Periodic sequences given by recurrence relations

Question

Question: Is there any sort of theory on periodic sequences given by recurrence relations? I cannot describe what makes the examples at the bottom interesting, or what I could possibly want to know about a general theory (if one exists). I hope they are more than just curiosities, but I cannot really tell where, in the mathematical world, they fit, or where I could go to learn anything about them.

What I know: (possibly a red herring, or running before crawling) To exclude sequences like $x \mapsto x + k \pmod p$ that are obviously periodic, the interesting examples I've seen so far have terms that are Laurent polynomials in the first two terms $a_1 = x$ and $a_2 = y$. This is even called the Laurent Phenomenon (I personally know very little about Laurent polynomials).

Based on my research (primarily Fomin and Reading's notes Root Systems and Generalized Associahedra and web searches), there are certain structures called cluster algebras (or, evidently, Laurent phenomenon algebras) that seem to have been created with these recurrence relations in mind, or as a motivation, or create them as a natural byproduct (I don't know).

Although I've taken some courses in combinatorics in which recurrence relations were covered, I really don't remember anything periodic happening, just the basic stuff (and I've forgotten most of that!).

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Motivation: In this question, a sequence $a_i$ is given by the recurrence relation $a_i = a_{i - 1}a_{i + 1}$, or equivalently, $a_{i + 1} = \frac{a_i}{a_{i - 1}}$. It is shown in several answers that if $a_1 = x$ and $a_2 = y$, the terms of the sequence are

$$\underbrace{x,\, y,\, \frac{y}{x},\, \frac{1}{x},\, \frac{1}{y},\, \frac{x}{y}}_{\text{period}},\, x,\, y,\, \ldots$$

and so is periodic with period of $6$.

This reminded me of Fomin and Reading's notes Root Systems and Generalized Associahedra. The first topic there is a sequence defined recursively by \begin{align} f_1 &= x,\\ f_2 &= y, \\ f_{i+1} &= \frac{f_i + 1}{f_{i - 1}}, \end{align} whose terms are $$\underbrace{x,\, y,\, \frac{y+1}{x},\, \frac{x+y+1}{xy},\, \frac{x+1}{y}}_{\text{period}},\, x,\, y,\, \ldots$$

that turns out to have period $5$.

Answer 1

Following our conversation in the comments, "periodic sequences given by recurrence relations" is very close to the behavior of a discrete-time dynamical system (which indeed is a recurrence relation) that arrives, starting from a initial condition $x_0$ to a periodic $n$-orbit cycle attractor, in other words, a stable cycle of points, repeating the visit to those points in the same order.

Caveat: please if somebody can enhance my answer, any correction is welcomed.

As in your case you are working with a one-dimensional recurrence relation (aka map, aka discrete-time dynamical system), there is no chaos (it is required at least two dimensions to obtain a chaotic dynamical system), so no chaotic attractors will appear associated to the system, but you can arrive to sequences of points from which the recurrence formula cannot escape (it is the attractor). So the attractor would be your "periodic sequence".

For instance, the most famous case is the Logistic map, which is very useful to understand the basic concepts of the discrete-time maps:$$x_{n+1}=r \cdot x_n(1-x_n)$$

Where you can decide the initial condition $x_0$ of the system and you can decide the value of the control parameter $r$. Depending on the value of $r$ you will arrive to different stable $n$-orbit solutions. So some of them will arrive depending on the value of $r$ to a $2$-orbit cycle, $3$, $4$, many... or you never arrive to one, which is also possible depending on the definition of the dynamical system.

To see the whole picture of what happens when $r$ changes, you can study the bifurcation diagrams. They basically represent a graph in which the $x$-axis is one of the control parameters and in the $y$-axis you put the value of the $n$-orbit points where the specific $r$ case arrive. This leads to a graph where you can study the evolution of the system depending on the value of $r$. E.g. here is the bifurcation diagram of the Logistic map (credits to Wikipedia):

Another example: if we assume that the Collatz conjecture is true, then it behaves like a discrete-time dynamical system (in $\Bbb N$): it does not matter the initial condition $x_0$: you will arrive to the $3$-orbit $\{1,4,2\}$. Since the moment you arrive to $1$ you cannot escape from $\{1,4,2\}$.

