Consider the sequence $x_{n+1} = 2x_{n}-\frac{1}{x_n},n\geq0 $.
- For $x_{0} = 0,87$ we have
$$
\begin{aligned}
X(1) &\approx 0,590574712643678 \\
X(2) &\approx -0,512116436915835\\
X(3) &\approx 0,928448055572567\\
X(4) &\approx 0,779829931731029\\
X(5) &\approx 0,277328986805082\\
X(6)&\approx -3,05116776974367\\
X(7)&\approx -5,77459217187698\\
X(8)&\approx -11,37601194123\\
X(9)&\approx-22,6641196146756\\
X(10)&\approx -45,2841166242379\\
X(11)&\approx -90,5461504504763\\
X(12)&\approx -181,081256809127\\
X(13)&\approx -362,156991235554\\
X(14)&\approx -724,311221237654\\
X(15)&\approx -1448,62106185332\\
X(16)&\approx -2897,24143339498\\
X(17)&\approx -5794,48252163406\\
X(18)&\approx -11588,9648706901\\
X(19)&\approx -23177,9296550913\\
X(20)&\approx -46355,8592670381\\
X(21)&\approx -92711,718512504\\\
X(22) &\approx -185423,437014222\\
X(23) &\approx -370846,874023051\\
X(24) &\approx -741693,748043405\\
X(25) &\approx -1483387,49608546
\end{aligned}
\begin{aligned}
X(26) &\approx -2966774,99217025\\
X(27) &\approx -5933549,98434016\\
X(28) &\approx -11867099,9686802\\
X(29) &\approx -23734199,9373602\\
X(30) &\approx -47468399,8747204\\
X(31) &\approx -94936799,7494408\\
X(32) &\approx -189873599,498882\\
X(33) &\approx -379747198,997763\\
X(34) &\approx -759494397,995526\\
X(35) &\approx -1518988795,99105\\
X(36) &\approx -3037977591,98211\\
X(37) &\approx -6075955183,96421\\
X(38) &\approx -12151910367,9284\\
X(39) &\approx -24303820735,8568\\
X(40) &\approx -48607641471,7137\\
X(41) &\approx -97215282943,4274\\
X(42) &\approx -194430565886,855\\
X(43) &\approx -388861131773,709\\
X(44) &\approx -777722263547,419\\
X(45) &\approx -1555444527094,84\\
X(46) &\approx -3110889054189,68\\
X(47) &\approx -6221778108379,35\\
X(48) &\approx -12443556216758,7\\
X(49) &\approx -24887112433517,4\\
X(50) &\approx -49774224867034,8
\end{aligned}
$$
and for $x_{0} = 0,88$ we have
$$
\begin{aligned}
X(1) &\approx 0,623636363636364\\
X(2) &\approx -0,356225815001326\\
X(3) &\approx 2,09475648880333\\
X(4) &\approx 3,71213052034011\\
X(5) &\approx 7,15487396107172\\
X(6) &\approx 14,1699830562016\\
X(7) &\approx 28,2693943978123\\
X(8) &\approx 56,503414850717\\
X(9) &\approx 112,989131656057\\
X(10) &\approx 225,969412903359\\
X(11) &\approx 451,934400429021\\
X(12) &\approx 903,866588147526\\
X(13) &\approx 1807,73206993709\\
X(14) &\approx 3615,46358669485\\
X(15) &\approx 7230,9268968\\
X(16) &\approx 14461,8536553051\\
X(17) &\approx 28923,7072414629\\
X(18) &\approx 57847,414448352\\
X(19) &\approx 115694,828879417\\
X(20) &\approx 231389,657750191\\
X(21) &\approx 462779,31549606\\
X(22) &\approx 925558,630989959\\
X(23) &\approx 1851117,26197884\\
X(24) &\approx 3702234,52395714\\
X(25) &\approx 7404469,047914
\end{aligned}
\begin{aligned}
X(26) &\approx 14808938,0958279\\
X(27) &\approx 29617876,1916557\\
X(28) &\approx 59235752,3833113\\
X(29) &\approx 118471504,766623\\
X(30) &\approx 236943009,533245\\
X(31) &\approx 473886019,06649\\
X(32) &\approx 947772038,13298\\
X(33) &\approx 1895544076,26596\\
X(34) &\approx 3791088152,53192\\
X(35) &\approx 7582176305,06384\\
X(36) &\approx 15164352610,1277\\
X(37) &\approx 30328705220,2554\\
X(38) &\approx 60657410440,5108\\
X(39) &\approx 121314820881,022\\
X(40) &\approx 242629641762,043\\
X(41) &\approx 485259283524,086\\
X(42) &\approx 970518567048,172\\
X(43) &\approx 1941037134096,34\\
X(44) &\approx 3882074268192,69\\
X(45) &\approx 7764148536385,38\\
X(46) &\approx 15528297072770,8\\
X(47) &\approx 31056594145541,5\\
X(48) &\approx 62113188291083\\
X(49) &\approx 124226376582166\\
X(50) &\approx 248452753164332
\end{aligned}
$$
What explains this?
Also, is it possible to determine the number $q$ located between $0.87$ and $0.88$ in the radical change of behavior occurs this sequence?
Best Answer
Extending the function $f : x \mapsto 2x- \frac 1x$ to the circle $\Bbb R \cup \{\infty\}$ by defining $f(0) = f(\infty) = \infty$, $f$ is a $2$-to-$1$ function with $3$ fixpoints, one attractive ($\infty$), and two repulsive ($1,-1$).
So, for most starting reals, the generated sequence will "converge" to $\infty$. Since $f$ is continuous, the set of reals that generate a sequence diverging to $+ \infty$ is an open subset, as well as the set of reals that generate a sequence diverging to $- \infty$.
those two open sets "come in contact" at the real numbers that generate a sequence eventually going to $0$ and then that stays on $\infty$. Here, you can plainly see that there is an $x \in (0.87 ; 0.88)$ such that $f^{6}(x) = 0$.
To find the precise value of that $x$, you need to compute the successive antecedents of $0$ by $f$ that lay in-between the two sequences.
$f^{-1}(y) = \frac{y \pm \sqrt {y^2+8}}4$, so the first one is $\frac {\sqrt 2}2$ (betwwen the two $X(5)$ values), then $\frac {\sqrt 2 + \sqrt {34}}8$ (between the two $X(4)$ values), and you can go on until you reach the incriminating $x \in (0.87;0.88)$.
Also, the set of all the antecedents of $0$ has accumulation points (starting with $1$ and $-1$, but they could be more), so it's very possible to find many incriminating values between the two initial values, by going back long enough in the antecedents tree.