All Questions
Tagged with differential-equations ds.dynamical-systems
85 questions with no upvoted or accepted answers
2
votes
0
answers
427
views
Lifting a quadratic system to a non-vanishing vector field on $S^{3}$ or $T^{1} S^{2}$
Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non-vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
- 356
2
votes
0
answers
124
views
Rotation number of perturbated equation
I have a differential equation on torus $(t,x)$ and well studied it's Arnold tongues for Poincare map of the circle $x(t=0) \to x(t=2\pi)$. The question is how changes rotation number when I add small ...
- 39
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
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
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
$$\mu_{x, T}(...
- 247
1
vote
0
answers
55
views
Periodic Orbit without Complex Eigenvalues
I am studying the following ODE system, representing a simple excitable circuit:
$$
\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(...
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
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
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
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
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
1
vote
0
answers
99
views
Long-term behavior of asynchronous, stochastic, numerical solution to a dynamical system
I am simulating the behavior of a dynamical system, say $$\dot{x} = f(Ax; \lambda), $$
with an Euler update, where $x\in \mathbb{R}^n$ and $\lambda$ are some parameters. In my scenario, $A\in \mathbb{...
- 131
1
vote
0
answers
77
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
1
vote
0
answers
44
views
Sufficient conditions for the continuity of an improper integral concerning the finite-time stability of a dynamical system
Consider the initial value problem
\begin{equation}\label{fainait ve}
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)), \;\; t \geq 0, \; \;\boldsymbol{f}(\boldsymbol{0}_n) = \boldsymbol{0}...
- 81
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
$$
...
- 5,405
1
vote
0
answers
38
views
A generalization of competitive systems
We consider the following standard partial order relation on $\mathbb{R}^n$:
We say $X=(x_1,x_2,\ldots,x_n)\leq (y_1,y_2,\ldots,y_n)=Y$ iff $\sum_{i=1}^k x_i \leq \sum_{i=1}^k y_i,\quad \forall k: 1\...
- 356
1
vote
0
answers
114
views
Final time maps of IVP's approximating functions $X\subseteq\mathbb{R}^n\to\mathbb{R}^n$
I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently:
Let $X$ be a compact $k$-...
- 121
1
vote
0
answers
80
views
Solutions of nonlinear equations with multiple parameters
In the course of analysing a particular three dimensional nonlinear dynamical system, I find the need to solve a nonlinear equation of the form:
$$ \mathcal{M}(x, \lambda) := x - f(x, \lambda_1, \...
- 213
1
vote
0
answers
95
views
A singular foliation analogy of the Riemann Hilbert problem
Note:
In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$.
...
- 356
1
vote
0
answers
276
views
Stability when linearization fails
The dynamics of the $j$th system:
\begin{equation}
\begin{split}
\dot{\overline r}_j &= h (\overline r_j)
\,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...
- 33
1
vote
0
answers
243
views
A (different) foliation arising from Hopf fibration
In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...
- 356
1
vote
0
answers
179
views
Smooth normal forms of vector fields (the path method)
I start by considering a polynomial vector field $$F=\varepsilon\frac{\partial}{\partial x}-(z^2+x)\frac{\partial}{\partial z}+0\frac{\partial}{\partial \varepsilon}.$$ Next I define a perturbation of ...
- 231
1
vote
0
answers
81
views
Single parameter bifurcations caused by a simple additive term
Note: I asked this question on Math.SE over two months ago, and it has not received any answers.
Motivation: A practical dynamical system is often described by an ODE that has a parameter that ...
1
vote
0
answers
65
views
Id monodromy in hamiltonian dynamics
In my problem I have non autonomous Hamiltonian which depends on 2 parameters (pretty close to oscillator Hamiltonian, $(a+b\cos t +1) p^2+(a+b\cos t-1)q^2$, $a,b$ - parameters). From numerical ...
- 11
1
vote
0
answers
462
views
Partial feedback linearization (Control theory)
I'm trying to understand a theorem about partial feedback linearization from the paper "On the largest feedback linearizable subsystem" by R. Marino (published in: Systems & Control Letters, ...
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
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
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
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
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
0
votes
0
answers
87
views
What kind of differential equation problem is this?
I have a function $f(x,t;k)$, a starting point $x_0$, a gradient $\operatorname{Grad}(f)$, and an equilibrium point $x^*$. I can adjust the parameter $k$ freely, and I know that for any $k$ the ...
0
votes
0
answers
320
views
Prove that origin is globally exponentially stable with Lyapunov Indirect Method
I'm wondering, if we have a nonlinear system governed by
$\dot{x} = Ax + g(x)$ where $||g(x)|| \leq \gamma ||x||^2$ and A is Hurwitz
how can we show that the origin is globally exponentially stable?...
- 11