All Questions
Tagged with ds.dynamical-systems differential-equations
216 questions
2
votes
0
answers
86
views
Gronwall-type bound for a mix-effect inequality?
This popped up in my research: we have the following mix-effect inequality that $\forall T \geq 1$
\begin{equation}\tag{*}
Y(T) - \frac{1}{100T^2}\int_1^T[\alpha^2 + e^{-(T - t)}]Y(t)dt
\lesssim \...
- 719
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
votes
1
answer
176
views
How do we approximate the pressure in the Boussinesq equations of hydrodynamics? [closed]
How do we approximate the pressure or the gradient of it in the Boussinesq equations of hydrodynamics ?
Is the pressure limited or can it be any amount?
- 11
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
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
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
1
answer
289
views
Proving positive invariance
I need to prove that set $D$(A picture for Set $D$) given by
$$D=\{(x,y):0\leq x\leq L_0,~0\leq y\leq X_0,~0\leq x+y \leq R_0\}\subseteq \mathbb{R}_+^2$$ of the system:
$$\dot{x}=k_1(R_0-x-y)(L_0-x)-...
- 11
2
votes
1
answer
160
views
Dynamical system described by coupled nonlinear differential equations
Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled nonlinear differential equations:
\begin{equation}
\begin{split}
...
- 197
1
vote
0
answers
36
views
Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms
Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...
- 5,405
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
2
votes
1
answer
239
views
Seeking a Lyapunov function for a SIR model with immunity loss
We add the immunity loss to the SIR model and obtain the following autonomous system.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\...
- 2,239
0
votes
1
answer
155
views
Conditions to determine sign of real roots
From a delay system, I obtain the following as part of a characteristic equation:
$$f(\lambda) = \lambda - a + be^{-c\lambda},$$
where $a, b,$ and $c$ are positive number and $a<b, ac<1$. My ...
- 513
7
votes
2
answers
901
views
Is this a new strange attractor?
I recently made some experiments in programming strange attractors, and I found this (very simple) equations, which create a nice strange attractor:
...
- 101
0
votes
0
answers
87
views
What kind of differential equation problem is this?
I have a function $f(x,t;k)$, a starting point $x_0$, a gradient $\operatorname{Grad}(f)$, and an equilibrium point $x^*$. I can adjust the parameter $k$ freely, and I know that for any $k$ the ...
2
votes
1
answer
158
views
A vector field $X$ on $\mathrm{GL}(n,\mathbb{R})$ with $\begin{cases} X.\mathrm{trace}=\mathrm{Det} \\X.\mathrm{Det}=-\mathrm{trace} \end{cases}$
Is there a vector field $X$ on $\operatorname{M}_n(\mathbb{R})$ or $\operatorname{GL}(n,\mathbb{R})$ with the following condition:
$$\begin{cases} X\cdot \operatorname{trace}=\operatorname{Det} \\X\...
- 356
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
0
votes
1
answer
139
views
flow, stable manifold and tangent
Given vector field $f: \mathbb{R}^2 \to \mathbb{R}^2$, with $f(0)=0$
ODE: $\dot{x}=f(x)$ generates a flow $\Phi^{t}$. so $\Phi^{t}(0)=0$ for all $t \in \mathbb{R}$
So time-one map $\Phi^1$ is diffeo....
- 553
2
votes
1
answer
133
views
Global first integral for certain $3$ dimensional system
A physicist colleague asks me the following question. I have no idea to answer him. Your answer is very appreciated.
Is there a global first integral on $\mathbb{R}^3$ for the following vector field?
...
- 356
4
votes
1
answer
247
views
Integrability/regularity of Lyapunov exponents
My question is about some basic properties of Lyapunov exponents. Sorry if this stuff is basic, I just don't know where to find the statements that I'm looking for.
Preliminaries. Let $X$ be a closed ...
- 1,231
2
votes
0
answers
44
views
Understand the condition of transcritical bifurcation (Crandall-Robinowitz) geometrically
Consider the dynamical system $\dot{x}=F(x,\lambda), x\in\mathbb{R^n}$, and let $F(0,\lambda)=0$ for some neighborhood of $\lambda_{0}$, the transcritical bifurcation arises if we have $w\frac{\...
