All Questions
Tagged with differential-equations ds.dynamical-systems
216 questions
5
votes
1
answer
212
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 807
4
votes
3
answers
288
views
A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit
Is there a polynomial vector field
$$P(x,y,z)\partial_x+Q(x,y,z)\partial_y+R(x,y,z)\partial_z$$
which has a closed orbit $K$ such that $K$ is a non trivial knot?
- 356
1
vote
0
answers
54
views
Are total curvature and the unknoting number of closed orbits of algebraic vector fields bounded uniformly by the degree of vector field?
I am interested in this question since 1999 when I heared the definition of a knot and I read the definition of unknoting and the total curvature of a knot.
To what extent can closed ...
- 356
2
votes
1
answer
153
views
What conditions on the rate matrix $Q$ ensure unique convergence in continuous-time Markov chains?
In the study of discrete-time Markov chains, the conditions under which all initial distributions converge to a unique stationary distribution are well-understood. Specifically, if the transition ...
- 807
3
votes
0
answers
183
views
Bounded solutions of nonlinear third-order ODEs
I am interested in understanding the behavior of solutions to certain nonlinear third-order ODEs. Specifically, I am curious about conditions that guarantee all solutions remain bounded for $t \in [0, ...
- 807
1
vote
0
answers
108
views
Stability of rigid bodies spinning around $z$-axis under gravity
Consider the problem of a rigid body rotating in 3D space under gravity with one point fixed. I am particularly curious about the equilibrium state where the body is spinning at a constant angular ...
- 807
0
votes
1
answer
77
views
Same occupation measure $\Rightarrow$ same trajectory
Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
The occupation ...
- 247
2
votes
0
answers
136
views
Progess on conjectures of Palis
I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...
- 247
5
votes
0
answers
263
views
Is the global solution to this ODE bounded?
Consider
$$\dot{\theta_i}=-\sum_{j=1}^nA_{ij}\sin(\theta_i-\theta_j),\ i\in\{1,2,\cdots,n\}$$
where $A_{ij}$ is adjacency matrix of a connected simple graph, and the vector $\theta=[\theta_1,\cdots,\...
- 405
3
votes
0
answers
50
views
Stability of indefinitely damped mechanical system with diagonal stiffness
I'm trying to find conditions for the asymptotic stability of the following linear system,
\begin{equation}
\mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0
\end{equation}
given the ...
0
votes
2
answers
69
views
Is the right-hand term of the dynamic system equivalent to the original system after being multiplied by a constant?
Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x,t),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z,t),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, ...
- 109
1
vote
0
answers
64
views
Physical measure of a dynamical system in terms of its density
Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by
$$\dot{x}(t)=f(x(t))$$
In ergodic theory, the occupation measure is
$$\mu_{x, T}(...
- 247
0
votes
0
answers
58
views
Role of basins of attraction in the Morse decomposition
Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by
$$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
- 247
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
$$\...
- 247
0
votes
0
answers
303
views
Proof that a first integral is not a constant function
Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions
$$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$
such that all of them are differentiable and ...
- 247
0
votes
0
answers
90
views
Numerical detection of Cantori
It is known that as parameters vary in Hamiltonian system, KAM tori can break [1,2].
How to construct numerically the breaking tori?
The most relevant paper that I could find is [3,4].
But it uses ...
- 593
0
votes
0
answers
70
views
Example of DS with a dense trajectory in the whole state space
Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure)
$$\dot{\mathbf{...
- 247
4
votes
1
answer
235
views
Dynamical analogue of Morse theory
Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property:
For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X_H$ on $H^{...
- 356
2
votes
1
answer
139
views
Can a chaotic trajectory solve an algebraic equation?
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$\dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n$$
we ...
- 247
1
vote
0
answers
55
views
Periodic Orbit without Complex Eigenvalues
I am studying the following ODE system, representing a simple excitable circuit:
$$
\dot{V}_m = I_{app} - (V_m - \alpha_f PL(V_m) + \alpha_s PL(V_s))
$$
$$
\tau_s \dot{V}_s = V_m - V_s
$$
where
$$
PL(...
