K) entropy. The conventional classification of fields based on Poincaré symmetry appears to be rooted in the chaotic regime of nonlinear dynamics far above the Standard Model scale.
... Two groups of explanations have been proposed to clarify the origins of the intermittency phenomena. First, intermittency may originate from Hamiltonian chaos (Zaslavsky & Zaslavskij, 2005) and hydrodynamical systems (Bessaih et al., 2015). Second, intermittent behaviors can also arise from small control parameter fluctuations around critical values (Contoyiannis et al., 2002;Hramov et al., 2014). ...
Koopman operator theory and the Hankel alternative view of the Koopman (HAVOK) model have been widely used to investigate the chaotic dynamics in complex systems. Although the statistics of intermittent dynamics have been evaluated in the HAVOK model, they are not adequate to characterize intermittent forcing. In this paper, we propose a novel method to characterize the intermittent phases, chaotic bursts, and local spectral-temporal properties of various intermittent dynamics modes using spectral decomposition and wavelet analysis. To validate our methods, we compared the sensitivity to noise level and sampling period of the HAVOK and our proposed method in the Lorenz system. Our results show that the prediction accuracy of lobe switching, and the intermittent forcing identifiability were highly sensitive to the sampling rate. While it is possible to maintain the desired accuracy in high noise-level cases with an appropriately selected rank in the HAVOK model, our proposed method is demonstrated to be more robust. To show the applicability of our proposed method, obstructive sleep apnea—a complex pathological disorder—was selected as a case study. The results show a strong association between active forcing and the hypopnea-apnea events. Our proposed method has been demonstrated to be a promising data-driven method to provide key insights into the dynamics of complex systems.
... Great interest has also been shown in studying the coupled systems of fractional-order differential equations, as such systems constitute the mathematical models of many physical phenomena occurring in bio-engineering [7], fractional dynamics [8], and financial economics [9], etc. Let us now dwell on some recent works on boundary value problems involving coupled fractional differential systems. ...
This paper is concerned with the existence of solutions for a new boundary value problem of nonlinear coupled (k,ψ)–Hilfer fractional differential equations subject to coupled (k,ψ)–Riemann–Liouville fractional integral boundary conditions. We prove two existence results by applying the Leray–Schauder alternative, and Krasnosel’skiĭ’s fixed-point theorem under different criteria, while the third result, concerning the uniqueness of solutions for the given problem, relies on the Banach’s contraction mapping principle. Examples are included for illustrating the abstract results.
It is known that both classical and Quantum Field Theory (QFT) are built on the fundamental principle of stationary action. The goal of this introductory work is to analyze the breakdown of stationary action under _nonadiabatic conditions_. These conditions are presumed to develop far above the Standard Model scale and favor the onset of Hamiltonian chaos and fractal spacetime. The nearly universal transition to nonadiabatic behavior is illustrated using a handful of representative examples. If true, these findings are likely to have far-reaching implications for phenomena unfolding beyond the Standard Model scale and in early Universe cosmology.
Recent research points out that the unavoidable approach to Hamiltonian chaos well above the Fermi scale leads to a spacetime having continuous (fractal) dimensions. Here we analyze a toy model of inflationary Universe comprising of a Higgs-like scalar in interaction with a pair of vector bosons. The model is manifestly nonintegrable as it breaks the perturbative unitarity of scattering processes and evolves towards Hamiltonian chaos and fractal spacetime.
The goal of this paper is to analyze the likely transition from integrability to Hamiltonian chaos in the primordial Universe. The transition is driven by curvature fluctuations and favors the onset of a spacetime endowed with continuous dimensions.
Discrete maps with uniformly distributed memory fading parameter are proposed. These maps are derived from equations with fractional derivatives and fractional integrals of uniformly distributed orders and periodic sequence of kicks that are described by Dirac delta-functions. The fractional differential and fractional integral equations, for which the discrete maps are derived, are nonlinear equations with fractional operators proposed by Nakhushev and Pskhu. The solutions of these fractional differential and integral equations of distributed orders are derived. It is emphasized that the fractional integrals and derivatives with a uniformly distributed order cannot be considered as mutually inverse operators for which the fundamental theorems hold. Fundamental theorems of fractional calculus for fractional operators with uniformly distributed order, which are proved by Pshu, are used to obtain discrete maps with uniformly distributed memory fading parameter. In this paper, we also derive discrete maps with memory from equations with FDs and FIs that are inverse operators to FDs and FIs of a distributed order. The all suggested discrete maps are derived from fractional differential and integral equations without an approximation.
In this paper, we consider the following perturbed fractional Hamiltonian systems
where \(\alpha \in (1/2,1],\,\,L \in C(\mathbb{R},{\mathbb{R}^{N \times N}})\) is symmetric and not necessarily required to be positive definite, \(W \in {C^1}(\mathbb{R}\times {\mathbb{R}^N},\mathbb{R})\) is locally subquadratic and locally even near the origin, and perturbed term \(G \in {C^1}(\mathbb{R} \times {\mathbb{R}^N},\mathbb{R})\) maybe has no parity in u. Utilizing the perturbed method improved by the authors, a sequence of nontrivial homoclinic solutions is obtained, which generalizes previous results.
