scholarly journals Simulation of Chaotic Oscillators of Fractional Order

2019 ◽  
pp. 11-17
Author(s):  
Alejandro Silva-Juárez ◽  
Miguel De Jesús Salazar-Pedraza ◽  
Juan Jorge Ponce-Mellado ◽  
Gustavo Herrera-Sánchez
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Arbitrary Order ◽  
Chaotic Behavior ◽  
Pid Controllers ◽  
Chaotic Oscillators ◽  
Non Linear ◽  
Derivatives Of ◽  
Corrective Method

In 1695 the theory of fractional calculus was introduced, but it only developed as a pure mathematical branch. Currently several research groups have focused on the control, the implementation of filters, PID controllers, synchronization, the implementation of circuits of chaotic systems of fractional order, etc. Currently, the number of applications of fractional calculus is increasing rapidly, these mathematical phenomena have allowed us to describe and model a real object more accurately than the classical "integer" methods. Along with the development of the fractional calculation, it was shown that many fractional-order nonlinear dynamic systems behave in a chaotic manner. This is the type of non-linear systems that are addressed in this research topic with the focus on derivatives of arbitrary order, where numerical simulations of chaotic behavior are presented in non-linear, fractional-order autonomous models. The case studies are six chaotic oscillators of fractional order; The systems of Lorenz, Rӧssler, Financiero, Lui, Chen and Lü, whose attractors are obtained by applying the definitions of the Grünwald-Letnikov definitions and the predictive corrective method of Adams-Bashforth-Moulton.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals Stabilization and Tracking Control of Inverted Pendulum Using Fractional Order PID Controllers

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Sunil Kumar Mishra ◽  
Dinesh Chandra
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Tracking Control ◽  
Inverted Pendulum ◽  
Fitness Function ◽  
Pid Controllers ◽  
Simulink Model ◽  
System A ◽  
Simulation Results ◽  
Fractional Order Pid

This work focuses on the use of fractional calculus to design robust fractional-order PID (PIλDμ) controller for stabilization and tracking control of inverted pendulum (IP) system. A particle swarm optimisation (PSO) based direct tuning technique is used to design two PIλDμcontrollers for IP system without linearizing the actual nonlinear model. The fitness function is minimized by running the SIMULINK model of IP system according to the PSO program in MATLAB. The performance of proposed PIλDμcontrollers is compared with two PID controllers. Simulation results are also obtained by adding disturbances to the model to show the robustness of the proposed controllers.

scholarly journals On Chebyshev and Riemann-Liouville Fractional Inequalities in q-Calculus

2019 ◽  
pp. 1-10
Author(s):  
Stephen. N. Ajega-Akem ◽  
Mohammed M. Iddrisu ◽  
Kwara Nantomah
Keyword(s):  
Fractional Calculus ◽  
Arbitrary Order ◽  
High Demand ◽  
Quantum Calculus ◽  
Integer Order ◽  
Model Complex ◽  
Derivatives Of ◽  
Increasing Functions ◽  
Anomalous Dynamics ◽  
New Inequalities

This paper presents some new inequalities on Fractional calculus in the context of q-calculus. Fractional calculus generalizes the integer order differentiation and integration to derivatives and integrals of arbitrary order. In other words, Fractional calculus explores integrals and derivatives of functions that involve non-integer order(s). Quantum calculus (q-Calculus) on the other hand focuses on investigations related to calculus without limits and in recent times, it has attracted the interest of many researchers due to its high demand of mathematics to model complex systems in nature with anomalous dynamics. This paper thus establishes some new extensions of Chebyshev and Riemann-Liouville fractional integral inequalities for positive and increasing functions via q-Calculus.

scholarly journals Mathematical models of the pneumatic cascade and a mem-brane pneumatic actuator described by the fractional calculus

2018 ◽  
Vol 19 (12) ◽  
pp. 526-531
Author(s):  
Mirosław Luft ◽  
Artur Nowocień ◽  
Daniel Pietruszczak
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Dynamic Systems ◽  
Dynamic Properties ◽  
Pneumatic Actuator ◽  
Programming Tool ◽  
Analysis And Synthesis ◽  
Continuous Dynamic ◽  
Derivatives Of

The paper presents the analysis of dynamic properties of pneumatic systems such aa pneumatic cascade and a membrane pneumatic actuator using differential equations of integer orders and differential equations with derivatives of non-integer orders. The analyzed systems were described in the domain of time by means of step characteristics and in terms of frequency with the help of Bode characteristics, i.e. logarithmic amplitude and phase characteristics. Each characteristic was determined on the basis of a differential equation with derivatives of non-integer order. To determine the characteristics, an irreplaceable programming tool was the interactive Simulink package built on the basis of the MATLAB programme, which allows the analysis and synthesis of continuous dynamic systems.

