All Questions
Tagged with nonlinear-dynamics recurrence-relations
8
questions
1
vote
0
answers
110
views
Long term behavior of a nonlinear dynamic system
Consider the following system of first order nonlinear difference equation:
$$
\begin{cases}
x_{k+1} = x_k + \alpha(1-x_ky_k^2) \\
y_{k+1} = y_k + \alpha(1-x_k^2y_k) \\
\end{cases}
$$
with a given ...
- 997
3
votes
1
answer
271
views
Given $(a_n)$ such that $a_1 \in (0,1)$ and $a_{n+1}=a_n+(\frac{a_n}{n})^2$. Prove that $(a_n)$ has a finite limit. [duplicate]
Given $(a_n)$ such that $a_1 \in (0,1)$ and $a_{n+1}=a_n+(\frac{a_n}{n})^2$. Prove that $(a_n)$ has a finite limit.
Clearly $a_n$ are increasing. Also, $$\frac{1}{a_{n+1}}=\frac{1}{a_n\left(1+\frac{...
- 55
0
votes
1
answer
197
views
Time Scale Calculus/Dynamic Equations on Time Scales
Studying the mathematics of dynamic phenomena seems to branch out in an overwhelming number of directions. There are univariate and multivariate varieties, continuous and discrete varieties, ...
- 1,922
4
votes
1
answer
94
views
If $x_{n+1}=x_n+x_n^{-m}, x_1>0, m \ge 2 $ then $x_{n}^{m+1} = (m+1)(n+\ln(n))+O_m(1) $
This is a generalization of
Divergence of a sequence proof
Show that
if
$x_{n+1}=x_n+\dfrac{1}{x_n^m},
x_1>0, m \ge 2
$
then
$x_{n}^{m+1}
= (m+1)(n+\ln(n))+O_m(1)
$.
(Note:
The notation $O_m(...)$
...
- 108k
0
votes
1
answer
915
views
How to determine the stability of a fixed point if the derivative at the point is equal to one? ($\,\left\lvert\, f'(x^{\ast})\right\rvert = 1$)
Context:
I am learning about 1-dimensional maps: For instance the logistic model of population growth.
Suppose I have the map $x_{n+1} = f(x_n)$. The point $x^{\ast}$ is called a fixed point of the ...
- 536
2
votes
2
answers
132
views
Tight upper bounds for a monotonically increasing non-linear recurrence
I have the following non-linear recurrence:
$$y_{n+1} = \sqrt{\frac{2}{1+y_n}}y_n,\quad y_0 \in[0,1]$$
Some basic thought shows that $0$ and $1$ are fixed points of this, and that $0$ is repelling ...
- 7,058
4
votes
2
answers
199
views
Solving the recurrence relation $y_{n+1}=y_n+a+\frac{b}{y_n}$
I am hoping to obtain a closed-form solution or an asymptotic result for the recurrence relation
$$y_{n+1}=y_n+a+\frac{b}{y_n}$$
Any help would be very much appreciated!
- 383
0
votes
1
answer
229
views
Can a 2 cycle occur in logistic map for r=4?
In web page http://mathworld.wolfram.com/LogisticMap.html it's written that for
$$x_{1,2}=\dfrac{1}{2} \Bigg[(1+r^{-1}) \pm r^{-1} \sqrt{(r-3)(r+1)} \Bigg] \tag 1 $$
a 2-cycle occurs. Although don't ...
- 103