Abstract
In the upcoming chapter we introduce recurrence relations. These are equations that define in recursive fashion, via suitable functions, the terms appearing in a real or complex sequence. The first section deals with some well-known examples that show how these relations may arise in real life, e.g., the Lucas Tower game problem or the death or life Titus Flavius Josephus problem. We then devote a large part of the chapter to discrete dynamical systems, namely recurrences of the form \(x_{n+1}=f(x_n)\) where f is a real valued function: in this context the sequence that solves the recurrence, starting from an initial datum, is called the orbit of the initial point. We thoroughly study the case where f is monotonic, and the periodic orbits. The last part of the chapter is devoted to the celebrated Sarkovskii theorem, stating that the existence of a periodic orbit of minimum period 3 implies the existence of a periodic orbit of arbitrary minimum period: we thus give to the reader the taste of a chaotic dynamical system, although that notion is not explicitly developed in this book.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Leonardo Pisano called Leonardo Fibonacci (1170–1240) .
- 2.
François Édouard Anatole Lucas (1842–1891).
- 3.
Titus Flavius Iosephus (37–100) .
- 4.
Sir Isaac Newton (1642–1727) .
- 5.
Oleksandr Mykolaiovych Sarkovskii (1936-).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Mariconda, C., Tonolo, A. (2016). Recurrence Relations. In: Discrete Calculus. UNITEXT(), vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-03038-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-03038-8_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03037-1
Online ISBN: 978-3-319-03038-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)