2
votes
0answers
83 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
0answers
80 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
1answer
115 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
1answer
171 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
1answer
45 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
2answers
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
2answers
159 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 ...
0
votes
1answer
118 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
1answer
190 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
2answers
309 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
1answer
243 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
1answer
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
1answer
115 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
0answers
145 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
1answer
418 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
4answers
315 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
4answers
338 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
2answers
599 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
2answers
1k 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
2answers
312 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
1answer
387 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
0answers
500 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
3answers
671 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
4answers
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
1answer
192 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
0answers
816 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 ...
3
votes
0answers
121 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
2answers
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
2answers
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 $f'> 0$. ...
21
votes
1answer
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
1answer
641 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
1answer
633 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
1answer
417 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
2answers
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
1answer
772 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
1answer
217 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 ...
66
votes
0answers
5k 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
2answers
355 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
1answer
259 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 ...