# Tagged Questions

**2**

votes

**0**answers

75 views

### Quick estimate of attractor of non-linear dynamical system [closed]

Say I have a system of form
$$
\frac{dy}{dt} = f(y),
$$
and it is know this system has an attractor. Can I quickly for given $\varepsilon$ guess some point, such in its $\varepsilon$- neighbourhood ...

**1**

vote

**0**answers

56 views

### Periodic solution of first order ODE

There is a famous result shows that for every continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$, the first order autonomous system
$$
\left\{
\begin{array}{l}
\dot{x}=f(x), \\
x(t_0)=x_0,
...

**2**

votes

**1**answer

80 views

### Global Solutions of Ordinary Differential Equations

Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
for every ...

**1**

vote

**1**answer

164 views

### Number of solutions of a system of equation!

Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...

**3**

votes

**1**answer

40 views

### Hopf bifurcation for systems where the dynamics is homogeneous of degree 1

Consider dynamical system in dimension 3
$$x'(t)=f(x(t),d)$$
where the dynamics f is homogeneous of degree 1 and there is exactly one
line of equilibrium points. This line is independent of the ...

**27**

votes

**2**answers

2k views

### Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...

**2**

votes

**2**answers

154 views

### Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...

**1**

vote

**1**answer

117 views

### Stability analysis of ODE

My questions concerns the stability analysis of the following dynamical system :
$\dfrac{d}{dt} a_{i}(t) = D_{i} + \displaystyle{\sum_{j=1}^{n}L_{ij}a_{j}(t) + \sum_{j=1}^{n}\sum_{k=1}^{n} C_{ijk} ...

**5**

votes

**1**answer

178 views

### Is Taylor expansion related to Helmholtz decomposition?

The Taylor expansion of a vector field $f(x)$ to the order of one is
$$f(x)=f(x_0)+Jf(x_0)\cdot\Delta x+o(\Delta x)$$
where $Jf$ is Jacobian of the vector field and $\Delta x=x-x_0$.
Suppose we ...

**1**

vote

**2**answers

295 views

### Replacing large-dimensional ODE systems with one PDE [closed]

Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?

**3**

votes

**1**answer

228 views

### A theory of bifurcation of braids ?

I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional ...

**2**

votes

**1**answer

64 views

### (A)periodicity and (In)dependence on the boundary condition for optimization problem related to ODE

The question is pair to MO117505 and translates some problem on error-correction codes to similar problem about differential operators. (See also If “force” is periodic does it imply “velocity” is ...

**1**

vote

**1**answer

106 views

### (A)periodicity and (In)dependence on the boundary condition for some discrete analog of ODE (convolutional codes)

(See also MO117508, MO116611). This post describes somewhat real problem with convolutional codes. Let me first try to give brief and vague formulation of the question, later give details.
Problem ...

**0**

votes

**0**answers

140 views

### Continuity of the Shadow of a Nondecreasing Function

So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...

**0**

votes

**1**answer

393 views

### Hölder continuity of uniform limit of piecewise constant functions

Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants ...

**1**

vote

**4**answers

307 views

### A Fractional Linear Transformation Class Property

Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where
$a,b,c,d \in R$, and ...

**7**

votes

**4**answers

336 views

### Identifying a system of ODEs

Studying the dynamics of the endpoints of an equilibrium measure (a minimizer of its logarithmic energy in an external field) I ran into the following system of differential equations (which I state ...

**6**

votes

**2**answers

577 views

### Linearly independent vector fields

Let $X_1,\dots,X_n$ be complete vector fields on $\mathbb R^n$ and suppose that $(X_1(p),\dots,X_n(p))$ is a basis for all $p \in \mathbb R^n$.
Question: Is it possible to choose a cube $C$ around ...

**11**

votes

**2**answers

917 views

### Nonvanishing of Jacobians implies global injectivity?

I am interested in obtaining injectivity of a $C^1$ map from the nonvanishing minors of its Jacobian matrix. Here is a brief history of the topic.
In 1953, Samuelson asked the following:
If the ...

**4**

votes

**2**answers

309 views

### mechanics: convergence to an equilibrium point

Hello,
this is a math forum, I know, but my question is about classical mechanics. I am looking for a general (but simple proof) of the very intuitive idea physicists have about the following ...

**0**

votes

**1**answer

373 views

### Simple system of ODEs with periodic coefficients

I am stuck with a little problem that I cannot solve mith the standard methods I learn at university. I have a system of coupled ODEs:
$f'(t) = P \cos(k t + \Phi_1) g(t)$
$g'(t) = Q \cos(k t + ...