The fractional Boltzmann and Fokker–Planck equations could be derived from the continuous-time random walks (CTRW) model. The usual random walks model corresponds to the ordinary diffusion governed by the equation with the integer-order derivatives, while the anomalous diffusion is described by the equation with the fractional-order derivatives. The fractional space derivative of the order \(\alpha < 2\) corresponds to the random walk model with the heavy-tailed power law distribution of the particle jumps \(\mathcal {P} (j > x) \propto x^{- \alpha }\) and describes the anomalous superdiffusion (e.g. Lévy walks)—the particles spread faster than the ordinary diffusion predicts. Similarly, fractional time derivative of the order \(\beta < 2\) corresponds to the random walk model with the heavy-tailed power law distribution of the particle waiting times \(\mathcal {P} (w > t) \propto t^{- \beta }\) and describes the anomalous subdiffusion.
This paper is concerned about the bi-spatial dynamics of the stochastic reaction–diffusion equations driven by a Wong–Zakai approximation process on the entire space \({\mathbb {R}}^n.\) We obtain the existence of pullback random \((L^2({\mathbb {R}}^n), L^p({\mathbb {R}}^n))\)-attractor and \((L^2({\mathbb {R}}^n), H^s({\mathbb {R}}^n))\)-attractor for any \(p>2, 0<s<1.\) In addition, we further prove the upper semi-continuity of these attractors of the approximate random equation under the topology of \(L^p({\mathbb {R}}^n)\) and \(H^s({\mathbb {R}}^n)\) as the size of approximation tends to zero.
The framework of fractional partial differential models is the first-rate hyperlink between mathematics and applied physics. This article project intends to utilize outcomes for the time-fractional Newell–Whitehead–Segel model in 2D-space including conservation laws, Lie point symmetry analysis, and series solutions. We introduce a particular fractional model that is free of the type of approximate methods, whilst, environmental flow and magnetohydrodynamics processes are considered to be the main real-world phenomena treated with such a model. Herein, the method of power series is exercised to provide an analytical solution to the current governing model. The idea and undertaking of the method lie in the assumption that the solution is a power series of coefficients that are determined by a recurrence relation obtained by substituting the series solution in the considered model. Also, the Riemann approach is utilized as a total derivative. Simulation effects are systematically demonstrated through a chain of check cases. Strong proofs with some related plots are performed to confirm the accuracy and fitness of the model version and the presented approach. Moreover, the laws of conservation depend on the existence of a Lagrangian of the fractional Newell–Whitehead–Segel model utilized. Ultimately, a few related comments and future proposals are epitomized.
• HIGHLIGHT
• The Lie point symmetries of the time-fractional Newell–Whitehead–Segel equation in two-dimensional space is utilized and derived.
• The technique of the power series is applied to conclude the explicit solutions for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space for the first time.
• The conservation laws for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space are built using a novel conservation theorem.
• Several graphical countenances were utilized to award a visual performance of the obtained solutions.
In this paper, we derive the conformable fractional wave equation and conformable fractional KdV equation from the ordinary Newton equation with deformed translational symmetry. At first, we obtain a solution to the dimensionless 1D conformable wave equation with fixed ends, and then we study the D'Alembert solution for full space. Next, we examine the dimensionless 2D conformable fractional wave equation in the deformed disk. Finally, we derive the conformable fractional KdV equation and explore its solution.
This paper equipped the time- and space-fractional Sobolev equation with the condition of the Caputo fractional derivative in \(\left(1+1\right)\)-dimensional space to be solved with the help of the reproducing kernel Hilbert space method. The aforesaid method depends on building two Hilbert spaces where the coefficients of fractional expansion are produced by using the generalized Gram Schmidt process. With the use of the Fourier functions expansion theorem, the numeric-analytic solutions are expressed by collection sets of orthonormal functions system in \(\mathfrak{A}\left(\mathcal{U}\right)\) and \(\mathfrak{B}\left(\mathcal{U}\right)\) spaces. In this flair, novel steps are fitted for the covering fractional Sobolev equation and the utilized numeric-analytic approach. To display the obtained results and theories, a variety of tables and graphics will be displayed and exhibited. The received effects imply that the technique is shrewd and has numerous capabilities balance for managing many fractional fashions rising in physics and applied mathematics by the Caputo class derivative. Future work and several notes are collected in the final part.
The main objective of this article is to use local fractional calculus (Calculus of arbitrary order) and the analytical Advanced Yang‐Fourier transforms method to solve the one‐dimensional fractal heat‐conduction problem in a fractal semi‐infinite bar.
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