All Questions
Tagged with ds.dynamical-systems ca.classical-analysis-and-odes
147 questions
0
votes
0
answers
34
views
Existence and uniqueness of heteroclinic solution of Allen–Cahn on $\mathbb R$ with driving-damping term
The Allen–Cahn equations on $\mathbb R$ are $u'' = u^3 - u$. It is well-known that all the solutions of this equation which satisfy the asymptotic boundary conditions $\lim_{x \to \pm \infty} u\left(x\...
- 395
1
vote
0
answers
59
views
Asymptotic behavior of positive solution to nonlinear scalar field equation
It is well-known that the radial positive solution
$u=u(r)$ to nonlinear scalar field equation $$-\Delta u+u=u^p\text{ in } ~\mathbb{R}^d, 1<p<\frac{d+2}{d-2}$$ has the following asymptotic ...
- 705
2
votes
2
answers
175
views
Great literature on discrete dynamical systems and/or qualitative theory of difference equations
I am asking for the great literature on topics of discrete dynamical systems and/or qualitative theory of difference equations especially aimed on pure mathematicians. Could you please give me some ...
0
votes
0
answers
112
views
Vector field connecting two points
I'm now working on somehow an inverse problem of an ODE:
Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t).
Now there is a ...
- 1
3
votes
0
answers
183
views
Bounded solutions of nonlinear third-order ODEs
I am interested in understanding the behavior of solutions to certain nonlinear third-order ODEs. Specifically, I am curious about conditions that guarantee all solutions remain bounded for $t \in [0, ...
- 807
3
votes
0
answers
56
views
Perturbation method for time-periodic singular system of ODEs
I am studying a problem arising in physics, and I managed to simplify it to a differential system (initial value problem) of the form:
$$
\begin{cases}
\dot{x} = \epsilon f_1(x,y,t) + \epsilon^2 f_2(...
- 131
2
votes
0
answers
57
views
Regularization for Newtonian n-body collisions in $\mathbb{R}^3$
In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
- 46
5
votes
0
answers
263
views
Is the global solution to this ODE bounded?
Consider
$$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$
where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
- 405
2
votes
1
answer
119
views
Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by $\phi_{f_{\theta(t)}(x)}^{\tau'}$?
Given a control family $F:=\{f_1,\dotsc,f_n\}$, and $\phi_f^\tau(x)$ is the flowmap of the dynamical system
$$
\begin{cases}
z'(t)=f(z),\\
z(0)=x,
\end{cases}
$$ at end time point $\tau$.
Suppose $a_i&...
- 109
0
votes
1
answer
39
views
Is the right-hand term of the autonomous dynamic system equivalent to the original system after being multiplied by a constant?
Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, and ...
- 109
0
votes
2
answers
69
views
Is the right-hand term of the dynamic system equivalent to the original system after being multiplied by a constant?
Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x,t),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z,t),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, ...
- 109
4
votes
0
answers
233
views
References for derivative w.r.t. initial condition of an ODE
Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ be measurable such that for all $n \in \mathbb N$ we have
$$
\sup_{t \ge 0} |b(t, 0)| + \sup_{t \ge 0} \sup_{x \in \mathbb R^d} |\nabla^n_x b (t, ...
- 835
11
votes
2
answers
852
views
What are the iterates of $x \mapsto 1 - \sqrt{1-x^2}$?
Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be ...
- 32.5k
21
votes
0
answers
416
views
Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
- 281
6
votes
1
answer
328
views
Solution of an ODE upon singular perturbation
The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.
The original system involves $N$ massless electric charges at position $\...
- 255
7
votes
1
answer
142
views
Hamiltonian-ization of a dynamic system
On affine space, a sufficiently smooth continuous-time Hamiltonian dynamic system $\dot p = \nabla_q H, \dot q = -\nabla_p H$ conserves $H$, preserves the volume form (e.g. if we are looking for ...
- 200
0
votes
0
answers
103
views
Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity
Problem:
Consider the autonomous ODE system
\begin{align*}
\dot{x} &= (1-x) (z-xy)\\
\dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\
\dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z
\end{...
- 291
4
votes
0
answers
49
views
Getting analytic center manifolds
The center manifold of a degenerate zero of an analytic vector field need not be unique nor analytic. But say I want it analytic. Does anyone know of additional conditions to be imposed on the ...
- 8,336
3
votes
1
answer
435
views
Showing convergence of an infinite ODE system
Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(0) \leq 1$ for all $n\in \mathbb N$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ ...
- 730
1
vote
0
answers
100
views
Generalizing an application of the Poincaré-Bendixson theorem
I was looking for applications of the Poincaré-Bendixson theorem and on this site I have found several examples almost all similar to this post. So I tried to make a quite natural generalization
$$
\...
- 143
2
votes
1
answer
371
views
Examples of ODEs with complex constant coefficients and applications to physics?
This question is asked on stackexchange: Are there examples for ODEs with complex coefficients with applications in physics?
but received no answers. I am reposting it here on the hope that it catches ...
- 852
1
vote
0
answers
70
views
What *piecewise* smooth curves/surfaces/hypersurfaces give rise to forward-invariant regions of dynamical systems?
Consider a set $\mathcal{B}\subset \mathbb{R}^n$ that is homeomorphic to a closed n-dimensional ball, and denote its boundary by $\mathcal{H}$. Assume that $\mathcal{H}$ is a "piecewise smooth&...
- 111
3
votes
1
answer
217
views
The energy of a semilinear ODE
I'm currently reading Caffarelli, Gidas, Spruck's paper "Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth". For some background, we ...
- 457
1
vote
0
answers
99
views
Limit cycles or stable solutions for k-dimensional piece-wise linear ODEs
As a branch of reinforcement learning, restless multi-armed bandits have been shown PSPACE-HARD but Whittle has offered an implementable solution called the Whittle Index Policy. Weber and Weiss ...
