scholarly journals Determining the symmetries of difference equations

2018 ◽  
Vol 474 (2219) ◽  
pp. 20180340 ◽  
Author(s):  
Pavlos Xenitidis
Keyword(s):  
Difference Equations ◽  
Functional Equations ◽  
Theoretical Framework ◽  
Systematic Method ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Symbolic Computations ◽  
Generalized Symmetries ◽  
Integrability Conditions ◽  
Algebraic Formulation

We derive the determining equations for the N th-order generalized symmetries of partial difference equations defined on d consecutive quadrilaterals on the lattice using the theory of integrability conditions. We provide their algebraic formulation and develop the necessary theoretical framework for their analysis along with a systematic method for solving functional equations of the form T (   f ) + A f + B = 0 . Our approach is algorithmic and can be easily implemented in symbolic computations. We demonstrate our approach by deriving the symmetries of various equations and discuss certain applications and extensions of the theory.

scholarly journals ON PARTIAL DIFFERENTIAL AND DIFFERENCE EQUATIONS WITH SYMMETRIES DEPENDING ON ARBITRARY FUNCTIONS

2016 ◽  
Vol 56 (3) ◽  
pp. 193 ◽  
Author(s):  
Giorgio Gubbiotti ◽  
Decio Levi ◽  
Christian Scimiterna
Keyword(s):  
Difference Equations ◽  
Lie Symmetries ◽  
Partial Differential ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Infinitesimal Generators ◽  
Discrete Equations ◽  
Generalized Symmetries

In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show a few examples, both in partial differential and partial difference equations where this happens. Moreover we show that the infinitesimal generators of generalized symmetries depending on arbitrary functions, both for continuous and discrete equations, effectively play the role of master symmetries.

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.

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