All Questions
Tagged with nonlinear-dynamics analysis
7
questions
2
votes
2
answers
160
views
Reduce System of 3 ODEs to 2 ODEs
I'm currently self-studying the book "Nonlinear Dynamics and Chaos". I got the following system
$$
\dot{x} = -kxy \\
\dot{y} = kxy - ly \\
\dot{z} = ly
$$
According to a comment on one of ...
- 33
3
votes
2
answers
83
views
Solving and plotting the 2-D Lorenz Equation
We are asked to first solve and then plot the phase plane of
\begin{align}
\begin{cases}
\dot{x}=\sigma x - \sigma y\\
\dot{y} = \rho x-y
\end{cases}, \ \sigma, \ \rho >0.
\end{align}
Now the ...
- 925
6
votes
0
answers
155
views
Nature of ODE $\dot x=x^2-\frac{t^2}{1+t^2}$
Discuss the equation $\dot{x}=x^2-\frac{t^2}{1+t^2}$.
Make a numerical analysis.
Show that there is a unique solution which asymptotically approaches the line $x=1$.
Show that all solutions below ...
- 9,098
2
votes
0
answers
43
views
Prove that there exists $\delta \in(0, \infty)$ such that when $y(0)=\delta$, the solution to the ODE for $y$ loses uniqueness in finite positive time
Consider $\dot{x}=f(x)$ where differentiation is with respect to $t$. Suppose there exist $x_0>0$, such that $f\left(x_0\right)=0$, and $\epsilon>0$ such that
$$
\left|\int_{x_0}^{x_0+\epsilon} \...
- 9,098
1
vote
1
answer
190
views
Prove the $k$-th power of the logistic map with parameter $\mu = 4$ has $2^k$ fixed points
I'm trying to solve an exercise in which I need to prove that the logistic map with parameter $\mu = 4$, $F_4:[0,1]\to[0,1]$, $F_4(x) = 4x(1-x)$, satisfies that for every positive integer $k$, $F_4^k$ ...
- 195
0
votes
0
answers
85
views
Is there an analytic solution for the following non-linear equation $x'(t) = a(t) + b\, x(t) + c\,x^3(t)$?
Here $b, c$ are constants. Of course there exists locally a solution by the Picard-Lindelöf theorem but I'm looking for an explicit expression.
In fact, this question comes from here but I changed the ...
- 3,679
2
votes
0
answers
93
views
A first order non-linear differential equation?
I'm trying to solve this non-linear differential equation :
$$ \frac{dy}{dx}= \frac{y^3}{(y+1)^2(y+2)^2} $$
with the boundary condition $y(x_0)=x_0$, $x>0$, and $y(x)$ being a positive function.
...
- 43