All Questions
Tagged with ds.dynamical-systems differential-equations
216 questions
2
votes
0
answers
108
views
Does a smooth dynamical system always come with a metric
Warning: My education in formal mathematics is very weak so I apologize for any confusions/errors in the following, please don't hesitate to correct me.
Question: Consider a smooth dynamical system $...
- 21
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
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
$$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) \...
- 356
10
votes
2
answers
350
views
Is this Riccati equation ("Josephson junction") always phase-locked at integer rotation numbers?
Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE
$$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$
Identifying ...
- 708
2
votes
2
answers
296
views
Planar polynomial vector field for a harmonic pair of polynomials
Has the system of ODEs
$$\frac{dx}{dt}=P(x,y)\\
\frac{dy}{dt}=Q(x,y)
$$
been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...
- 7,127
5
votes
2
answers
255
views
General term formulas for nonlinear recurrence sequences
It seems to be a well known question: in which cases will there be general term formulas for sequences like $p_n=a p_{n-1} ^2 +b p_{n-1} +c$ where $a, b, c$ are real or complex numbers and n is ...
- 51
7
votes
2
answers
259
views
Can a periodically additively perturbed sinusoidal vector field on the circle have a stable periodic orbit of higher least period?
I have heard that differential equations on $\mathbb{S}^1$ of the form
\begin{equation} \hspace{40mm} \dot{\theta}(t) \ = \ A\sin(\theta(t)) + g(t) \hspace{4mm} \mathrm{mod} \ 2\pi, \hspace{40mm} (1) \...
- 708
1
vote
1
answer
289
views
Center-localized oscillating modes with exponential decay tails, solved from coupled ODE
Two coupled non-linear differential equations in a radial $r$-direction in the region $r \in [0, \infty)$:
$$
-a\big(\partial_r^2+\frac{\partial_r}{r}-\frac{n^2}{r^2}+c\big) U(r)+
B(r) (\partial_r-...
- 10.5k
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
4
votes
2
answers
196
views
Polynomial vector field tangent to a given analytic simple closed curve
Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{R}^2$ which surrounds origin.
Is there a polynomial vector field on the plane which is tangent to $\gamma$? In the other word, ...
- 356
1
vote
1
answer
89
views
The entire parametrization of leaves of singular holomorphic foliation of $\mathbb{C}P^2$
What is an example of an entire non constant holomorphic function $\gamma: \mathbb{C} \to \mathbb{C}P^2$ such that the image of $\gamma$ is a leaf of a singular holomorphic foliation of ...
- 356
0
votes
1
answer
178
views
Closed orbit for vector field $f(\bar{z})$ where $f$ is holomorphic function
Edit : According to the comments of Michael Renardy and Christian Remling I revise the question as follows:
Is there a vector field $X$ on an open set $U\subseteq \mathbb{R}^2$ such ...
- 356
2
votes
1
answer
55
views
The number of limit cycles of a quadratic vector field with a unique singularity
Is there a uniform upper bound for the number of limit cycles of a quadratic vector field which has a unique singular point in the plane?
- 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
2
votes
1
answer
296
views
Isochronization of quadratic vector fields with center
What is a classification of all quadratic vector fields
$$\begin{cases}
x'=P(x,y)\\
y'=Q(x,y)
\end{cases}\qquad (V)$$
with a center at origin such that $$\left(\frac{x^2+y^2}{yP(x,y)-xQ(x,y)}\...
- 356
3
votes
1
answer
195
views
An explicit formula for a flat metric compatible to certain polynomial vector field with center
Let $X$ be the following vector field on the plane:
$$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$
The vector field $ (X)$ has a non isochronous center at the origin.The ...
- 356
-1
votes
1
answer
95
views
transforming a Ricatti equation into a generalised Ricatti equation [closed]
C̶o̶n̶s̶i̶d̶e̶r̶ ̶a̶ ̶R̶i̶c̶a̶t̶t̶i̶ ̶e̶q̶u̶a̶t̶i̶o̶n̶ ̶o̶f̶ ̶t̶h̶e̶ ̶f̶o̶r̶m̶
$$ y' + y^2 = S(x), \qquad \qquad \qquad (1)$$
w̶h̶e̶r̶e̶ ̶$̶S̶(̶x̶)̶$̶ ̶i̶s̶ ̶a̶ ̶m̶e̶r̶o̶m̶o̶r̶p̶h̶i̶c̶ ...
- 347
2
votes
1
answer
65
views
Boundedness of particle motion with time-varying force
Consider the differential equation
$$ m \ddot{x} + k \dot{x} = - W_t x $$
where
$m$ and $k$ are nonnegative.
$x_t \in \mathbb{R}^n$
$W_t$ is a matrix that satisfies $$ \alpha I \succeq W_t \...
- 23
3
votes
0
answers
139
views
Two semi stable limit cycles with disjoint interior
What is a precise example of a quadratic vector field on the plane with at least one semi stable limit cycles?
Furthermore, is there a quadratic polynomial vector field on the plane with two ...
- 356
2
votes
0
answers
236
views
A cubic system with two nested limit cycles with opposite orientations(2)
The second part of Hilbert's 16th problem not only concerns "The number of limit cycles of a polynomial vector field", but also the position and configuration of of those limit cycles with respect to ...