Answer 2

The related question is finding functions such that their composition returns the argument: $$f(f(x))=x$$ Simple examples are: $$f(x)=1-x$$ $$f(x)=\frac{1}{x}$$ $$f(x)=\frac{1-x}{1+x}$$

They are called self-inverse functions, because by definition of inverse function:

$$f^{-1}(f(x))=x$$

A more general example:

$$f(x)=(1-x^a)^{1/a}$$

Self-inverse functions always give period $2$, but we can also search for functions such that: $$f(f(f(x)))=x$$ and so on. All of this allows for a 1st order recurrence relation to be periodic, instead of 2nd order which the OP provides.

A simple case of 1st order recurrence with period $N$ will be

$$z_{n+1}=z_n^{\omega_N}$$

where

$$\omega_N=\sqrt[N]{1} \neq 1$$

For example $\omega_3=e^{ \pm 2 \pi i/3}$ will give a recurrence with period $3$.

Answer 3

Here is something interesting. Define $\;a_n := f(n\; r)\;$ where $\;r\;$ is a constant, $\;f(x)=f(x+1)\;$ for all $x$,$\;f$ is a period $1$ function. If $\;r\;$ is rational then the sequence $\{a_n\}$ is purely periodic. If not, then the sequence is not periodic unless $\;f(x)\;$ is constant, but the function $\;f\;$ can be uniquely recovered from the sequence if $\;f\;$ is continuous, and even though $\{a_n\}$ is not periodic, still it is uniquely associated with the function $\;f\;$ which is periodic.

Generalized Somos sequences lead to such sequences. For example, let Somos-4 be defined by $$\;s_0=s_1=s_2=s_3=1\; \textrm{and} \;s_n = (s_{n-1}s_{n-3} + s_{n-2}s_{n-2})/s_{n-4}.\;$$ Now define the 2nd quotient sequence $a_n := (s_{n-1}s_{n+1})/(s_ns_n).\;$ Associated is the function $$ f(x) := 1 - \wp(\omega_2(x-1/4)+\omega_1 + u)$$ where $\;u=.543684160\dots,\;r=.3789172825\dots,\;g_2=4,\; g_3=-1\;$ and the Weierstrass periods are $\;\omega_1=-2.451389\dots,\; \omega_2=2.993458\dots.$

The sequence satisfies $\;a_1\!=\!a_2\!=\!1,\; a_{n+1}\!=\!(a_n + 1)/(a_na_na_{n-1}).\;$ Compare to the Lyness 5-cycle.

The above example can be greatly generalized to produce interesting sequence defined by rational recurrence relations and which are associated with periodic functions. For example, Somos-5, Somos-6, Somos-7 sequences and their generalization also work when we use the 2nd quotient sequences of them. Note that it is not immediately obvious that the associated functions $f$ exist. Note also that the sequences all satisfy the Laurent phenomenon -- an unexpected property.

Unlike the special cases $\;a_n=a_{n-1}/a_{n-2}\;$ and $\;a_n=(a_{n-1}+1)/a_{n-2}\;$ which are purely periodic, these generalized sequences are associated with functions $f$ where $r$ depends on the initial values of the sequence and only periodic if $r$ is rational. For a very good example of this please read MSE question 1584296 about generalizing these two special cases, and which I also answered.

Finally, if you have time, you may be interested in the Ph.D. Thesis of Jonny Griffiths, Lyness Cycles, Elliptic Curves, and Hikorski Triples which goes into a lot of details, has proofs, references, a wide range of topics, and gives elementary examples such as a 10-cycle and 12-cycle.

Answer 4

This is mainly a consideration more then an answer, but could be useful in discussing this interesting subject.

It is known that there are "similarities" in the solutions to Ordinary Differential Equations (ODE) and to Finite Difference Equations (FDE).
Actually, FDE can be used, under proper conditions, to compute approximated solutions to the ODE.

Since a recurrence is essentially a FDE, than a FDE that mimicks a ODE that admits periodic solutions might also give a periodic solution, with appropriate initial conditions.

