Chaos in a Cancer Model via Fractional Derivatives with Exponential Decay and Mittag-Leffler Law

Gómez-Aguilar, J. F.; López, G.; Alvarado-Martínez, V.M.; Băleanu, Dumitru; Khan, Hasib

doi:10.3390/e19120681

Cited by 76 publications

(32 citation statements)

References 42 publications

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“…This imposes some restriction on system (1). However, due to the important dynamics of the solutions of system (1), it is significant to solve the system (1) analytically or numerically [28]. Because of the novelty and the newness of this topic there are a few articles on this subject.…”

Section: Introductionmentioning

confidence: 99%

A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel

et al. 2018

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In this paper, we solve a system of fractional differential equations within a fractional derivative involving the Mittag-Leffler kernel by using the spectral methods. We apply the Chebyshev polynomials as a base and obtain the necessary operational matrix of fractional integral using the Clenshaw-Curtis formula. By applying the operational matrix, we obtain a system of linear algebraic equations. The approximate solution is computed by solving this system. The regularity of the solution investigated and a convergence analysis is provided. Numerical examples are provided to show the effectiveness and efficiency of the method.

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Section: Introductionmentioning

confidence: 99%

A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel

et al. 2018

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“…Under certain assumptions on the physical problem, they proved existence of solutions, and, by means of an iterative algorithm, numerical evidence of chaos is shown when 0:25 , α , 0:3 and 0:4 , α , 0:5 for the usual set of parameters σ ¼ 10, ρ ¼ 8 3 , and β ¼ 28. In [42] a three-dimensional fractional-order dynamical system for cancer growth is proposed replacing the standard derivatives in the evolution equations:…”

Section: Some Simple Systems Exhibiting Chaosmentioning

confidence: 99%

A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems

Troparevsky¹,

Seminara²,

Fabio³

2020

Nonlinear Systems -Theoretical Aspects and Recent Applications

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Fractional differential equations can describe the dynamics of several complex and nonlocal systems with memory. They arise in many scientific and engineering areas such as physics, chemistry, biology, biophysics, economics, control theory, signal and image processing, etc. Particularly, nonlinear systems describing different phenomena can be modeled with fractional derivatives. Chaotic behavior has also been reported in some fractional models. There exist theoretical results related to existence and uniqueness of solutions to initial and boundary value problems with fractional differential equations; for the nonlinear case, there are still few of them. In this work we will present a summary of the different definitions of fractional derivatives and show models where they appear, including simple nonlinear systems with chaos. Existing results on the solvability of classical fractional differential equations and numerical approaches are summarized. Finally, we propose a numerical scheme to approximate the solution to linear fractional initial value problems and boundary value problems.

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“…It is possible to find at least 35 works [ 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 , 82 , 83 , 84 , 85 , 86 , 87 , 88 , 89 , 90 , 91 ] in the Entropy journal that relate chaos theory in different areas; however, there are only five articles, from all areas, where reproducibility was mentioned [ 92 , 93 , 94 , 95 , 96 ]. From those aforementioned papers, only Funabashi [ 95 ] presented some slight relationship with the reproducibility of nonlinear dynamics fields.…”

Section: Related Workmentioning

confidence: 99%

A Note on the Reproducibility of Chaos Simulation

Nazare

Nepomuceno

Martins

et al. 2020

Entropy

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An evergreen scientific feature is the ability for scientific works to be reproduced. Since chaotic systems are so hard to understand analytically, numerical simulations assume a key role in their investigation. Such simulations have been considered as reproducible in many works. However, few studies have focused on the effects of the finite precision of computers on the simulation reproducibility of chaotic systems; moreover, code sharing and details on how to reproduce simulation results are not present in many investigations. In this work, a case study of reproducibility is presented in the simulation of a chaotic jerk circuit, using the software LTspice. We also employ the OSF platform to share the project associated with this paper. Tests performed with LTspice XVII on four different computers show the difficulties of simulation reproducibility by this software. We compare these results with experimental data using a normalised root mean square error in order to identify the computer with the highest prediction horizon. We also calculate the entropy of the signals to check differences among computer simulations and the practical experiment. The methodology developed is efficient in identifying the computer with better performance, which allows applying it to other cases in the literature. This investigation is fully described and available on the OSF platform.

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Chaos in a Cancer Model via Fractional Derivatives with Exponential Decay and Mittag-Leffler Law

Cited by 76 publications

References 42 publications

A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel

A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel

A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems

A Note on the Reproducibility of Chaos Simulation

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