scholarly journals Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated M-Derivative

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 626 ◽  
Author(s):  
Jesús Emmanuel Solís-Pérez ◽  
José Francisco Gómez-Aguilar
Keyword(s):  
Numerical Analysis ◽  
Exponential Decay ◽  
Power Law ◽  
Chaotic Attractor ◽  
Lagrange Interpolation ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Fractional Operators ◽  
Conformable Derivative ◽  
Sensitive Dependence

In this research, novel M-truncated fractional derivatives with three orders have been proposed. These operators involve truncated Mittag–Leffler function to generalize the Khalil conformable derivative as well as the M-derivative. The new operators proposed are the convolution of truncated M-derivative with a power law, exponential decay and the complete Mittag–Leffler function. Numerical schemes based on Lagrange interpolation to predict chaotic behaviors of Rucklidge, Shimizu–Morioka and a hybrid strange attractors were considered. Additionally, numerical analysis based on 0–1 test and sensitive dependence on initial conditions were carried out to verify and show the existence of chaos in the chaotic attractor. These results showed that these novel operators involving three orders, two for the truncated M-derivative and one for the fractional term, depict complex chaotic behaviors.

On generalized fractional operators and a gronwall type inequality with applications

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5457-5473 ◽  
Author(s):  
Yassine Adjabi ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equations ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Type Inequality ◽  
Weighted Spaces ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Derivatives

In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

scholarly journals ANALYSIS OF FRACTAL–FRACTIONAL MALARIA TRANSMISSION MODEL

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040041 ◽  
Author(s):  
J. F. GÓMEZ-AGUILAR ◽  
T. CÓRDOVA-FRAGA ◽  
THABET ABDELJAWAD ◽  
AZIZ KHAN ◽  
HASIB KHAN
Keyword(s):  
Malaria Transmission ◽  
Exponential Decay ◽  
Power Law ◽  
Fixed Point Theory ◽  
Control Strategies ◽  
Point Theory ◽  
Transmission Model ◽  
Fractional Operators ◽  
Uniqueness And Existence ◽  
Exponential Decay Law

In this paper, the malaria transmission (MT) model under control strategies is considered using the Liouville–Caputo fractional order (FO) derivatives with exponential decay law and power-law. For the solutions we are using an iterative technique involving Laplace transform. We examined the uniqueness and existence (UE) of the solutions by applying the fixed-point theory. Also, fractal–fractional operators that include power-law and exponential decay law are considered. Numerical results of the MT model are obtained for the particular values of the FO derivatives [Formula: see text] and [Formula: see text].

SENSITIVE DEPENDENCE OF LIFETIMES OF CHAOTIC TRANSIENT ON NUMERICAL ACCURACY FOR A MODEL WITH DRY FRICTION AND FREQUENCY DEPENDENT DRIVING AMPLITUDE

1996 ◽  
Vol 10 (03n05) ◽  
pp. 153-159 ◽  
Author(s):  
V. PAAR ◽  
N. PAVIN
Keyword(s):  
Exponential Decay ◽  
Initial Conditions ◽  
Numerical Accuracy ◽  
Frequency Dependent ◽  
Sensitive Dependence ◽  
Exponential Decay Law ◽  
Numerical Precision ◽  
Non Linear System ◽  
Chaotic Transients ◽  
Chaotic Transient

Chaotic transients are investigated for a one-dimensional model, which is a driven non-linear system having frequency dependent driving amplitude, with weak dissipation consisting of Coulomb and quadratic friction. It is shown that lifetimes of chaotic transients are sensitively dependent on numerical accuracy when the initial condition is fixed and approximately follow the exponential decay law, which cannot be distinguished from the well-known exponential decay law for dependence on initial conditions when the numerical precision is fixed.

scholarly journals ATANGANA–SEDA NUMERICAL SCHEME FOR LABYRINTH ATTRACTOR WITH NEW DIFFERENTIAL AND INTEGRAL OPERATORS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040044
Author(s):  
ABDON ATANGANA ◽  
SEDA İĞRET ARAZ
Keyword(s):  
Numerical Simulations ◽  
Exponential Decay ◽  
Power Law ◽  
Real World ◽  
Numerical Scheme ◽  
Integral Operators ◽  
Fractional Operators ◽  
Mathematical Concepts ◽  
Newton Polynomial ◽  
Real World Problems

