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37Bxx Dynamical systems and ergodic theory -- Topological dynamics

68R01 Computer science -- Discrete mathematics in relation to computer science -- General

65T50 Numerical analysis -- Numerical methods in Fourier analysis -- Discrete and fast Fourier transforms

37C05 Dynamical systems and ergodic theory -- Smooth dynamical systems: general theory -- Smooth mappings and diffeomorphisms En-ligne: Springerlink | Zentralblatt | MathSciNet

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The book is written for advanced undergraduate majors.

Chapter 1 introduces the subject of difference equations with Fibonacci numbers. Chapter 2 moves on to homogeneous linear equations. Chapters 3 and 4 introduce general finite difference equations and generating functions. Chapter 5 focuses on nonnegative difference equations and the roots of their characteristic polynomials. Chapter 6 is devoted to Leslie matrices and Chapter 7 to matrix difference equations. Chapter 8 explores recurrences modm, and Chapter 9 tours the subject of computational complexity, including fast Fourier transforms. Chapter 10 introduces nonlinear difference equations from a dynamical systems perspective, including Sarkovskiĭ's theorem, local and global stability, and chaos. Appendices include worked examples, complex numbers, a review of linear algebra, and Marden's method for determining whether roots are in the unit circle.

This book is well written and easy to read, with a large number of exercises (but I could not find any answers to selected problems). It is not a comprehensive or up-to-date treatment of difference equations, but rather gives a broad introductory tour of recurrences from the perspectives of many disciplines, including computational points of view. Although most of the problems and examples do not require computer simulations, the text gives samples of pseudocode so that students can implement algorithms for more involved problems.

This book is not written for those primarily interested in the dynamical systems and modeling points of view, nor does it contain or point to the wealth of recent advances in difference equations. For example, I found no references to the Journal of Difference Equations and Applications, nor did I see acknowledgments or citations of the leaders (such as Elaydi and Ladas) in that field. The bibliography is quite old, in fact, making the book seem like an excellent collection of class notes that has been published after years of classroom refining. ... (MathSciNet)

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