All Questions
Tagged with differential-equations ds.dynamical-systems
216 questions
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
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
28
votes
2
answers
2k
views
Codimension of the range of certain linear operators
Added:8/15/2024 What about holomorphic or real analytic version? Please see the comment discussions on this post.
Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We ...
- 356
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
20
votes
3
answers
690
views
Escaping from infinitely many pursuers
The fugitive is at the origin. They move at a speed of $1$. There's a guard at $(i,j)$ for all $i,j\in \mathbb{Z}$ except the origin. A guard's speed is $\frac{1}{100}$. The fugitive and the guards ...
- 2,619
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
14
votes
2
answers
390
views
Is there a singularity theorem in higher-dimensional Newtonian gravity?
In classical Newtonian gravity with 3 spatial dimensions, it's hard to get two particles to exactly collide, since at short distance the centrifugal force (~1/$r^3$) beats the gravitational attraction ...
- 273
14
votes
1
answer
2k
views
The perturbation of non-Hamiltonian algebraic vector fields
In this question, we are interested in the number of limit cycles which appears in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
\...
- 356
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 $\...
- 91.8k
12
votes
3
answers
2k
views
Vector field with holomorphic flow
Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.Question Show the flow of $JX$ is by biholomorphisms.
I know one reference ...
- 6,995
11
votes
1
answer
1k
views
How to eliminate secular terms for perturbed non-oscillatory equations?
Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely
$$x(t)=a_0+b_0e^{-t}+\epsilon(...
- 1,731
10
votes
2
answers
350
views
Is this Riccati equation ("Josephson junction") always phase-locked at integer rotation numbers?
Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE
$$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$
Identifying ...
- 708
9
votes
1
answer
481
views
Existence of a vector field with a finite number of limit cycles.
The following question is related to the Seifert conjecture.
Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
- 4,736
9
votes
2
answers
648
views
An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)
Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question.
Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x,y)+wQ(...
- 356
8
votes
2
answers
1k
views
What is the current status on methods to find limit cycles?
What are the current best methods to show analytically the existence of a limit cycle in a $n$-dimensional system of the form:
$$
\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\vec{f}(\vec{x})
$$
Where $...
- 117
8
votes
0
answers
256
views
Structural Stability on Compact $2$-Manifolds with Boundary
I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...
- 333
7
votes
2
answers
387
views
Is it possible to prove unboundedness of 3rd order ODE?
Consider the 3rd order ODE
$$\dddot{x}+A\ddot{x}-\dot{x}^{2}+x=0$$ where $\dot{x}\equiv \frac{dx}{dt},\ddot{x}\equiv \frac{d^{2}x}{dt^{2}}, etc$. $A$ is a constant.
If we multiply this equation by $\...
- 213
7
votes
2
answers
901
views
Is this a new strange attractor?
I recently made some experiments in programming strange attractors, and I found this (very simple) equations, which create a nice strange attractor:
...
- 101
7
votes
2
answers
732
views
A second order nonlinear ODE
In my research (in differential geometry) I recently came across the following nonlinear second order ode:
$$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$
It ...
- 277
7
votes
2
answers
259
views
Can a periodically additively perturbed sinusoidal vector field on the circle have a stable periodic orbit of higher least period?
I have heard that differential equations on $\mathbb{S}^1$ of the form
\begin{equation} \hspace{40mm} \dot{\theta}(t) \ = \ A\sin(\theta(t)) + g(t) \hspace{4mm} \mathrm{mod} \ 2\pi, \hspace{40mm} (1) \...
- 708
7
votes
2
answers
318
views
Planar flow with bounded orbits and a single equilibrium point
Is there a $C^1$ flow $\varphi_t$ defined on ${\bf R}^2$ with a single fixed point $0$ and such that for all x,
$$\lim_{t\rightarrow +\infty}\varphi_t(x) = 0,$$
$$\lim_{t\rightarrow -\infty}\varphi_t(...
- 18.7k
7
votes
2
answers
641
views
Canard limit cycle for certain singularly perturbed system (Is there a contradictory situation?)
From the figures of page 478 and 479 of this paper one find that the author probably means that we have a (canard) limit cycle for the system
$$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) \...
- 356
7
votes
1
answer
930
views
(In)stability of a two-dimensional dynamical system
Consider the following system of coupled differential equations
\begin{eqnarray*}
\dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t)...
- 2,712
7
votes
2
answers
2k
views
The integral of torsion
I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced ...
- 356
7
votes
0
answers
369
views
On the solvability of a nonlinear differential system
A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system ...
- 79
6
votes
2
answers
701
views
On Wilson's claim that Lyapunov function level sets are not exotic spheres
In Wilson's paper "The structure of the level surfaces of a Lyapunov function," he states in Corollary 1.3 that the level sets of a smooth Lyapunov function are diffeomorphic to a standard sphere. (...
- 501
6
votes
2
answers
213
views
Continuity of the period for a periodic dynamical system
Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 1)$ a velocity field such that every solution $(x_t)_{t\geq 0}$ of $(d/dt)x_t=v(x_t)$ is periodic. Denote, for a non-stationary point $x\in\...
- 449
6
votes
1
answer
382
views
Convergence of dynamical system on the sphere
Let $A(x)$ be a symmetric negative semi-definite matrix which depends continuously on the parameter $x\in\mathbb{R}^{d}$. We consider the differential equation
$$\dot{x} = (I-xx^*)A(x)x$$
on the unit ...
