Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order

2017 ◽  
Vol 487 ◽  
pp. 1-21 ◽  
Author(s):  
A. Coronel-Escamilla ◽  
J.F. Gómez-Aguilar ◽  
L. Torres ◽  
R.F. Escobar-Jiménez ◽  
M. Valtierra-Rodríguez
Keyword(s):  
Chaotic Systems ◽  
Variable Order ◽  
Fractional Operators

scholarly journals Applications of Distributed-Order Fractional Operators: A Review

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 110
Author(s):  
Wei Ding ◽  
Sansit Patnaik ◽  
Sai Sidhardh ◽  
Fabio Semperlotti
Keyword(s):  
Fractional Calculus ◽  
Real World ◽  
Transport Processes ◽  
Model Systems ◽  
Variable Order ◽  
Fractional Operators ◽  
Research Activity ◽  
Road Map ◽  
Real World Applications ◽  
Distributed Order

Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

Robust control for fractional variable-order chaotic systems with non-singular kernel

2018 ◽  
Vol 133 (1) ◽  
Author(s):  
C. J. Zuñiga-Aguilar ◽  
J. F. Gómez-Aguilar ◽  
R. F. Escobar-Jiménez ◽  
H. M. Romero-Ugalde
Keyword(s):  
Robust Control ◽  
Chaotic Systems ◽  
Singular Kernel ◽  
Variable Order

Sliding mode control for a class of variable-order fractional chaotic systems

2020 ◽  
Vol 357 (15) ◽  
pp. 10127-10158 ◽  
Author(s):  
Jingfei Jiang ◽  
Dengqing Cao ◽  
Huatao Chen
Keyword(s):  
Sliding Mode Control ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Variable Order ◽  
Mode Control

Synthesis of multifractional Gaussian noises based on variable-order fractional operators

2011 ◽  
Vol 91 (7) ◽  
pp. 1645-1650 ◽  
Author(s):  
Hu Sheng ◽  
Hongguang Sun ◽  
YangQuan Chen ◽  
TianShuang Qiu
Keyword(s):  
Variable Order ◽  
Fractional Operators

scholarly journals Time-Delay Synchronization and Anti-Synchronization of Variable-Order Fractional Discrete-Time Chen–Rossler Chaotic Systems Using Variable-Order Fractional Discrete-Time PID Control

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2149
Author(s):  
Joel Perez Padron ◽  
Jose Paz Perez ◽  
José Javier Pérez Díaz ◽  
Atilano Martinez Huerta
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Discrete Time ◽  
Pid Control ◽  
Chaotic Systems ◽  
Discrete Systems ◽  
Variable Order ◽  
Proportional Integral Derivative ◽  
Discrete Time Delay ◽  
Proportional Integral Derivative Control

In this research paper, we solve the problem of synchronization and anti-synchronization of chaotic systems described by discrete and time-delayed variable fractional-order differential equations. To guarantee the synchronization and anti-synchronization, we use the well-known PID (Proportional-Integral-Derivative) control theory and the Lyapunov–Krasovskii stability theory for discrete systems of a variable fractional order. We illustrate the results obtained through simulation with examples, in which it can be seen that our results are satisfactory, thus achieving synchronization and anti-synchronization of chaotic systems of a variable fractional order with discrete time delay.

Modeling Nonlinear Oscillators via Variable-Order Fractional Operators

2019 ◽  
Author(s):  
Sansit Patnaik ◽  
Fabio Semperlotti
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Models ◽  
Contact Problems ◽  
Variable Order ◽  
Mathematical Tool ◽  
Fractional Operators ◽  
State Variables ◽  
Governing Equations ◽  
Seamless Transition ◽  
Physical Phenomena

Abstract Fractional derivatives and integrals are intrinsically multiscale operators that can act on both space and time dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or variable order (VO) where, in the latter case, the order can also be function of either independent or state variables. When using VO differential governing equations to describe the response of dynamical systems, the order can evolve as a function of the response itself therefore allowing a natural and seamless transition between largely dissimilar dynamics (e.g. linear, nonlinear, and even contact problems). Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. In this study, we present the possible application of VO operators to a class of nonlinear lumped parameter models that has great practical relevance in mechanics and dynamics. Specific examples include hysteresis and contact problems for discrete oscillators. Within this context, we present a methodology to define VO operators capable of capturing such complex physical phenomena. Despite using simplified lumped parameters nonlinear models to present the application of VO operators to mechanics and dynamics, we provide a more qualitative discussion of the possible applications of this mathematical tool in the broader context of continuous multiscale systems.

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