In this paper, we present a new numerical scheme for a model involving new mathematical concepts that are of great importance for interpreting and examining real world problems. Firstly, we handle a Labyrinth chaotic problem with fractional operators which include exponential decay, power-law and Mittag-Leffler kernel. Moreover, this problem is solved via Atangana-Seda numerical scheme which is based on Newton polynomial. The accuracy and efficiency of the method can be easily seen with numerical simulations.

UNUSUAL CHAOTIC ATTRACTORS IN NONSMOOTH DYNAMIC SYSTEMS

2005 ◽  
Vol 15 (12) ◽  
pp. 4081-4086 ◽  
Author(s):  
U. GALVANETTO
Keyword(s):  
Mechanical System ◽  
Chaotic Attractor ◽  
Initial Conditions ◽  
One Dimensional ◽  
Field Of Definition ◽  
Sensitive Dependence ◽  
Irregular Motion ◽  
Small Set ◽  
Chaotic Nature ◽  
One Dimensional Maps

The present paper describes an unusual example of chaotic motion occurring in a nonsmooth mechanical system affected by dry friction. The mechanical system generates one-dimensional maps the orbits of which seem to exhibit sensitive dependence on initial conditions only in an extremely small set of their field of definition. The chaotic attractor is composed of zones characterized by very different rates of divergence of nearby orbits: in a large portion of the chaotic attractor the system motion follows a regular pattern whereas the more usual irregular motion affects only a small portion of the attractor. The irregular phase reintroduces the orbit in the regular zone and the sequence is repeated. The Lyapunov exponent of the map is computed to characterize the steady state motions and confirm their chaotic nature.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

scholarly journals An investigation on dam settlement during and end of construction using instrumentation data and numerical analysis

2021 ◽  
Vol 3 (3) ◽  
Author(s):  
Behrang Beiranvand ◽  
Mehdi Komasi
Keyword(s):  
Numerical Analysis ◽  
Multivariate Regression ◽  
Initial Conditions ◽  
Horizontal Displacement ◽  
Behavioral Model ◽  
Critical Section ◽  
Earth Dam ◽  
Dam Construction ◽  
Good Match ◽  
Consolidation Settlement

AbstractIn the present study, using instrumentation data regarding vertical and horizontal displacement of the dam have been analyzed. Also, the largest and most critical section of the Marvak earth dam is modeled with the behavioral model of the Mohr–Coulomb by GeoStudio software. Numerical modeling of the dam has been done considering the actual embankment conditions and to analyze the changes of the immediate settlement during construction and the consolidation settlement just after construction and initial impounding. The outcomes of instrumentation and numerical analysis at the end of Marvak dam construction showed a settlement between 20 and 500 mm. The results show that the settlement will occur during the construction at the upper levels and the end of construction at the middle levels of the dam. By comparing observed and predicted data, multivariate regression and the explanation coefficient criterion (R2) was found to be R2 = 0.9579, which shows a very good correlation between observed and predicted data, and represents a good match for the settlement points and their location with the initial conditions of the design, and the behavior of the dam in terms of the settlement is found to be stable.

IS SENSITIVE DEPENDENCE ON INITIAL CONDITIONS NATURE’S SENSORY DEVICE?

1992 ◽  
Vol 02 (01) ◽  
pp. 193-199 ◽  
Author(s):  
RAY BROWN ◽  
LEON CHUA ◽  
BECKY POPP
Keyword(s):  
Initial Conditions ◽  
Sensory Perception ◽  
Sensory Features ◽  
Sensitive Dependence ◽  
Nonlinear Devices

In this letter we illustrate three methods of using nonlinear devices as sensors. We show that the sensory features of these devices is a result of sensitive dependence on parameters which we show is equivalent to sensitive dependence on initial conditions. As a result, we conjecture that sensitive dependence on initial conditions is nature’s sensory device in cases where remarkable feats of sensory perception are seen.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

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