Form Symmetries and Reduction of Order in Difference Equations
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Features
- Uses form symmetries to define semiconjugate factorizations of recursive higher order difference equations and general nonrecursive higher order difference equations
- Presents many of the main concepts and results using simple algebraic structures, such as groups and fields
- Applies the methods to large classes of difference equations, including nonautonomous and nonhomogenous linear equations, quadratic difference equations, and separable difference equations
- Includes numerous new results for the first time in print
- Contains a large number of examples and exercises that illustrate key ideas and enhance understanding
Summary
Form Symmetries and Reduction of Order in Difference Equations presents a new approach to the formulation and analysis of difference equations in which the underlying space is typically an algebraic group. In some problems and applications, an additional algebraic or topological structure is assumed in order to define equations and obtain significant results about them. Reflecting the author’s past research experience, the majority of examples involve equations in finite dimensional Euclidean spaces.
The book first introduces difference equations on groups, building a foundation for later chapters and illustrating the wide variety of possible formulations and interpretations of difference equations that occur in concrete contexts. The author then proposes a systematic method of decomposition for recursive difference equations that uses a semiconjugate relation between maps. Focusing on large classes of difference equations, he shows how to find the semiconjugate relations and accompanying factorizations of two difference equations with strictly lower orders. The final chapter goes beyond semiconjugacy by extending the fundamental ideas based on form symmetries to nonrecursive difference equations.
With numerous examples and exercises, this book is an ideal introduction to an exciting new domain in the area of difference equations. It takes a fresh and all-inclusive look at difference equations and develops a systematic procedure for examining how these equations are constructed and solved.
Table of Contents
Introduction
Difference Equations on Groups
Basic definitions
One equation, many interpretations
Examples of difference equations on groups
Semiconjugate Factorization and Reduction of Order
Semiconjugacy and ordering of maps
Form symmetries and SC factorizations
Order-reduction types
SC factorizations as triangular systems
Order-preserving form symmetries
Homogeneous Equations of Degree One
Homogeneous equations on groups
Characteristic form symmetry of HD1 equations
Reductions of order in HD1 equations
Absolute value equation
Type-(k,1) Reductions
Invertible-map criterion
Identity form symmetry
Inversion form symmetry
Discrete Riccati equation of order two
Linear form symmetry
Difference equations with linear arguments
Field-inverse form symmetry
Type-(1,k) Reductions
Linear form symmetry revisited
Separable difference equations
Equations with exponential and power functions
Time-Dependent Form Symmetries
The semiconjugate relation and factorization
Invertible-map criterion revisited
Time-dependent linear form symmetry
SC factorization of linear equations
Nonrecursive Difference Equations
Examples and discussion
Form symmetries, factors, and cofactors
Semi-invertible map criterion
Quadratic difference equations
An order-preserving form symmetry
Appendix: Asymptotic Stability on the Real Line
References
Index
Notes and Problems appear at the end of each chapter.
