Tangential Chaos

Dan Sitaru has posted the following problem and its solution at the CutTheKnotMath facebook page:

Solve in real numbers:

$\begin{cases} yx^{4}+4x^{3}+y=6x^{2}y+4x\\ zy^{4}+4y^{3}+z=6y^{2}z+4y\\ xz^{4}+4z^{3}+x=6z^{2}x+4z \end{cases}$


Let $x=\tan a,$ $\displaystyle a\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right).$ Then

$\displaystyle y=\frac{4x-4x^{3}}{x^{4}-6x^{2}+1}=\tan 4a.$

Thus, $\displaystyle z=\tan 16a$ and $\displaystyle x=\tan 64a,$ implying $\tan a=\tan 64a$ from which $\displaystyle a=\frac{k\pi}{63},$ with $k$ an integer. But, since $\displaystyle a\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),$ $k=0,\pm 1,\ldots,\pm 31.$

It follows that

$(x,y,z)\in\left\{\displaystyle\left(\tan\frac{k\pi}{63},\tan\frac{4k\pi}{63},\tan\frac{16k\pi}{63}\right):\; k\in\{0,\pm 1,\ldots,\pm31\}\right\}.$


Let $\displaystyle f(x)=\frac{4x-4x^{3}}{x^{4}-6x^{2}+1}.$ Then the solution to the system $y=f(x),$ $z=f(y),$ $x=f(z)$ could be seen as having iterations on $f$ run into $3\mbox{-cycle}$ which, reminds (if only spuriously) of Sharkovsky’s theorem (see also Period Three Implies Chaos) means that the iteration on function $f$ have cycles of any length and are, in principle, chaotic. Dan's solution makes it obvious that the substitution $x=\tan a$ will solve $n\mbox{-cycles}$ for any $n=2,3,4,\ldots$ Moreover, the union of all such solutions is the countable set of numbers in the form $\displaystyle\frac{k\pi}{4^{n}-1},$ where $|k|\lt 4^n/2.$ Iterations that start with any other point will be chaotic.

Quite obviously the same can be said of a simpler function $\displaystyle f(x)=\frac{2x}{1-x^{2}},$ that, for example, could be converted to a system of three much simpler equations:

$\begin{cases} y-2x=x^{2}y\\ z-2y=y^{2}z\\ x-2z=z^{2}x. \end{cases}$

Systems of Iterated Equations

  1. Iterations on Monotone Functions
  2. Graphing Equations Is Useful
  3. Graphing Equations Is Useful, II
  4. Graphing Equations Is Useful, III
  5. Graphing Equations Is Useful, IV
  6. Graphing Equations Is Useful, V
  7. Tangential Chaos
  8. Equation in Radicals as a System of Equations
  9. Two Conditions for a Triangle to Be Equilateral

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