- 799
6
votes
2
answers
862
views
A dynamical system defined by the Riemann zeta function
Let $\zeta$ be the classical Riemann zeta function.
We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector ...
- 356
6
votes
2
answers
277
views
Reference needed: $C^r$ convergence of Euler's method
Let $U\subset R^n$ be open, $F\colon U\to \mathbb{R}^n$ a $C^\infty$ vector field, and $x(t)$ the solution of
$$x’ = F(x)$$
with initial condition $x(0) = y$, which we assume defined at least for $t\...
- 491
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
6
votes
0
answers
342
views
Had this theorem in Tresser's article been proven somewhere?
The article in question is About Some Theorems by L.P. Sil'nikov by Charles Tresser. I am interested in the theorem C from page 453 and a particular application of such theorem which is illustrated ...
- 165
6
votes
0
answers
469
views
An algebraic Hamiltonian vector field with a finite number of periodic orbits (2)
Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...
- 356
6
votes
0
answers
201
views
The geometric shape of domains of flows
Consider a smooth (non-compact) manifold $M$ with a vector field $X$. Then we know that there is a open neighbourhood $U \subseteq M \times \mathbb{R}$ of $M \times \{0\}$ such that on $U$ the flow $\...
- 8,098
5
votes
1
answer
703
views
Updated background on Hilbert 16th problem?
What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
- 356
5
votes
2
answers
255
views
General term formulas for nonlinear recurrence sequences
It seems to be a well known question: in which cases will there be general term formulas for sequences like $p_n=a p_{n-1} ^2 +b p_{n-1} +c$ where $a, b, c$ are real or complex numbers and n is ...
- 51
5
votes
2
answers
442
views
Rotation number of composition
Let $f,g:S^1 \to S^1$ be orientation-preserving homeomorphisms. Consider the lift $F,G:\mathbb R \to \mathbb R$. Let $\rho(G)$ and $\rho(F)$ be a rotation numbers. What we can say about rotation ...
- 51
5
votes
1
answer
1k
views
Numerical calculation of Arnold tongue
Hello.
I am working on investigation of family of dynamical systems on the torus
$$\dot{x}=\cos(x)+b\cos(t)+a$$
$$\dot{t}=1$$
and it's Poincare map $$P:(x,0) \rightarrow (P(x),2\pi=0)$$
I need to find ...
- 53
5
votes
1
answer
164
views
A non-geodesible foliation of $S^3$ or $S^2\times S^1$
Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation?
If the answer is ...
- 356
5
votes
2
answers
647
views
Flow of a nowhere vanishing complete vector field
Let X be a nowhere vanishing complete vector field on a manifold M, $\gamma: \mathbb{R} \to M$ be its flow with $\gamma(0)=p \in M$ and suppose it is not periodic. If $\gamma(\mathbb{R})$ is closed, ...
- 103
5
votes
1
answer
212
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 807
5
votes
1
answer
205
views
Bias of DS literature to polynomial ODEs
In the literature on continuous time dynamical system, we generally deal with an open set $U \subset \mathbb{R}^n$ and a vector field $F: U \rightarrow \mathbb{R}^n$ and define a DS by the ODE
$$\...
- 247
5
votes
1
answer
177
views
Can you use Oseledet's theorem to numerically approximate the Lyapunov spectra?
Let's say you have a set of first order differential equations with known Jacobian $J$. Let $x_0, x_1, ..., x_n$ be sampled points on the trajectory near the attractor.
Let $T_n = J(x_{n-1})J(x_{n-2})....
- 188
5
votes
2
answers
256
views
Can a holomorphic vector field have an attractor homoclinic loop?
It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post
Orbits space of real-analytic planar foliations
One can ...
- 356
5
votes
3
answers
643
views
What quantities are conserved under a general gradient-flow $\dot X(t) = -\nabla L(X(t))$?
Let $L:\mathbb R^N \to \mathbb R$ be a continuously differential function with gradient $x \mapsto \nabla L(x)$ and consider induced gradient-flow
$$
\dot X(t) = -\nabla L(X(t)).
$$
Question. Is ...
- 6,853
5
votes
2
answers
258
views
Noninvariance for a specific nonlinear oscillator
Consider the nonlinear system
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix} x_2(t) \\ -4x_1(t) + x_1^2(t) \end{pmatrix},
\end{align*}
which admits ...
user45183
5
votes
1
answer
184
views
A non vanishing vector field on $S^3$ with a periodic attractor
Is there a non vanishing real analytic vector field $X$ on $S^3$ such that $X$ has an attractor periodic orbit(An asymptotically stable periodic orbit) ? What about the smooth case?
- 356
5
votes
1
answer
414
views
Fredholm index vs. Limit cycle theory
Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$.
Let $B $ be ...
- 356
5
votes
0
answers
263
views
Is the global solution to this ODE bounded?
Consider
$$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$
where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
- 405
5
votes
0
answers
173
views
Solve nonlinear, forced and damped Duffing oscillator
I am trying to solve a Duffing type equation by using Van Der Paul's method:
\begin{align}
\ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t)
\end{align}
with $$x(t) = Re[A(t) \...
- 69