Tagged Questions
1
vote
1answer
84 views
Lipschitz condition on nonlinear ODE
Suppose we have the ODE $$x''=-a\sin{x}.$$
Then let $$x'=y$$ and $$y'=-a\sin{x}.$$ So $$\mathbf X = \begin{pmatrix} y \\ -a\sin{x} \end{pmatrix}.$$ Im confused about how to show a Lipschitz ...
2
votes
2answers
74 views
does it have unique fixed point?
$p:C[0,1]\rightarrow C[0,1]$ defined by $p(f(x))=\int_{0}^{x} (x-t)f(t)dt$, well, I am getting all constant functions are fixed points, but the answer says that it has unique fixed point. I got ...
1
vote
1answer
63 views
Fixed Points of a system of diff.equations
I have
$f'_1(t)=-af_1(t)f_2(t)+bf_3(t)$
$f'_2(t)=f_1(t)$
$2f'_3(t)=-f_1(t)$
How is it possible to evaluate fixed points of this system of equations and afterwards the stability of these points. I ...
0
votes
1answer
46 views
IVP- Has at most one solution
Suppose $f(t, x)$ is nonincreasing with respect to $x$ for all $t \geq 0$ and $x \in \mathbb{R}$. Prove that the IVP problem
$$
\left\{
\begin{array}{l}
x'(t)=f(t,x) \\
x(t_0)=x_0
\end{array}
\right.
...
3
votes
1answer
140 views
Fixed point: a consequence of symmetry?
I'm studying a dynamical system with $\mathbf{D}_{3}$ symmetry (the symmetry group of an equilateral triangle), which is given by:
$\begin{align*}
d\mathbf{x}_{0}/dt &= ...
