Difference Equations: From Rabbits to Chaos
Authors: Paul Cull, Mary Flahive, Robby Robson
Publication date: 2005
Publisher: Springer New York
Summary
Difference Equations: From Rabbits to Chaos is an undergraduate introduction to difference equations and recurrence relations, written particularly for students of mathematics and computer science. Beginning with familiar examples such as the Fibonacci sequence, the book develops the theory of homogeneous linear recurrences, finite-difference equations, generating functions, nonnegative systems, and matrix difference equations. The approach is strongly computational: techniques are presented so that small examples can be solved by hand while larger problems can be implemented algorithmically.
The book gradually connects recurrence relations with several major areas of mathematics. It uses linear algebra, graph theory, formal power series, combinatorics, Perron–Frobenius theory, and matrix methods, while applications include Leslie population models, pseudorandom-number generation, modular recurrences, integer factorization, and computational complexity. One particularly attractive feature is the way discrete mathematics, algorithms, and modelling are brought together rather than treating difference equations purely as an abstract subject.
The later chapters move toward more sophisticated computational and nonlinear phenomena, including modular recurrences, complexity questions and nonlinear recurrence relations—the source of the “chaos” in the subtitle. The text contains numerous worked examples and more than 250 exercises, some of which lead naturally toward further investigation or research problems. Reviewers have particularly noted its accessible style, computational emphasis, and usefulness as either an upper-level undergraduate textbook or a reference on recurrence relations.
Key topics
Key takeaways
1. Difference equations are the discrete analogue of differential equations. Instead of describing continuous change, they describe how quantities evolve step by step, for example through a recurrence such as
xn+1=f(xn).x_{n+1}=f(x_n).
2. The book is especially valuable at the intersection of mathematics and computer science. Recurrences are connected to algorithms, complexity, cryptography, population modelling and numerical computation.
3. It is more computational than purely theoretical. The emphasis is on actually solving recurrences, constructing algorithms and understanding concrete examples rather than developing only an abstract general theory.
4. Difficulty level: roughly advanced undergraduate / beginning graduate, though much of the early material is accessible after courses in calculus, discrete mathematics and elementary linear algebra.
BOOK
Authors: Paul Cull, Mary Flahive, Robby Robson
Publication date: 2005
Publisher: Springer New York
Summary
Difference Equations: From Rabbits to Chaos is an undergraduate introduction to difference equations and recurrence relations, written particularly for students of mathematics and computer science. Beginning with familiar examples such as the Fibonacci sequence, the book develops the theory of homogeneous linear recurrences, finite-difference equations, generating functions, nonnegative systems, and matrix difference equations. The approach is strongly computational: techniques are presented so that small examples can be solved by hand while larger problems can be implemented algorithmically.
The book gradually connects recurrence relations with several major areas of mathematics. It uses linear algebra, graph theory, formal power series, combinatorics, Perron–Frobenius theory, and matrix methods, while applications include Leslie population models, pseudorandom-number generation, modular recurrences, integer factorization, and computational complexity. One particularly attractive feature is the way discrete mathematics, algorithms, and modelling are brought together rather than treating difference equations purely as an abstract subject.
The later chapters move toward more sophisticated computational and nonlinear phenomena, including modular recurrences, complexity questions and nonlinear recurrence relations—the source of the “chaos” in the subtitle. The text contains numerous worked examples and more than 250 exercises, some of which lead naturally toward further investigation or research problems. Reviewers have particularly noted its accessible style, computational emphasis, and usefulness as either an upper-level undergraduate textbook or a reference on recurrence relations.
Key topics
- Fibonacci numbers and recurrence sequences
- Homogeneous linear recurrence relations
- Finite-difference equations
- Generating functions
- Nonnegative difference equations
- Leslie population matrix models
- Matrix recurrence equations
- Modular recurrences and pseudorandom sequences
- Computational complexity
- Nonlinear recurrences and chaotic behaviour
Key takeaways
1. Difference equations are the discrete analogue of differential equations. Instead of describing continuous change, they describe how quantities evolve step by step, for example through a recurrence such as
xn+1=f(xn).x_{n+1}=f(x_n).
2. The book is especially valuable at the intersection of mathematics and computer science. Recurrences are connected to algorithms, complexity, cryptography, population modelling and numerical computation.
3. It is more computational than purely theoretical. The emphasis is on actually solving recurrences, constructing algorithms and understanding concrete examples rather than developing only an abstract general theory.
4. Difficulty level: roughly advanced undergraduate / beginning graduate, though much of the early material is accessible after courses in calculus, discrete mathematics and elementary linear algebra.
BOOK
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