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2 votes
0 answers
57 views

Regularization for Newtonian n-body collisions in $\mathbb{R}^3$

In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
user12994's user avatar
4 votes
0 answers
233 views

References for derivative w.r.t. initial condition of an ODE

Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ be measurable such that for all $n \in \mathbb N$ we have $$ \sup_{t \ge 0} |b(t, 0)| + \sup_{t \ge 0} \sup_{x \in \mathbb R^d} |\nabla^n_x b (t, ...
Akira's user avatar
  • 825
71 votes
3 answers
5k views

Does iterating the derivative infinitely many times give a smooth function whenever it converges?

I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let $d = \frac{d}{dx}$, and let $f \in C^{\infty}(\mathbb{R})$ be such that for every real $x$, $$g(...
Paul Cusson's user avatar
  • 1,763
6 votes
2 answers
295 views

Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium?

Consider a continuous ODE, $$\dot x = f(x), f \in C^1$$ $\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For ...
Concu Bine's user avatar
3 votes
1 answer
613 views

Searching for the proof of a certain claim in Arnold's ODE book from 1992

I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition). On pages 12-13 he writes the following: Consider the following ordinary ...
Alan's user avatar
  • 1,594
13 votes
1 answer
661 views

Poincaré on analytic dependence on parameters of solutions of linear differential equations

There is the following important General Principle: if a parameter enters in a linear differential equation additively, for example $$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$ where the parameter is $\...
Alexandre Eremenko's user avatar
3 votes
1 answer
202 views

Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations

Consider the following autonomous system of differential equations: $$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$ where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
valle's user avatar
  • 884
3 votes
0 answers
135 views

Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form $$\ddot{...
digiboy1's user avatar
2 votes
0 answers
283 views

Uniqueness of analytic center manifold

In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
aristote's user avatar
5 votes
0 answers
224 views

Unit eigenvalue of the linearized Poincare return map

Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare ...
user65812's user avatar
4 votes
1 answer
354 views

Reference request: Invariant sets of dynamical systems

(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...
ttb's user avatar
  • 185
1 vote
0 answers
216 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, \...
Mohammad Khosravi's user avatar
1 vote
1 answer
207 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 ...
Mohammad Khosravi's user avatar
18 votes
2 answers
5k 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 ...
Syang Chen's user avatar
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 ...
Eric Blair's user avatar
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 $$\...
Tom LaGatta's user avatar
  • 8,512