21
votes
0
answers
416
views
Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
- 281
3
votes
1
answer
134
views
Analyticity of central stable manifolds
Let $X$ be a real analytic vector field defined on $\mathbb{R}^n$. Assume the origin $0 \in \mathbb{R}^n$ is a zero of $X$. Assume, furthermore, that we know that the center-stable manifold (in the ...
- 1,409
3
votes
1
answer
253
views
Stability analysis of equilibrium point of non-linear ODE system with Jacobian going to infinity
Let's say I have a nonlinear system of ODEs, where one of equations looks like:
$$
\frac{dX_i}{dt} = a_1X_0+\dotsb+a_j\frac{X_j^{0.5}}{X_j^{0.5}+b_j^{0.5}}+\dotsb.
$$
And equilibrium point is 0. I ...
- 31
0
votes
0
answers
103
views
Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity
Problem:
Consider the autonomous ODE system
\begin{align*}
\dot{x} &= (1-x) (z-xy)\\
\dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\
\dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z
\end{...
- 291
2
votes
0
answers
72
views
On bounded solutions of a given fourth-order linear ODE
Consider the fourth-order linear ODE
$$
\label{eq1}
v^{(4)} + \frac{-C_2 - 2\alpha \phi}{C_4}v'' + \frac{4\alpha \phi'}{C_4}v' + \frac{k_1 + 3k_3\phi^2 -2\alpha \phi''}{C_4}v = 0.
$$
Without getting ...
- 121
3
votes
0
answers
101
views
Turing reaction diffusion equations and neural networks
Suppose you have a Turing-type reaction-diffusion system
$$
\begin{cases}
\partial_t \phi = & f(\phi, \psi) + D_\phi \nabla^2\phi \\
\partial_t \psi = & g(\phi, \psi) + D_\psi \nabla^2\psi
\...
1
vote
0
answers
48
views
Smoothness of unstable manifold near (non?)-hyperbolic fixed point w.r.t. generator of the flow
Consider a dynamical system $\dot{x} = f(x,p)$, where $p \in R^n$ is a bunch of parameters and $f$ is a smooth function, both in $x$ and $p$. Given an initial value problem where $x(0) = x_0$, it can ...
- 11
3
votes
0
answers
93
views
Regularity of center manifold
Consider a $C^r$ vector field $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $r \geq 1$. Let $\bar x$ be a critical point of $f$, that is, $f(\bar x) = 0$.
Suppose that the spectrum of $\mathrm{D}f(\...
- 31
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
1
vote
0
answers
53
views
Basin of attraction comparative statics* using local energy functions?
Let $\dot{\boldsymbol{x}}=\boldsymbol{F}(\boldsymbol{x};p)$ be an autonomous dynamical system defined on $[a,b]^n$ ($-\infty<a<B<\infty)$; $p\in\mathbb{R}$ is some fixed parameter. Suppose ...
1
vote
1
answer
151
views
Analytically characterizing basins of attraction boundaries and sizes
While I understand that doing the above is not possible in general, I would like to know more about how to proceed when it is possible. That is, what are the common methods people use to analytically ...
3
votes
2
answers
415
views
Asymptotic behavior of system of differential equations
Let $f:[0,1]^2 \rightarrow \mathbb{R}^2,$ where $f_1(x,y) = g(y)-x$ and $f_2(x,y) = g(x)-y.$ Here $g(\cdot)$ is a strictly decreasing polynomial function such that $g(0)=1$ and $g(1)=0.$ I am ...
- 63
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
2
votes
0
answers
167
views
Why were these constants picked in this Lyapunov function and how did the author arrive at the final form of the Lyapunov function?
Consider the following paper:
"A note on global stability for a tuberculosis model" by Gao and Huang: https://doi.org/10.1016/j.aml.2017.05.004
The methodology is understood in this paper ...
- 185
3
votes
0
answers
74
views
A foliation version of S.Husseini counter example in fixed point theory
In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977
"The Products of Manifolds with the f.p.p. Need Not have the f.p.p"
who gave an example of two ...
