Fractional Attractors Research Articles

Palatini [formula omitted]: A new framework for inflationary attractors

Palatini F(R) gravity proved to be a powerful tool in order to realize asymptotically flat inflaton potentials. Unfortunately, it also inevitably implies higher-order inflaton kinetic terms in the Einstein frame that might jeopardize the evolution of the system out of the slow-roll regime. We prove that a F(R+X) gravity, where X is the inflaton kinetic term, solves the issue. Moreover, when F is a quadratic function such a choice easily leads to a new class of inflationary attractors, fractional attractors, that generalizes the already well-known polynomial α-attractors.

Nonlinear analysis of a four-dimensional fractional hyper-chaotic system based on general Riemann-Liouville-Caputo fractal-fractional derivative.

In this study, a four-dimensional fractional hyperchaotic model is analyzed based on general Riemann–Liouville–Caputo (RLC) fractal–fractional derivative (FFD). A series of new operators are constructed using three different elements, namely, the general Mittag–Leffler function, exponential decay, and power law. The operators have two parameters: One is considered as fractional order and the other as fractal dimension. The Qi hyperchaotic fractional attractor is modeled by using these operators, and the models are solved numerically using a very efficient numerical scheme. Meanwhile, the existence and uniqueness of solutions have been investigated to justify the physical adequacy of the model and the numerical scheme proposed in the resolution. The numerical simulations for some specific fractional order and fractal dimension are presented. Furthermore, these results obtained via generalized Caputo–Fabrizio and fractal–fractional derivative show some crossover effects, which is due to non-index law property. Finally, these obtained from generalized fractal–fractional derivative show very strange and new attractors with self-similarities.

Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

NEW ESTIMATES OF INTEGRAL INEQUALITIES VIA GENERALIZED PROPORTIONAL FRACTIONAL INTEGRAL OPERATOR WITH RESPECT TO ANOTHER FUNCTION

In this paper, the newly proposed concept of the generalized proportional fractional integral operator with respect to another function [Formula: see text] has been utilized to generate integral inequalities using convex function. This new concept will have the option to reduce self-similitudes in the fractional attractors under investigation. We discuss the implications and other consequences of the integral inequalities concerning the generalized proportional fractional integral operator with respect to another function [Formula: see text] are derived here and these outcomes permit us specifically to generalize some classical inequalities. Certain intriguing subsequent consequences of the fundamental hypotheses are also figured. It is to be supposed that this investigation will provide new directions in the quantum theory of capricious nature.

An experimental synthesis methodology of fractional-order chaotic attractors

A novel synthesis methodology of fractional-order chaotic systems, from the level of nonlinear systems until their experimental verification using microcontrollers, is presented. Firstly, the integer-order behavioral model of the Lorenz’s, Rossler’s, Chen’s, Liu’s, Saturated Nonlinear Function Series at one- and two-direction systems is briefly reviewed. Secondly, a first-order transfer function that approximates the behavior of a fractional-order integrator, based on continued fraction expansion method, is substituting the integer-order integrator inside the revised previously chaotic systems. Thirdly, the minimum phase Al-Alaoui’s transformation method is used for synthesizing all fractional-order chaotic systems in the discrete domain and are programmed in MATLAB and in the Arduino DUE development board. For a fair comparison, the minimum phase Al-Alaoui’s algorithm not only is used for solving the integer-order chaotic systems and it is also programmed in MATLAB and Arduino DUE board, but same initial conditions are also used for both interger- and fractional-order chaotic systems. Finally, experimental results of the integer- and fractional-order chaotic oscillators are shown. The results obtained not only allow a simple synthesis methodology of fractional-order chaotic attractors and their experimental evidence on reconfigurable hardware, but also demonstrate the viability of fractional attractors to be used in various applications such as secure communications, robot control, cryptography and so on.

Application of the Euler and Runge–Kutta Generalized Methods for FDE and Symbolic Packages in the Analysis of Some Fractional Attractors

Abstract This paper applies the Euler and the fourth-order Runge–Kutta methods in the analysis of fractional order dynamical systems. In order to illustrate the two techniques, the numerical algorithms are applied in the solution of several fractional attractors, namely the Lorenz, Duffing and Liu systems. The algorithms are implemented with the aid of Mathematica symbolic package. Furthermore, the Lyapunov exponent is obtained based on the Euler method and applied with the Lorenz fractional attractor.

Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α- Families of Maps

Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α- Families of Maps

Theoretical Analysis and Circuit Verification for Fractional-Order Chaotic Behavior in a New Hyperchaotic System

A novel nonlinear four-dimensional hyperchaotic system and its fractional-order form are presented. Some dynamical behaviors of this system are further investigated, including Poincaré mapping, parameter phase portraits, equilibrium points, bifurcations, and calculated Lyapunov exponents. A simple fourth-channel block circuit diagram is designed for generating strange attractors of this dynamical system. Specifically, a novel network module fractance is introduced to achieve fractional-order circuit diagram for hardware implementation of the fractional attractors of this nonlinear hyperchaotic system with order as low as 0.9. Observation results have been observed by using oscilloscope which demonstrate that the fractional-order nonlinear hyperchaotic attractors exist indeed in this new system.

Fractional Maps and Fractional Attractors. Part I: α-Families of Maps

Fractional Maps and Fractional Attractors. Part I: α-Families of Maps

Fractional dissipative standard map

Using kicked differential equations of motion with derivatives of noninteger orders, we obtain generalizations of the dissipative standard map. The main property of these generalized maps, which are called fractional maps, is long-term memory. The memory effect in the fractional maps means that their present state of evolution depends on all past states with special forms of weights. Already a small deviation of the order of derivative from the integer value corresponding to the regular dissipative standard map (small memory effects) leads to the qualitatively new behavior of the corresponding attractors. The fractional dissipative standard maps are used to demonstrate a new type of fractional attractors in the wide range of the fractional orders of derivatives.