All Questions
Tagged with ds.dynamical-systems differential-equations
216 questions
0
votes
1
answer
792
views
phase portrait of system of differential equations
Is there a full classification of the phase portraits of the following systems of differential equations
\begin{equation}
\cases{
\dot x=a_{11}x+a_{12}y+a_{13}z \\
\dot y=a_{2 1}x+a_{22}y+a_{23}z\\
...
- 301
1
vote
1
answer
901
views
How to deduce the existence of stationary points from fixed points of evolution maps?
This is probably a very elementary question. Nevertheless I decided to post it on MO. Consider a smooth manifold $M$ and a smooth complete vector field $v:M\rightarrow TM$. Consider an autonomous ...
- 965
3
votes
1
answer
381
views
First order PDE, singular at a point
I am pretty sure this should be text book material, but I couldn't find this anywhere; maybe I just don't know where to look.
Problem: Suppose we have a smooth vector field $X = a_i x^i \partial_i + ...
- 9,853
1
vote
0
answers
462
views
Partial feedback linearization (Control theory)
I'm trying to understand a theorem about partial feedback linearization from the paper "On the largest feedback linearizable subsystem" by R. Marino (published in: Systems & Control Letters, ...
3
votes
0
answers
2k
views
Bessel functions in wave propagation and scattering
Is there a way to scale $J_n(\cdot)$ (Bessel of first kind) and $H_n(\cdot)$ (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher values of n) and small arguments....
- 41
3
votes
0
answers
1k
views
(Approximate) analytic solutions to the Mathieu equation
I'm trying to solve the driven Mathieu equation
$x''+\beta x'+(a-2q\cos{\Omega t})\frac{\Omega^2}{4}x=f(t)$
for both zero and non-zero $\beta$.
I can write down an analytic solution using the ...
- 31
4
votes
1
answer
4k
views
Omega-limit set of the omega-limit set
Consider a dynamical system given by its flow $\phi(t,x)$, where $t \in R$, $x \in R^n$ and $\phi: R \times R^n \to R^n$ is (say) differentiable.
The $\omega$-limit set, $\omega(p)$, of a point $p \...
- 531
17
votes
5
answers
2k
views
2- and 3-body problems when gravity is not inverse-square
Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing
as $1/d^p$ for distance separation $d$ and some power $p$.
Two questions:
Presumably the 2-body ...
- 151k
4
votes
1
answer
1k
views
Limit of a discrete time dynamical system
I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have ...
- 53
9
votes
1
answer
481
views
Existence of a vector field with a finite number of limit cycles.
The following question is related to the Seifert conjecture.
Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
- 4,736
3
votes
1
answer
2k
views
A formula for the Jacobian of a flow
Let $U : \mathbb R^d \to \mathbb R^d$ be a smooth vector field, and let $F_t : \mathbb R \times \mathbb R^d \to \mathbb R^d$ be the corresponding smooth flow, defined by the differential equation $$\...
- 8,512
3
votes
3
answers
865
views
Analytic ODE with complex time
Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi_t(x)...
- 303
6
votes
1
answer
508
views
Estimating the flow when we know the vector field
Suppose we have a $C^k$ vector field $v$ and let $\phi_t$ be the corresponding flow. I have estimates on $v$ and its derivatives: $|v| < C_0$, $|Dv| < C_1$, $|D^2v| < C_2$, ... $|D^kv| < ...
- 303
101
votes
1
answer
8k
views
Dropping three bodies
Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...
- 8,336
0
votes
1
answer
189
views
Difference Equations & Possible Limits
The answer to this may well be in some elementary textbook - a reference might be more useful than a short answer here.
If we look at the behaviour of a point in R n under matrix multiplication, we ...
164
votes
14
answers
40k
views
What is an integrable system?
What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "...
- 24.7k