0
votes
1
answer
147
views
Why all the coefficients of the center manifold of this system are zeros?
I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the ...
- 159
0
votes
0
answers
222
views
Convergence of ODE solutions almost everywhere to a stable equilibrium point
Theorem: Suppose ${\bf g} :\mathbb{R}^n \mapsto \mathbb{R}^n$ is continuously differentiable, there exists a set $\mathcal{A} \subset \mathbb{R}^n$ such that $\bf g$ is uniformly Lipschitz on $\...
- 1
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(...
- 1,763
3
votes
1
answer
95
views
Inductive proof that $\dot{M}_{n+1}=-M_{n+1}+W^{(n+2)}(0)+vM_{n+2}$
The motivation for the following is to convert the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=-kx+\beta\int_{-\infty}^t W'(x(t)-x(s))e^{s-t}ds,
\end{equation}
into a ...
- 79
7
votes
0
answers
369
views
On the solvability of a nonlinear differential system
A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system ...
- 79
2
votes
0
answers
153
views
Stability test for LTV systems by differential Lyapunov inequalities
Consider a linear time-varying system:
\begin{equation}
\dot x(t) = A(t) x(t), \tag{$*$}
\end{equation}
where $A(t)$ is a time-varying block matrix defined as
$$
A(t) =
\begin{bmatrix}
0 & I\\
-\...
- 55
6
votes
2
answers
295
views
Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium?
Consider a continuous ODE,
$$\dot x = f(x), f \in C^1$$
$\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For ...
- 205
13
votes
1
answer
3k
views
Behavior of $n^\alpha \sin^{\circ\, n}(n^{-\alpha}x)$
I'll write it formally: Let $\sin^{\circ\, 1}(x) = \sin(x)$ and $\sin^{\circ n+1}(x) = \sin\bigl(\sin^{\circ n}(x)\bigr)$ for $n\in \Bbb N$ with $n>1$.
What is the limit as $n \to \infty$?
It's ...
1
vote
0
answers
157
views
Selecting a suitable Lyapunov function for the following systems?
i) SI MODEL
Consider
\begin{align}
\frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex]
\frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I
\end{align}
Where $N=S+I$ is the total population.
If ...
- 185
3
votes
1
answer
613
views
Searching for the proof of a certain claim in Arnold's ODE book from 1992
I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition).
On pages 12-13 he writes the following:
Consider the following ordinary ...
- 1,594
6
votes
2
answers
437
views
Matrix-valued ordinary differential equation with symmetry
I am considering the following equation
$$\begin{pmatrix} -\frac{d}{dx} + \lambda \sin(2\pi x) & \lambda - \lambda \cos(2\pi x) \\ -\lambda-\lambda \cos(2\pi x) & -\frac{d}{dx} - \lambda \sin(...
- 192
3
votes
0
answers
73
views
What is known about discrete versions of the spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities?
Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of ...
- 111
1
vote
0
answers
181
views
Subset of the domain of attraction
Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$
$$
\frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t))
$$
be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
- 61
2
votes
2
answers
210
views
Oscillator with discrete number of amplitudes?
I know the question is a little vague, but I would like to know if someone can direct me to what kind of oscillator (if exist), that can follow the next behavior.
I manually create the gif to try to ...
9
votes
1
answer
800
views
Why the least action principle is always (?) used in this particular form?
The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order)...
- 21.8k
5
votes
1
answer
544
views
Is the series $\sum_{n=1}^{\infty} \sin(n^4)\sin(4^n)$ convergent or divergent?
Is the series $$ \sum_{n=1}^{\infty} \sin(n^4)\sin(4^n) $$ convergent or divergent?
I tried expanding the sine functions and got no clue, and any test that I know of isn't helping me with this series. ...
- 159
1
vote
0
answers
76
views
What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?
When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
- 183
3
votes
0
answers
42
views
Approximation of vector field by vector fields with all trajectories closed
Let $\vec j$ be a smooth compactly supported vector field in $\mathbb{R}^3$ such that $div(\vec j)=0$. Are there known either necessary or sufficient conditions such that there exists a sequence of ...
- 21.8k
1
vote
1
answer
287
views
Regular singular point of non-linear ODE: $\dot{x}(t) + t^{-1}Ax(t) = Q(x(t))$
Consider a system of ordinary differential equations of the form
$$
\dot{x}(t) + \frac{1}{t}Ax(t) = Q(x(t))
$$
where $x(t) \in \mathbb{C}^n$, $A \in \mathrm{Mat}_{n\times n}(\mathbb{C})$ is a constant ...
- 1,383
5
votes
1
answer
348
views
A differential inequality involving gradient and laplacian
Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$.
What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...
- 311
2
votes
1
answer
160
views
Dynamical system described by coupled nonlinear differential equations
Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled nonlinear differential equations:
\begin{equation}
\begin{split}
...
- 197
1
vote
0
answers
36
views
Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms
Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...
- 5,405
2
votes
2
answers
1k
views
Monotonic and bounded sequences throughout mathematics [closed]
When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
- 4,943
11
votes
2
answers
883
views
Do infinitely nested radicals have any applications?
There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
- 4,943
7
votes
2
answers
731
views
A second order nonlinear ODE
In my research (in differential geometry) I recently came across the following nonlinear second order ode:
$$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$
It ...
- 277
1
vote
0
answers
67
views
Alternate characterization of floquet multipliers: Floquet theory
Given an autonomous ode $\dot{x}=f(x)$ in $\mathbb{R}^n$ possessing a period-p time-periodic solution $\bar x(t)$, one can use the so-called variational equation about $\bar x$ to study its stability. ...
- 11