Chaotification of First-Order Partial Difference Equations

2020 ◽  
Vol 30 (15) ◽  
pp. 2050229
Author(s):  
Wei Liang ◽  
Haihong Guo
Keyword(s):  
Dynamical Systems ◽  
Difference Equations ◽  
Discrete Dynamical Systems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
First Order ◽  
Controlled Systems ◽  
The Difference

This paper is concerned with chaotification of first-order partial difference equations. Two criteria of chaos for the difference equations with general controllers are established, and all the controlled systems are proved to be chaotic in the sense of Li–Yorke or of both Li–Yorke and Devaney by applying the coupled-expanding theory of general discrete dynamical systems. The controllers used in this paper can be easily constructed, facilitating the chaotification of first-order partial difference equations. For illustration, two illustrative examples are provided.

CHAOTIFICATION FOR A CLASS OF FIRST-ORDER PARTIAL DIFFERENCE EQUATIONS

2008 ◽  
Vol 18 (03) ◽  
pp. 717-733 ◽  
Author(s):  
WEI LIANG ◽  
YUMING SHI ◽  
CHAO ZHANG
Keyword(s):  
Dynamical Systems ◽  
Computer Simulations ◽  
Difference Equations ◽  
System Size ◽  
Discrete Dynamical Systems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
First Order ◽  
Controlled Systems ◽  
Finite Dimensional

This paper is concerned with chaotification for a class of first-order partial difference equations, in which the system size is finite or infinite. Nine new chaotification schemes for the class of first-order partial difference equations with general controllers, mod-operation, and sawtooth functions are established, respectively. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. In addition, five new chaotification schemes for general discrete dynamical systems in finite-dimensional real spaces and l∞are established. Two illustrative examples are provided with computer simulations.

Anti-Control of Chaos for First-Order Partial Difference Equations via Sine and Cosine Functions

2019 ◽  
Vol 29 (10) ◽  
pp. 1950140 ◽  
Author(s):  
Wei Liang ◽  
Zihan Zhang
Keyword(s):  
Boundary Condition ◽  
Dynamical Systems ◽  
Difference Equations ◽  
Discrete Dynamical Systems ◽  
Control Of Chaos ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
First Order ◽  
Sine And Cosine Functions ◽  
Cosine Functions

In this paper, anti-control of chaos for first-order partial difference equations with nonperiod boundary condition is studied. Three new chaotification schemes for first-order partial difference equations with sine and cosine functions are established, respectively. It is proved that all the systems are chaotic in the sense of both Devaney and Li–Yorke by applying coupled-expanding theory of general discrete dynamical systems. Two illustrative examples are provided with computer simulations

Chaotic Dynamics of Partial Difference Equations with Polynomial Maps

2021 ◽  
Vol 31 (09) ◽  
pp. 2150133
Author(s):  
Haihong Guo ◽  
Wei Liang
Keyword(s):  
Dynamical Systems ◽  
Computer Simulations ◽  
Difference Equations ◽  
Chaotic Dynamics ◽  
Discrete Dynamical Systems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Polynomial Maps ◽  
Expansion Theory ◽  
Theoretical Results

In this paper, chaotic dynamics of a class of partial difference equations are investigated. With the help of the coupled-expansion theory of general discrete dynamical systems, two chaotification schemes for partial difference equations with polynomial maps are established. These controlled equations are proved to be chaotic either in the sense of Li–Yorke or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with computer simulations for demonstration.

scholarly journals Chaotification for Partial Difference Equations via Controllers

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Wei Liang ◽  
Yuming Shi ◽  
Zongcheng Li
Keyword(s):  
Dynamical Systems ◽  
Computer Simulations ◽  
Difference Equations ◽  
Discrete Dynamical Systems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Theoretical Results

Chaotification problems of partial difference equations are studied. Two chaotification schemes are established by utilizing the snap-back repeller theory of general discrete dynamical systems, and all the systems are proved to be chaotic in the sense of both Li-Yorke and Devaney. An example is provided to illustrate the theoretical results with computer simulations.

scholarly journals Generalized quantum exponential function and its applications

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4907-4922
Author(s):  
Burcu Silindir ◽  
Ahmet Yantir
Keyword(s):  
Exponential Function ◽  
Difference Equations ◽  
Taylor Series ◽  
Infinite Product ◽  
Existence And Uniqueness Theorem ◽  
Exponential Functions ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
First Order ◽  
Dynamic Wave

This article aims to present (q; h)-analogue of exponential function which unifies, extends hand q-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q,h)-analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations.

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