A Technique for Studying a Class of Fractional-Order Nonlinear Dynamical Systems

2017 ◽  
Vol 27 (09) ◽  
pp. 1750144 ◽  
Author(s):  
Gamal M. Mahmoud ◽  
Ahmed A. M. Farghaly ◽  
A. A.-H. Shoreh
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Fractional Order ◽  
Electromagnetic Waves ◽  
Nonlinear Dynamical Systems ◽  
Main Idea ◽  
Order System ◽  
Dielectric Polarization ◽  
Nonlinear Dynamical ◽  
Before And After

In this work, we propose a technique to study nonlinear dynamical systems with fractional-order. The main idea of this technique is to transform the fractional-order dynamical system to the integer one based on Jumarie’s modified Riemann–Liouville sense. Many systems in the interdisciplinary fields could be described by fractional-order nonlinear dynamical systems, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, heat conduction, resistance-capacitance-inductance (RLC) interconnect and electromagnetic waves. To deal with integer order dynamical system it would be much easier in contrast with fractional-order system. Two systems are considered as examples to illustrate the validity and advantages of this technique. We have calculated the Lyapunov exponents of these examples before and after the transformation and obtained the same conclusions. We used the integer version of our example to compute numerically the values of the fractional-order and the system parameters at which chaotic and hyperchaotic behaviors exist.

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

Generalized Wright stability for distributed fractional-order nonlinear dynamical systems and their synchronization

2019 ◽  
Vol 97 (1) ◽  
pp. 413-429 ◽  
Author(s):  
Gamal M. Mahmoud ◽  
Tarek Aboelenen ◽  
Tarek M. Abed-Elhameed ◽  
Ahmed A. Farghaly
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical

NONLINEAR DYNAMICAL SYSTEM AND CHAOS SYNCHRONIZATION

2008 ◽  
Vol 18 (05) ◽  
pp. 1531-1537 ◽  
Author(s):  
AYUB KHAN ◽  
PREMPAL SINGH
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Numerical Techniques ◽  
Nonlinear Dynamical ◽  
Chaotic Dynamical Systems ◽  
Response System ◽  
Identical Response

Chaos synchronization of nonlinear dynamical systems has been studied through theoretical and numerical techniques. For the synchronization of two identical nonlinear chaotic dynamical systems a theorem has been constructed based on the Lyapunov function, which requires a minimal knowledge of system's structure to synchronize with an identical response system. Numerical illustrations have been provided to verify the theorem.

scholarly journals Aggregation of a Class of Large-Scale, Interconnected, Nonlinear Dynamical Systems

2000 ◽  
Author(s):  
Swaroop Darbha ◽  
K. R. Rajagopal
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Large Scale ◽  
Nonlinear Dynamical Systems ◽  
Linear Time ◽  
Hierarchical Systems ◽  
Nonlinear Dynamical ◽  
Analysis And Design ◽  
Linear Time Invariant ◽  
Time Invariant

Abstract In a previous paper, we discussed the characteristics of a “meaningful” average of a collection of dynamical systems, and introduced as well as contructed a “meaningful” average that is not usually what is meant by an “ensemble” average. We also addressed the associated issue of the existence and construction of such an average for a class of interconnected, linear, time invariant dynamical systems. In this paper, we consider the issue of the construction of a meaningful average for a collection of a class of nonlinear dynamical systems. The construction of the meaningful average will involve integrating a nonlinear differential equation, of the same order as that of any member of the systems in the collection. Such an “average” dynamical system is not only attractive from a computational perspective, but also represents the macroscopic behavior of the interconnected dynamical systems. An average dynamical system can be used in the analysis and design of hierarchical systems.

On Periodic Motions in the First-Order Nonlinear Systems

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yeyin Xu ◽  
Zhaobo Chen
Keyword(s):  
Dynamical System ◽  
Fourier Series ◽  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Nonlinear Dynamical System ◽  
Periodic Motions ◽  
Nonlinear Dynamical ◽  
First Order

In this paper, analytical solutions of periodic motions in the first-order nonlinear dynamical system are discussed from the finite Fourier series expression. The first-order nonlinear dynamical system is transformed to the dynamical system of coefficients in the Fourier series. From investigation of such dynamical system of coefficients, the analytical solutions of periodic motions are obtained, and the corresponding stability and bifurcation of periodic motions will be determined. In fact, this method provides a frequency-response analysis of periodic motions in nonlinear dynamical systems, which is alike the Laplace transformation of periodic motions for nonlinear dynamical systems. The harmonic frequency-amplitude curves are obtained for different-order harmonic terms in the Fourier series. Through such frequency-amplitude curves, the nonlinear characteristics of periodic motions in the first-order nonlinear system can be determined. From analytical solutions, the initial conditions are obtained for numerical simulations. From such initial conditions, numerical simulations are completed in comparison of the analytical solutions of periodic motions.