- 135
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
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
2
votes
1
answer
123
views
Keeping track of limit cycles via certain second order differential operator
Inspired by the two posts which are linked bellow we ask the following question:
Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
- 356
1
vote
1
answer
160
views
A singular holomorphic foliation of $\mathbb{C}^2$ with a bounded leaf
Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.
- 356
4
votes
0
answers
100
views
Flow lines of a real analytic vector field convergent to a point
Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ ...
- 1,409
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
1
vote
1
answer
136
views
Trajectory leaving a set
Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...
- 143
2
votes
1
answer
132
views
A special oscillatory orbit in space
Edit: According to the comment of Prof. Eremenko I revise the question.
19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...
- 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
2
votes
0
answers
150
views
Global solution of second order ODE defined on riemannian manifold
Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the ...
- 335
2
votes
0
answers
149
views
Can a local extremum of a function be an asymptotically stable equilibrium of corresponding gradient dynamics?
Let's first describe the setup: we consider a(say smooth enough) function $f: \mathbb{R}^d \to \mathbb{R}$ and write it as $(x,y) \to f(x,y)$, where $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
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
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
2
votes
0
answers
105
views
Bifurcations due to a nonlinearity parameter
Suppose we want to analyze the behavior of the system
$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,
$$
...
- 243
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:
$$ \mathcal{M}(x, \lambda) := x - f(x, \lambda_1, \...
- 213
2
votes
1
answer
155
views
Lotka Volterra existence of Caratheodory solution
I strive to prove that the following system of differential equations:
$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$
has a unique Caratheodory solution ...
- 1,759
0
votes
1
answer
168
views
Does differentiating an integro-differential equation results in equivalent stability of the solution?
I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation:
...
5
votes
1
answer
703
views
Updated background on Hilbert 16th problem?
What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
- 356
5
votes
0
answers
140
views
Algebraic independence of limit cycles of Lienard equation
It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle.
According to this fact, we search for a related ...
- 356
7
votes
1
answer
930
views
(In)stability of a two-dimensional dynamical system
Consider the following system of coupled differential equations
\begin{eqnarray*}
\dot{x}_1(t) & = & -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t)...
- 2,712
5
votes
0
answers
234
views
Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$
I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...
- 333
3
votes
0
answers
194
views
If the sum of everywhere linearly independent vector fields are periodic, are the component vector fields periodic?
I feel like the above must be true but embarrassingly cannot seem to prove it. Take linearly independent, commuting vector fields $X$ and $Y$ on a manifold and corresponding flows $\Phi^t_X$, $\Phi^...
- 979
2
votes
0
answers
190
views
Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems
I have a query regarding two equalities in the lemma in the book.
But first I'll provide two definitions that one needs for this lemma.
Definition 4.2.4: Consider the vector field $f(x,t)$ with $f:\...
- 1,594
2
votes
0
answers
59
views
Stability of ODEs with exponentials in the vector field
What is known about fine stability properties of ODEs of the following kind?
$$ \dot{x} = Ax + b + \phi(x),\quad x\in \mathbb{R}^d ,$$
where $d\geq 1$; $A$ is a constant matrix with all e.v. having ...
- 225
1
vote
2
answers
222
views
Behavior of a non-linear differential equation
Let us consider the following differential equation
$$
\dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}.
$$
My question. Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the ...
- 2,712
3
votes
2
answers
264
views
ODE with Holder drift - Cauchy-Peano theorem
Consider the following ODE:
$$
x′(t)=b(x(t)),\quad x(0)=x_0.
$$
If $b$ is bounded and Holder continuous, then the Cauchy-Peano theorem ensures that there exists a solution to the above equation (but ...
- 411
2
votes
1
answer
134
views
On local attractivity of a coupled non-linear differential equation
Consider a dynamical system described by the following coupled non-linear differential equation
\begin{align}
\dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\
\dot{x}_2(...
- 2,712
1
vote
0
answers
95
views
A singular foliation analogy of the Riemann Hilbert problem
Note:
In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$.
...
- 356
2
votes
0
answers
59
views
Region of attraction of simple ODE with perturbation
Consider the following simplest example:
$$\dot{x} = x(x-1)(x+1)$$ $[-1,1]$ is the ROA.
Now consider the two dimensional case:
\begin{equation}
\begin{aligned}
&\dot{x} = x(x-1)(x+1)\\
&...
- 345
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