A Modified Second-Order Collatz Equation as a Mathematical Model of Bipolar Disorder

Abstract

We propose, for the sake of dialogue, that the following system of difference equations serve as a phenomenological model of bipolar disorder, a psychiatric illness characterized by cycles or recurrent episodes of severe disturbances in mood (i.e., in being happy or sad, emotions at opposite poles of the spectrum):

$$ \left\{ \begin{array}{l} x_{n+1}=(ax_{n}+b) \bmod m, \\ \\ z_{n+1}=\left\{ \begin{array}{ll} \displaystyle {\frac{-z_{n}-z_{n-1}}{2}}, &{} \text {if } z_{n}+z_{n-1} \text { is even}, \\ \displaystyle {-z_{n}-z_{n-1}}, &{} \text {if } z_{n}+z_{n-1} \text { is odd} \end{array}\right. +s\updelta (x_{n}), \end{array}\right. $$

where $a, b, m\in \mathbf {N}$, $x_{0}\in \mathbf {N}\cup \{0\}$, $s\in \mathbf {Z}-\{0\}$, $z_{-1}, z_{0}\in \mathbf {Z}$, and

$$ \updelta (x)=\left\{ \begin{array}{ll} 0, &{} \text {if } x\ne d\in \{0, 1, \ldots , m-1\}, \\ &{} \\ 1, &{} \text{ if } x=d. \end{array}\right. $$

The first equation in the system is a linear congruential sequence, used to generate a pseudo-random sequence of numbers; and the second equation is a modified (with the addition of $s\updelta (x_{n})$) version of one of the sixteen Collatz difference equations investigated by Amleh and colleagues in 1998. Let c be an odd scalar. While every solution of the (unmodified) Collatz equation is eventually periodic in the three-cycle $(c, 0, -c)$ or $(-c, 0, c)$, we observe (and conjecture) that every solution $\{z_{n}\}_{n=0}^{\infty }$ of the system above is also eventually periodic, but not in the three-cycle $(c, 0, -c)$ or $(-c, 0, c)$ and instead contains infinitely many recurrences of the interrupted three-cycle $(c, 0, -c)$ or $(-c, 0, c)$. Thus, a solution $\{z_{n}\}_{n=0}^{\infty }$ of the system is intended to represent the recurrent episodes of mood disturbance seen in an individual with bipolar disorder.