- 356
3
votes
0
answers
165
views
Flat Riemannian metrics adapted to quadratic vector fields with center
Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$
Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
- 356
4
votes
1
answer
366
views
A cubic system with two nested limit cycles with opposite orientations
What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C_1,C_2$ of the system surrounds an annular region $R$ such that $...
- 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
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
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
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
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
0
answers
309
views
Is the closed orbit of the Van der Pol equation a stable periodic orbit?
We consider the Van der Pol vector field $$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$$ on $\mathbb{R}^2.$
It is well known that this equation has a unique limit ...
- 356
5
votes
0
answers
281
views
Basin of attraction of gradient flow
Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \...
- 151
6
votes
0
answers
342
views
Had this theorem in Tresser's article been proven somewhere?
The article in question is About Some Theorems by L.P. Sil'nikov by Charles Tresser. I am interested in the theorem C from page 453 and a particular application of such theorem which is illustrated ...
- 165
4
votes
1
answer
541
views
A vector field on the tangent bundle which is not equivalent to any second order ODE
A second order differential equation on a manifold $M$ is a vector field $X$ on $TM$ which is not only a section of the vector bundle $T(T(M)) \to TM $ with the obvious structure, ...
- 356
2
votes
0
answers
226
views
Geometric ergodicity of dynamical system
I'm working with dynamical systems defined by ODEs and SDEs, in this latter case gradient systems in particular, a special case of Ito diffusions.
I've read that under reasonable assumptions this ...
- 205
6
votes
2
answers
862
views
A dynamical system defined by the Riemann zeta function
Let $\zeta$ be the classical Riemann zeta function.
We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector ...
- 356
3
votes
1
answer
202
views
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
Consider the following autonomous system of differential equations:
$$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$
where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...
- 884
2
votes
0
answers
129
views
Is the interpolating Hamiltonian flow of an exact near-identity symplectic map globally defined?
It is well-known that an analytic near-identity map $\bar{x} = F_{\epsilon}(x) = x + \epsilon f(x) + O(\epsilon^{2})$ may be embedded into the flow of a differential equation, and if that map is ...
- 21
2
votes
0
answers
211
views
A particular case of of the higher dimensional Poincare Bendixson theorem
We consider the planar polynomial vector field $$(*) \;\;\;\begin{cases} \dot x= P(x,y) \newline \dot y =Q(x,y)\end{cases}$$
We replace the real variables $x,y$ with complex variables $x:=x_{1}+...
- 356
3
votes
1
answer
540
views
The Matrix form of the Van der Pol equation
Motivated by the classical Van der Pol equation which has a unique periodic attractor, we consider the following differential equation on $M_{2}(\mathbb{R})\times M_{2}(\mathbb{R}):$
$$(*)\;\;\;\...
- 356
1
vote
0
answers
70
views
What's best result for normal form theory of non-autonomous dynamical system [closed]
Given an autonomous dynamical system, we could find its rest points and then try to understand the grems of the vector field on the neighborhoods of the rest points. In particular, there is the famous ...
- 71
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
2
votes
0
answers
191
views
Geometric properties of solutions of Hamiltonian system
Context : We are interested in the following dynamic with state $(q,\varphi)$
$$
\dot q = \varepsilon F(q,\varphi), \quad \dot \varphi = \omega(q) + \varepsilon G(q,\varphi)
$$
($\varepsilon >0$ ...
- 141
3
votes
1
answer
247
views
When is a limit cycle generated by a Hamiltonian oval stable?
Consider a real polynomial $H$ of degree $n+1$ in the plane. A closed, connected component of a level curve $H=t$ is denoted by $\gamma(t)$ and called an oval of $H$. Let $\omega$ be a real 1-form ...
- 31
2
votes
1
answer
253
views
Differential inequalities for a strictly diagonal dominant system of linear ODEs
Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...
- 135
4
votes
2
answers
197
views
Periodic orbits in the plane
Consider a vector field $F:\mathbb{R}^2\rightarrow \mathbb{R}^2$ of the following form $F(y_1,y_2)=(y_2,\mu(y_1))$, where $\mu\in\mathscr{C}^1(\mathbb{R})$ has appropriate growth so that the solutions ...
- 3,425
1
vote
1
answer
92
views
Is this non-linear system of differential equations tractable by other means than numeric approximation and dynamic analysis?
Is there any way to solve the following system of non-linear differential equations exactly?
$x'(t) = x\times(y - \frac{1}{3(t + C)})$
$y'(t) = -\frac{1}{3}x^2 - \frac{y}{t + C}$
Here $x$ and $y$ ...
- 113
-4
votes
1
answer
872
views
Existence and uniqueness of solutions for a system of first order PDEs [closed]
Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs:
A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{...
- 103
1
vote
1
answer
268
views
Is there an entire solution for the Van der pol equation?
Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation?
$$\begin{cases}\dot{x}=y-x^{3}\\\dot y=-...
- 356
4
votes
0
answers
116
views
Dynamics of pairwise distances in the $n$-body problem
Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
...
4
votes
0
answers
149
views
Connection between cardiac equations and untangling knots?
I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul Sutcliffe. ...
- 151k
0
votes
1
answer
88
views
underdamped oscillation with quadratic decay
I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form:
...
- 169
0
votes
1
answer
66
views
What is the relationship between solutions for the parameterised second order differential equations
Let us consider the following parameterised complex-valued second order differential equations, and $u(x,\lambda)$ be the solution for
$$
u''+u'-i\lambda V(x)u=0, \, x\in [0,1],
$$
What is the ...
- 407