- 356
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
0
votes
0
answers
72
views
$\mathbb{R}^n$-flow, cross-section and Whitney theorem
For a $\mathbb{R}$-flow (X, $\Phi_{\mathbb{R}}$), the (local) cross-section is well defined (recall that a subset $S\subset X$ is a cross section of time $\xi>0$ if $S\cap \Phi_{[-\xi, \xi]}(x)=\{x\...
- 675
1
vote
0
answers
184
views
Lie group flows [closed]
I recently came across a way to think of an ordinary differential equation on a smooth manifold $M$ is as a Lie group homomorphism $\phi : (\mathbb{R}, +) \rightarrow \operatorname{Diff}(M)$ where $\...
- 11
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
1
vote
1
answer
657
views
Finding the eigenvalues and eigenvectors of Jacobian at equilibrium point of nonlinear ODEs
Consider the vector field $V:\mathbb{R}^4\rightarrow\mathbb{R}^4$, defined by
\begin{equation}
V(x,v,M_0,M_1)=(v,\kappa^{-1}(\beta M_0-v-kx),-M_0+v M_1,-M_1+1-vM_0),
\end{equation}
such that $\...
- 79
1
vote
0
answers
67
views
Solution to recurrence relation from integro-differential dynamical system?
Consider the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=2\int_0^t J_1(x_t-x_s)e^{-\epsilon(t-s)}ds.\tag{1}
\end{equation}
such that $\kappa,\epsilon\in\mathbb{R}$, $t\in\...
- 79
2
votes
0
answers
153
views
Stability test for LTV systems by differential Lyapunov inequalities
Consider a linear time-varying system:
\begin{equation}
\dot x(t) = A(t) x(t), \tag{$*$}
\end{equation}
where $A(t)$ is a time-varying block matrix defined as
$$
A(t) =
\begin{bmatrix}
0 & I\\
-\...
- 55
6
votes
2
answers
213
views
Continuity of the period for a periodic dynamical system
Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 1)$ a velocity field such that every solution $(x_t)_{t\geq 0}$ of $(d/dt)x_t=v(x_t)$ is periodic. Denote, for a non-stationary point $x\in\...
- 449
0
votes
0
answers
58
views
The solutions of a system of differential equations
Let $P(x,y) = \frac{x}{y}^{\frac{x^2}{y-x}}$ for $x \neq y$ and using the proper limits $P(x,y)=e^{-x}$ for $x=y $, $P(x,y)=0$ for $x\neq0, y=0,$ and $P(x,y)=1$ for $x=0, y\neq0.$
Consider this system ...
- 31
20
votes
3
answers
690
views
Escaping from infinitely many pursuers
The fugitive is at the origin. They move at a speed of $1$. There's a guard at $(i,j)$ for all $i,j\in \mathbb{Z}$ except the origin. A guard's speed is $\frac{1}{100}$. The fugitive and the guards ...
- 2,619
5
votes
1
answer
177
views
Can you use Oseledet's theorem to numerically approximate the Lyapunov spectra?
Let's say you have a set of first order differential equations with known Jacobian $J$. Let $x_0, x_1, ..., x_n$ be sampled points on the trajectory near the attractor.
Let $T_n = J(x_{n-1})J(x_{n-2})....
- 188
7
votes
2
answers
318
views
Planar flow with bounded orbits and a single equilibrium point
Is there a $C^1$ flow $\varphi_t$ defined on ${\bf R}^2$ with a single fixed point $0$ and such that for all x,
$$\lim_{t\rightarrow +\infty}\varphi_t(x) = 0,$$
$$\lim_{t\rightarrow -\infty}\varphi_t(...
- 18.7k
1
vote
0
answers
61
views
Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?
Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties?
The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...
- 356
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
1
vote
0
answers
62
views
Lyapunov theory in coupled nonlinear dynamic system with input
Suppose I have the following nonlinear coupled dynamic system
\begin{align*}
&\dot{x}_1 = f_1(x_1,x_2)\\
&\dot{x}_2 = f_2(x_2) + u
\end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{...
- 345