**3**

votes

**0**answers

487 views

### Find a second integral for Arnold's example

Consider Arnold's example for Arnold diffusion 1964.
$$H=I_1^2/2+I_2^2/2+\epsilon(1-\cos\theta_2)(1+\mu(\sin\theta_1+\sin t)) $$
We can first make it a system of three degrees of freedom.
Then we ...

**4**

votes

**3**answers

576 views

### Homogeneous linear differential equation system with simple periodical coefficient matrix

Hello, I encountered the following system of linear first-order differential equations:
$y'(z)=A(z) y(z)$
where
$y(z): R \rightarrow R^2$ and
$A(z)=\begin{pmatrix}
0 & B Cos(\alpha z + \Phi_b) ...

**11**

votes

**4**answers

1k views

### Routh-Hurwitz for eigenvalues

The Routh-Hurwitz criterion provides a convenient test, even for hand calculation, of whether a polynomial with real coefficients has all its roots in the left half plane. I'm wondering about a ...

**2**

votes

**1**answer

189 views

### Zeros of linear partial fractions

I am trying to find some general properties of the zeros of
$P(z) = \sum_{i=1}^n \frac{\alpha_i}{z+z_i}$,
with $\sum_{i} \alpha_i = 0$, $z_i \in [-M\; 0], i=1,\ldots,n$ and all $\alpha_i$ and ...

**3**

votes

**0**answers

777 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 ...

**4**

votes

**0**answers

116 views

### Asymptotic rearrangement

I had some trouble coming up with a good title for this question. Here is the setup. Suppose you have two infinite sets of (positive real, say) numbers $\{a_k\}$ and $\{b_k\}$ such that the ...

**6**

votes

**2**answers

380 views

### Functions which form continuous curve with its own iterations

The following function
$$f(x)=-2 \cos \left(\sqrt{2} \arccos \left(\frac{x-1}{2}\right)\right)+1$$
has interesting property to form a continuous curve with its own integer iterations. The following ...

**13**

votes

**2**answers

1k views

### The vector field of a given flow

Let $f:(0,1)\rightarrow(0,1)$ be a map with some regularity (${\mathcal C}^1$, ${\mathcal C}^2$, ${\mathcal C}^\infty$, analytic ?). We assume that $f(t)> t$ for every $t$, and that $f'> 0$.
...

**18**

votes

**1**answer

1k views

### Analogues of Luzin's theorem

If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of ...

**3**

votes

**1**answer

635 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 ...

**2**

votes

**1**answer

603 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 ...

**6**

votes

**1**answer

411 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| < ...

**11**

votes

**2**answers

1k views

### How did Gauss discover the invariant density for the Gauss map?

The Gauss map is defined on $(0,1)$ by the formula
$$
f(x)=\frac1x-\Big\lfloor\frac1x\Big\rfloor
$$
Then the density
$$
\rho(x)=\frac{1}{\log2(1+x)}
$$
is $f$-invariant.
It appeared in Gauss' diary. ...

**25**

votes

**1**answer

755 views

### A question of Erdős on equidistribution

In his book Metric Number Theory, Glyn Harman mentions the following problem he attributes to Erdős:
Let $f(\alpha)$ be a bounded measurable function with period 1. Is it true that
...

**5**

votes

**1**answer

213 views

### Is there a notion of “Morse index” for geodesics in a manifold with indefinite metric that is well-behaved under cutting and gluing?

More generally, I'm interested in the situation of Lagrangian mechanics. And actually my question is local, so you can work on $\mathbb R^n$ if you like. I will begin with some background on ...

**64**

votes

**0**answers

4k 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 ...

**5**

votes

**2**answers

351 views

### Asymptotics of iterated polynomials

Let the sequence $u_1, u_2, \ldots$ satisfy $u_{n+1} = u_n - u_n^2 + O(u_n^3)$. Then it can be shown that if $u_n \to 0$ as $n \to \infty$, then $u_n = n^{-1} + O(n^{-2} \log n)$. (See N. G. de ...

**2**

votes

**1**answer

256 views

### ODE system question

Consider a system of the form: dx/dt = f(x,y) , dy/dt=g(x,y), with the property that the associated ODE dy/dx = g(x,y)/f(x,y) has a unique solution to IVP y(0)=0.
Also, f(x,y) is smooth every except ...