Timeline for Convergence of the sequence $s_{n+1}=s_n^2-s_{n-1}^2$
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| May 16 at 3:05 | answer | added | KhashF | timeline score: 3 | |
| May 15 at 20:36 | comment | added | look at me | There is also an unstable 5 cycle wolframalpha.com/… | |
| May 15 at 20:32 | answer | added | Robert Israel | timeline score: 9 | |
| May 15 at 19:39 | comment | added | Robert Israel | I think that the requirement $s_0 = \sqrt{x}$ is a red herring. You really should be looking at the dynamics of the map $F: (x,y) \to (y, y^2 - x^2)$ on $\mathbb R^2$. It is clear that $(0,0)$ is the only fixed point, and it is an attractor. There is an unstable $3$-cycle $(1,0)$, $(0,-1)$, $(-1,1)$. | |
| May 15 at 14:16 | comment | added | KhashF | The sequence diverges for $x\geq\left(\frac{1+\sqrt{5}}{2}\right)^2\approx 2.618033$: This amounts to $s_1\geq s_0+1$. By induction, if $s_i\geq s_{i-1}+1$ for $i=1,\dots,n$ (which implies that $s_0,\dots,s_n\geq 1$), then $s_{n+1}=(s_n-s_{n-1})(s_n+s_{n-1})\geq s_n+s_{n-1}\geq s_n+1$. Hence $s_i\geq s_{i-1}+1$ holds for $i=n+1$ too. | |
| May 15 at 12:27 | comment | added | Liviu Nicolaescu | $$s_{n+1}+s_n+\cdots +s_2+s_1=s_n^2.$$ Hence if $s_n$ converges, so does the series, and thus $s_n$ can only converge to $0$. | |
| May 15 at 8:16 | comment | added | Emil Jeřábek | As for telling whether it converges: once there is $n$ such that $s_{n-1},s_n\in(-1,1)$ (or: $|s_{n-1}|,|s_n|<1/2$ in the $\mathbb C$ case), the sequence will converge to $0$. | |
| May 15 at 6:31 | comment | added | Pietro Majer | So do you want to study the recursion in $\mathbb R$ or in $\mathbb C$? | |
| May 15 at 4:34 | comment | added | Fedor Petrov | Does it ever converge to a value other than 0? - no, if the limit equals $a$, we get $a=a^2-a^2=0$ | |
| May 15 at 4:28 | comment | added | Daniel Weber | It converges to zero | |
| May 15 at 3:53 | comment | added | look at me | @CommandMaster I just used google, which has not nearly as much precision. Also did it converge at zero or did it loop? | |
| May 15 at 3:23 | comment | added | Daniel Weber | Using a thousand digits of precision it seems like both 1.670718 and 1.670719 converge, although 1.7 still diverges, so it's not monotonic | |
| May 15 at 3:14 | comment | added | Daniel Weber | If the system is chaotic then how can you tell whether it actually converges or diverges? It seems likely that numerical errors accumulate, and might shift it from diverging to converging or the other way around | |
| May 15 at 3:09 | history | edited | Daniel Weber |
Added dynamical systems tag
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| May 15 at 3:08 | comment | added | Daniel Weber | It seems useful to consider $f(x,y) = (y, y^2-x^2)$ | |
| S May 15 at 3:00 | history | suggested | CommunityBot | CC BY-SA 4.0 |
spelling and MathJax
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| May 15 at 2:42 | review | Suggested edits | |||
| S May 15 at 3:00 | |||||
| May 15 at 1:42 | history | edited | look at me | CC BY-SA 4.0 |
added 57 characters in body
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| May 15 at 1:29 | history | asked | look at me | CC BY-SA 4.0 |