Therefore, as an example of linear equations, to $$y''+y=0\quad \to \quad y(x)=A \sin{x+\phi}$$ we can associate a slight different FDE $$ \Delta ^{\,2} y(n) + \Delta y(n) + y(n) = y(n + 2) - y(n + 1) + y(n) = 0\quad \to \quad y(n) = A\cos \left( {n{\pi \over 6} + \alpha } \right) $$

We have in fact $$ \eqalign{ & y(n) = A\cos \left( {n{\pi \over 6} + \alpha } \right) = A\left( {\cos \alpha \cos \left( {n{\pi \over 6}} \right) - \sin \alpha \sin \left( {n{\pi \over 6}} \right)} \right) \cr & \Delta y(n) = A\left( { - \left( {{{\cos \alpha + \sqrt 3 \sin \alpha } \over 2}} \right)\cos \left( {n{\pi \over 6}} \right) + \left( {{{\sin \alpha - \sqrt 3 \cos \alpha } \over 2}} \right)\sin \left( {n{\pi \over 6}} \right)} \right) \cr & \Delta ^{\,2} y(n) = A\left( {\left( {{{ - \cos \alpha + \sqrt 3 \sin \alpha } \over 2}} \right)\cos \left( {n{\pi \over 6}} \right) + \left( {{{\sin \alpha + \sqrt 3 \cos \alpha } \over 2}} \right)\sin \left( {n{\pi \over 6}} \right)} \right) \cr & \Delta ^{\,3} y(n) = y(n) \cr} $$ so that we could also use $$ \Delta ^{\,3} y(n) = y(n) $$

For non-linear equations "similarities" are quite less straight but ODEs can provide an indication.

The disciplines of Digital Signal Processing and of Dynamical Systems provide various tools to analize the response of circuits in the dicrete time domain, which are the practical realization of recurrent relations. Included are the mathematical tools to parallel the discrete time and continuous time behaviour, Laplace and z-Transforms for instance (refer to this Wikipedia article for starting and look for references).

In summary, all the linear and non-linear physical models that provides an oscillating or resonating behaviour will translate into homogeneous or non-homogeneous ODEs and FDEs whose solutions include periodic continuous or discrete functions: a simple or double pendulum, a ball in a bowl of any convex shape, a particle in a gravitational field, an acoustic or EMW resonator, etc.
And finally, to mention an intrinsically discrete time oscillator, consider any system governed by a periodic Markov chain.

Answer 5

Suppose you have a sequence of distinct elements $b_0,\ldots,b_{n-1}$ and let

$$a_{k+1} = \sum_{i = 0}^{n-1} b_{i+1} \prod_{j\neq i}\frac{a_k - b_j}{b_i - b_j}.$$

Note that if we have $a_k = b_i$, all terms in the sum vanish except the one for $b_{i+1}$, where the product is just 1, so $a_{k+1} = b_{i+1}$. This shows that if we set $a_1 = b_1$, the sequence will be periodic with terms $b_0,\ldots,b_{n-1}$.

The RHS of the recurrence relation is a degree $n-1$ polynomial in $a_k$. Perhaps this characterizes these sequences?

The idea comes from Lagrange interpolation.

Answer 6

The classic example of that periodic sequence is the periodic part of the quotents sequence in the Euclidean algorithm for a square irrationals in the form of $$x_{n+1} = \frac 1{x_n - [x_n]},$$ where $$x_n = \frac{a_n\sqrt M + b_n}{d_n},\tag1$$ because every square irrational can be presented as periodic continued fraction.

At the same time, this recurrent relation generates periodic natural sequences $a_n, b_n, d_n$ and $c_n= [x_n],$ because $$x_{n+1} = \dfrac1{\dfrac{a_n\sqrt M + b_n}{d_{n}} - c_n} = \frac {d_n}{a_n\sqrt M + b_n - c_nd_n} = \dfrac{a_n\sqrt M + c_nd_n - b_n}{a_n^2M -(b_n - c_nd_n)^2}$$ also can be presented in the form (1).