Differential and integral relations in the class of multi-index Mittag-Leffler functions

2018 ◽  
Vol 21 (1) ◽  
pp. 254-265 ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Integration ◽  
Multi Index ◽  
Generalized Fractional Calculus ◽  
Special Cases ◽  
Integral Relations ◽  
Derivatives Of ◽  
Prabhakar Function

Abstract As recently observed by Bazhlekova and Dimovski [1], the n-th derivative of the 2-parametric Mittag-Leffler function gives a 3-parametric Mittag-Leffler function, known as the Prabhakar function. Following this analogy, the n-th derivative of the (2m-index) multi-index Mittag-Leffler functions [6] is obtained, and it turns out that it is expressed in terms of the (3m-index) Mittag-Leffler functions [10, 11]. Further, some special cases of the fractional order Riemann-Liouville and Erdélyi-Kober integrals of the Mittag-Leffler functions are calculated and interesting relations are proved. Analogous relations happen to connect the 3m-Mittag-Leffler functions with the integrals and derivatives of 2m-Mittag-Leffler functions. Finally, multiple Erdélyi-Kober fractional integration operators, as operators of the generalized fractional calculus [5], are shown to relate the 2m- and 3m-parametric Mittag-Leffler functions.

scholarly journals Chaos Suppression in a Gompertz-like Discrete System of Fractional Order

2020 ◽  
Vol 30 (03) ◽  
pp. 2050049
Author(s):  
Marius-F. Danca ◽  
Michal Fečkan
Keyword(s):  
Lyapunov Exponent ◽  
Fractional Calculus ◽  
Numerical Integration ◽  
Fractional Order ◽  
Discrete System ◽  
Control Algorithm ◽  
Impulsive Control ◽  
Chaotic Behavior ◽  
Discrete Fractional Calculus ◽  
Chaos Suppression

In this paper, we introduce the fractional-order variant of a Gompertz-like discrete system. The chaotic behavior is suppressed with an impulsive control algorithm. The numerical integration and the Lyapunov exponent are obtained by means of the discrete fractional calculus. To verify numerically the obtained results, beside the Lyapunov exponent, the tools offered by the 0-1 test are used.

scholarly journals Arbitrary-order economic production quantity model with and without deterioration: generalized point of view

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mostafijur Rahaman ◽  
Sankar Prasad Mondal ◽  
Ali Akbar Shaikh ◽  
Ali Ahmadian ◽  
Norazak Senu ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Production Rate ◽  
Fractional Order ◽  
Arbitrary Order ◽  
Optimization Techniques ◽  
Point Of View ◽  
Economic Production Quantity ◽  
Economic Production ◽  
Epq Model ◽  
Production Quantity

AbstractThe key objective of this paper is to study and discuss the application of fractional calculus on an arbitrary-order inventory control problem. Using the concepts of fractional calculus followed by fractional derivative, we construct different possible models like generalized fractional-order economic production quantity (EPQ) model with the uniform demand and production rate and generalized fractional-order EPQ model with the uniform demand and production rate and deterioration. Also, we show that the classical EPQ model is the particular case of the corresponding generalized fractional EPQ model. This greatly facilitates the researcher a novel tactic to analyse the solution of the EPQ model in the presence of fractional index. Furthermore, this attempt also provides the solution obtained through the optimization techniques after using the real distinct poles rational approximation of the generalized Mittag-Leffler function.

scholarly journals General Fractional Integrals and Derivatives of Arbitrary Order

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 755
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Arbitrary Order ◽  
Fractional Integrals ◽  
Basic Properties ◽  
Suitable Generalization ◽  
Fractional Integrals And Derivatives ◽  
Fundamental Theorems ◽  
Derivatives Of

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

On a Caputo-type fractional derivative

2019 ◽  
Vol 10 (2) ◽  
pp. 81-91 ◽  
Author(s):  
Daniela S. Oliveira ◽  
Edmundo Capelas de Oliveira
Keyword(s):  
Differential Operator ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fundamental Theorem ◽  
Arbitrary Order ◽  
Fractional Operator ◽  
Derivatives Of ◽  
Generalized Fractional Derivative

Abstract In this paper, we present a new differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative recently proposed by Katugampola. The generalized fractional derivative, when convenient limits are considered, recovers the Riemann–Liouville and the Hadamard derivatives of arbitrary order. Our differential operator recovers as limiting cases the arbitrary order derivatives proposed by Caputo and by Caputo–Hadamard. Some properties are presented as well as the relation between this differential operator of arbitrary order and the Katugampola generalized fractional operator. As an application we prove the fundamental theorem of fractional calculus associated with our operator.

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