All Questions
Tagged with differential-equations ds.dynamical-systems
216 questions
3
votes
0
answers
130
views
Approximating solutions of non-linear differential equations
I have met a system of non-linear equations as follows,
$$\frac{\mathbb{d}y_k}{\mathbb{d}t}=-(1-\alpha)y_k\sum_s{s^az_s}-\alpha y_kz_k,$$
$$\frac{\mathbb{d}z_k}{\mathbb{d}t}=(1-\alpha)y_k\sum_s{s^az_s}...
- 181
3
votes
0
answers
2k
views
Bessel functions in wave propagation and scattering
Is there a way to scale $J_n(\cdot)$ (Bessel of first kind) and $H_n(\cdot)$ (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher values of n) and small arguments....
- 41
3
votes
0
answers
1k
views
(Approximate) analytic solutions to the Mathieu equation
I'm trying to solve the driven Mathieu equation
$x''+\beta x'+(a-2q\cos{\Omega t})\frac{\Omega^2}{4}x=f(t)$
for both zero and non-zero $\beta$.
I can write down an analytic solution using the ...
- 31
2
votes
2
answers
296
views
Planar polynomial vector field for a harmonic pair of polynomials
Has the system of ODEs
$$\frac{dx}{dt}=P(x,y)\\
\frac{dy}{dt}=Q(x,y)
$$
been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...
- 7,127
2
votes
1
answer
158
views
A vector field $X$ on $\mathrm{GL}(n,\mathbb{R})$ with $\begin{cases} X.\mathrm{trace}=\mathrm{Det} \\X.\mathrm{Det}=-\mathrm{trace} \end{cases}$
Is there a vector field $X$ on $\operatorname{M}_n(\mathbb{R})$ or $\operatorname{GL}(n,\mathbb{R})$ with the following condition:
$$\begin{cases} X\cdot \operatorname{trace}=\operatorname{Det} \\X\...
- 356
2
votes
1
answer
196
views
Is autonomous dynamical system equivalent to one single higher-order ode?
We know that a higher-order ode can be converted to dynamical system by replacing each higher-order derivative by a new variable. What about inverse problem? Does a dynamical system convert to a ...
- 280
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
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
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
2
votes
2
answers
494
views
Making a system of second-order ODEs chaotic
Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions $X = (x_1, x_2, ..., x_N)$, we have
$...
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
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.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\...
- 2,239
2
votes
1
answer
132
views
A special oscillatory orbit in space
Edit: According to the comment of Prof. Eremenko I revise the question.
19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...
- 356
2
votes
1
answer
134
views
On local attractivity of a coupled non-linear differential equation
Consider a dynamical system described by the following coupled non-linear differential equation
\begin{align}
\dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\
\dot{x}_2(...
- 2,712
2
votes
1
answer
55
views
The number of limit cycles of a quadratic vector field with a unique singularity
Is there a uniform upper bound for the number of limit cycles of a quadratic vector field which has a unique singular point in the plane?
- 356
2
votes
1
answer
253
views
Differential inequalities for a strictly diagonal dominant system of linear ODEs
Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...
- 135
2
votes
1
answer
325
views
The type of a Riemann surface arising from a polynomial vector field
Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$
It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...
- 356
2
votes
2
answers
564
views
Linear dynamical systems: interpretation of Frobenius eigenvector
Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...
- 143
2
votes
1
answer
139
views
Can a chaotic trajectory solve an algebraic equation?
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$\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$$
we ...
- 247
2
votes
1
answer
123
views
Keeping track of limit cycles via certain second order differential operator
Inspired by the two posts which are linked bellow we ask the following question:
Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...
- 356
2
votes
1
answer
296
views
Isochronization of quadratic vector fields with center
What is a classification of all quadratic vector fields
$$\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
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 $$ \alpha I \succeq W_t \...
- 23
2
votes
1
answer
90
views
Hopf bifurcation for systems where the dynamics is homogeneous of degree 1
Consider dynamical system in dimension 3
$$x'(t)=f(x(t),d)$$
where the dynamics f is homogeneous of degree 1 and there is exactly one
line of equilibrium points. This line is independent of the ...
- 21
2
votes
1
answer
412
views
Non-linear 1st order difference equation
I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$
I have tried various substitutions, simplifications ...
- 173
2
votes
0
answers
136
views
Progess on conjectures of Palis
I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...
- 247
2
votes
0
answers
72
views
On bounded solutions of a given fourth-order linear ODE
Consider the fourth-order linear ODE
$$
\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.
$$
Without getting ...
- 121
2
votes
0
answers
167
views
Why were these constants picked in this Lyapunov function and how did the author arrive at the final form of the Lyapunov function?
Consider the following paper:
"A note on global stability for a tuberculosis model" by Gao and Huang: https://doi.org/10.1016/j.aml.2017.05.004
The methodology is understood in this paper ...
- 185
2
votes
0
answers
153
views
Stability test for LTV systems by differential Lyapunov inequalities
Consider a linear time-varying system:
\begin{equation}
\dot x(t) = A(t) x(t), \tag{$*$}
\end{equation}
where $A(t)$ is a time-varying block matrix defined as
$$
A(t) =
\begin{bmatrix}
0 & I\\
-\...
- 55
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
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
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 $x \in \mathbb{R}^{d_x}$, $y \in \mathbb{R}^{d_y}$ ...
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
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
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
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
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
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
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
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
2
votes
0
answers
280
views
Uniqueness of analytic center manifold
In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
- 21
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
1
answer
1k
views
Global Solutions of Ordinary Differential Equations
Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
for every ...
- 123
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
2
answers
466
views
Two questions related to $TS^{2}$ as a holomorphic manifold
We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the following question. ...
- 356
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
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
1
vote
1
answer
476
views
Two limit cycles which lie on the same leaf
Edit 1: For a related discussion see this MSE post
I apologize in advance, if this question is obvious:
1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...
- 356