Is the sequence $a_{n+1}=a_n-\frac{1}{a_n}$, $a_0=2$ bounded?

The sequence again for convenience $$ a_{n+1}=a_n-\frac{1}{a_n},\;a_0=2 $$ My friend asked me this question and I do not know how to tackle it. It's clear it does not have a limit, but I am not sure whether it is unbounded; it seems to oscillate with a large amplitude when you simulate it numerically.

I also can't seem to get anywhere with generating functions, but I also don't know how to use them for nonlinear recurrence relations.

Solution 1:

Note that there is a 2-cycle if we begin with $$ x = \pm \frac{1}{\sqrt 2} \approx \pm 0.7071, $$ as we then get $$ x - \frac{1}{x} = -x. $$ $$ f(x) = x - \frac{1}{x}. $$ Important: $f$ is odd, $$ f(-x) = -f(x). $$ $$ \mbox{If} \; \; \; f(x) = -x, \; \; \; \mbox{then} \; \; \; f(f(x)) = x.$$

We also get 4-cycles at the roots of $$ 2 x^4 - 4 x^2 + 1, $$ as then $$ f(f(x)) = -x, $$ $$ \mbox{If} \; \; \; f(f(x)) = -x, \; \; \; \mbox{then} \; \; \; f(f(f(f(x)))) = x.$$

The presence of cycles of larger and larger degree tends to support the hypothesis of chaotic behavior elsewhere.

6-cycles at the roots of $$ 2x^8 - 11x^6 + 17x^4 - 8x^2 + 1 $$ $$ \mbox{If} \; \; \; f(f(f(x))) = -x, \; \; \; \mbox{then} \; \; \; f(f(f(f(f(f(x)))))) = x.$$ Now that I think of it, this diagram also shows some 3-cycles.

Seems to me that these points involved in $2k$-cycles might be dense in the real line. If so, that would be strong evidence.

Wildly unstable. I did the first 25 steps with the given $x$ seed value in rational number arithmetic,

    1: 1.5 =                       3/2

    2: 0.8333333333333333 =                       5/6

    3: -0.3666666666666666 =                    -11/30

    4: 2.36060606060606 =                   779/330

    5: 1.93698603493212 =             497941/257070

    6: 1.420720051612811 = 181860254581/128005692870

    7: 0.7168516161213897 = 16687694789137362648661/23279147893155496537470

    8: -0.6781372177053623 = -263439569256003706800705587722279993788907979/388475314992168993748220639081347493631827670

    9: 0.7964905919638025 = 81512663708476146329709015825571064954724426915346799560162522434680208602364731247764459/102339769648127358726761918460732576814168548432921287355299929744910591862606847215978930

   10: -0.4590170186589809 = -3829114106780645548860005128366999929762127231121785938212801344907004576697195186639451276755270737038953266911319233153169994507036998538685318746477046140028242936033060382219/8341987227330719589550045299118644973579016226280953918450224539222226868008655704094557970730123521406125932252315167316321272393191426438953060083145494237274901280505746848870

   11: 1.719551442531198 = 

   12: 1.138004432499333 = 

   13: 0.2592732330050055 = 

   14: -3.597661740227243 = 

   15: -3.319703423907924 = 

   16: -3.018471695555874 = 

   17: -2.687178213005645 = 

   18: -2.315040631969854 = 

   19: -1.883082770759831 = 

   20: -1.352038668223384 = 

   21: -0.6124148516101889 = 

   22: 1.020465208974159 = 

   23: 0.0405199926102738 = 

   24: -24.63865528804984 = 

   25: -24.5980686574765 =