On the Differential Geometry of Flows in Nonlinear Dynamical Systems

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Reference Surface ◽  
Change Rate ◽  
Nonlinear Dynamical ◽  
Geometrical Relation ◽  
Reference Flow ◽  
Normal Distance ◽  
Contact Flows

In order to investigate the geometrical relation between two flows in two dynamical systems, a flow for an investigated dynamical system is called the compared flow and a flow for a given dynamical system is called the reference flow. A surface on which the reference flow lies is termed the reference surface. The time-change rate of the normal distance between the reference and compared flows in the normal direction of the reference surface is measured by a new function (i.e., G function). Based on the surface of the reference flow, the kth-order G functions are introduced for the noncontact and lth-order contact flows in two different dynamical systems. Through the new functions, the geometric relations between two flows in two dynamical systems are investigated without contact between the reference and compared flows. The dynamics for the compared flow with a contact to the reference surface is briefly addressed. Finally, the brief discussion of applications is given.

CLOSURES OF FRACTAL SETS IN NONLINEAR DYNAMICAL SYSTEMS WITH SWITCHED INPUTS

2001 ◽  
Vol 11 (08) ◽  
pp. 2205-2215 ◽  
Author(s):  
RYOICHI WADA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical ◽  
Fractal Sets ◽  
Fractal Set

This paper studies closures of fractal sets observed in nonlinear dynamical systems excited stochastically by switched inputs. The Duffing oscillator and the forced dumped pendulum are analyzed as examples. The dynamics of the system is characterized by a fractal set in the phase space. We can numerically construct a closure that encloses the fractal set. Furthermore, it is shown that the closure is a limit cycle attractor of a dynamical system defined by the switching manifold.

scholarly journals On Algorithm of Integrability Classification of the Nonlinear Dynamical Systems via Computer Algebra Methods

2021 ◽  
pp. 7-12
Author(s):  
Bohdan Fil ◽  
Yaroslav Pelekh ◽  
Myroslava Vovk ◽  
Halyna Beregova ◽  
Tatiana Magerovska ◽  
...  
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conservation Laws ◽  
Computer Algebra ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Nonlinear Dynamical ◽  
Wolfram Mathematica ◽  
De Vries

There is developed an algorithm to classify integrable nonlinear dynamical systems via Wolfram Mathematica. The hierarchy of conservation laws for the nonlinear dynamical system can be cal-culated by this algorithm. There are demonstrated some modifications of nonlinear Korteweg-de Vries equations integrated by inverse scatering method.

On the Differential Geometry of Flows in Nonlinear Dynamical Systems

2007 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Small Time ◽  
Time Interval ◽  
Reference Surface ◽  
Nonlinear Dynamical ◽  
Reference Flow ◽  
Contact Flows ◽  
Order Contact

In this paper, in order to investigate the relation between two flows given in two dynamical systems, a flow for an investigated dynamical system is called the compared flow and a flow for a given dynamical system is called the reference flow. A surface on which the reference flow lies is termed the reference surface. For a small time interval, the change rate of the normal distance between the reference and compared flows on the normal direction of the reference surface is measured by a new function (i.e., G-function). Based on the surface of the reference flow, the kth -order G-functions for the non-contact and lth-order contact flows in two different dynamical systems are introduced. Through the new functions, the geometric relations between two flows in two dynamical systems imposed in the same phase space are investigated without contact between the reference and compared flows. The compared flow passing through, returning back from and paralleling to the surface of the reference flow is discussed first. The tangency and passability for a compared flow to the surface of a reference flow with the lth-order contact are presented. The dynamics for the compared flow with such a contact to the reference surface is briefly addressed. Finally, the brief discussion of applications is given. The contact and tangential singularity between two flows are different. The kth-order contact of the compared flow to the reference surface indicates that the compared flow to the reference surface is of the kth-order singularity. However, the compared flow with the kth-order singularity to the reference surface does not mean that the compared flow to the reference surface is of the kth-order contact.

A YING–YANG THEORY IN NONLINEAR DISCRETE DYNAMICAL SYSTEMS

2010 ◽  
Vol 20 (04) ◽  
pp. 1085-1098 ◽  
Author(s):  
ALBERT C. J. LUO
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Hénon Map ◽  
Henon Map ◽  
Nonlinear Dynamical ◽  
Iterative Maps

This paper presents a Ying–Yang theory for nonlinear discrete dynamical systems considering both positive and negative iterations of discrete iterative maps. In the existing analysis, the solutions relative to "Yang" in nonlinear dynamical systems are extensively investigated. However, the solutions pertaining to "Ying" in nonlinear dynamical systems are investigated. A set of concepts on "Ying" and "Yang" in discrete dynamical systems are introduced to help one understand the hidden dynamics in nonlinear discrete dynamical systems. Based on the Ying–Yang theory, the periodic and chaotic solutions in nonlinear discrete dynamical system are discussed, and all possible, stable and unstable periodic solutions can be analytically predicted. A discrete dynamical system with the Henon map is investigated, as an example.

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