Numerical solution of a fractal-fractional order chaotic circuit system

The dynamical system has an important research area and due to its wide applications many researchers and scientists working to develop new model and techniques for their solution. We present in this work the dynamics of a chaotic model in the presence of newly introduced fractal-fractional operators. The model is formulated initially in ordinary differential equations and then we utilize the fractal-fractional (FF) in power law, exponential and Mittag-Leffler to generalize the model. For each fractal-fractional order model, we briefly study its numerical solution via the numerical algorithm. We present some graphical results with arbitrary order of fractal and fractional orders, and present that these operators provide different chaotic attractors for different fractal and fractional order values. The graphical results demonstrate the effectiveness of the fractal-fractional operators.

Download Full-text

A fractional order epidemic model for the simulation of outbreaks of Ebola

Advances in Difference Equations ◽

10.1186/s13662-021-03272-5 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Weiqiu Pan ◽

Tianzeng Li ◽

Safdar Ali

Keyword(s):

Numerical Solution ◽

Root Mean Square ◽

Fractional Order ◽

Relative Error ◽

Numerical Solutions ◽

Real Data ◽

Mean Square ◽

Order Model ◽

The Real ◽

Fractional Orders

AbstractThe Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number $R_{0}$ R 0 , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error $g(U^{*})=0.4146$ g ( U ∗ ) = 0.4146 are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is $g(U^{*})=0.2744$ g ( U ∗ ) = 0.2744 . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.

Download Full-text

Multistability and Coexisting Attractors in a New Circulant Chaotic System

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501748 ◽

2019 ◽

Vol 29 (13) ◽

pp. 1950174 ◽

Cited By ~ 6

Author(s):

Karthikeyan Rajagopal ◽

Akif Akgul ◽

Viet-Thanh Pham ◽

Fawaz E. Alsaadi ◽

Fahimeh Nazarimehr ◽

...

Keyword(s):

Fractional Order ◽

Order System ◽

Cyclic Symmetry ◽

Order Model ◽

Coexisting Attractors ◽

Circuit Realization ◽

Chaotic Flow ◽

Fractional Order Model ◽

Dynamical Properties ◽

Fractional Orders

In this paper, a new four-dimensional chaotic flow is proposed. The system has a cyclic symmetry in its structure and shows a complicated, chaotic attractor. The dynamical properties of the system are investigated. The system shows multistability in an interval of its parameter. Fractional order model of the proposed system is discussed in various fractional orders. Bifurcation analysis of the fractional order system shows that it has a kind of multistability like the integer order system, which is a very rare phenomenon. Circuit realization of the proposed system is also carried out to show that it is usable for engineering applications.

Download Full-text

Bifurcation and Chaos in Integer and Fractional Order Two-Degree-of-Freedom Shape Memory Alloy Oscillators

Complexity ◽

10.1155/2018/8365845 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

Karthikeyan Rajagopal ◽

Anitha Karthikeyan ◽

Prakash Duraisamy ◽

Riessom Weldegiorgis

Keyword(s):

Shape Memory Alloy ◽

Shape Memory ◽

Fractional Order ◽

Dynamical Behavior ◽

Degree Of Freedom ◽

Order Model ◽

Bifurcation And Chaos ◽

Fractional Order Model ◽

Two Degree Of Freedom ◽

Fractional Orders

A two-degree-of-freedom shape memory oscillator derived using polynomial constitutive model is investigated. Periodic, quasiperiodic, chaotic, and hyperchaotic oscillations are shown by the shape memory alloy based oscillator for selected values of the operating temperatures and excitation parameters. Bifurcation plots are derived to investigate the system behavior with change in parameters. A fractional order model of the shape memory oscillator is presented and dynamical behavior of the system with fractional orders and parameters are investigated.

Download Full-text

Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel

AIMS Mathematics ◽

10.3934/math.2022046 ◽

2022 ◽

Vol 7 (1) ◽

pp. 756-783

Author(s):

Muhammad Farman ◽

◽

Ali Akgül ◽

Kottakkaran Sooppy Nisar ◽

Dilshad Ahmad ◽

...

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Fixed Point Theory ◽

Control Strategies ◽

Point Theory ◽

Fractional Operators ◽

Order Model ◽

Epidemiological Analysis ◽

Fractional Order Model ◽

Sumudu Transform

<abstract> <p>This paper derived fractional derivatives with Atangana-Baleanu, Atangana-Toufik scheme and fractal fractional Atangana-Baleanu sense for the COVID-19 model. These are advanced techniques that provide effective results to analyze the COVID-19 outbreak. Fixed point theory is used to derive the existence and uniqueness of the fractional-order model COVID-19 model. We also proved the property of boundedness and positivity for the fractional-order model. The Atangana-Baleanu technique and Fractal fractional operator are used with the Sumudu transform to find reliable results for fractional order COVID-19 Model. The generalized Mittag-Leffler law is also used to construct the solution with the different fractional operators. Numerical simulations are performed for the developed scheme in the range of fractional order values to explain the effects of COVID-19 at different fractional values and justify the theoretical outcomes, which will be helpful to understand the outbreak of COVID-19 and for control strategies.</p> </abstract>

Download Full-text