0
votes
0answers
40 views

what's the recent trend in the field of dynamical system? [on hold]

I'm studying dynamical system,and I want to know some research topics in the field of dynamical system,such as bifurcations and chaos theory.Could you recommend me some materials or references?Thanks ...
2
votes
2answers
31 views

If $f(x) \cdot x < 0$ for all $x \in \partial B_R(0)$, then the IVP $x' = f(x)$, $x(0) = x_0$ has a global solution.

I have a homework problem that asks If $f : \mathbb{R}^n \to \mathbb{R}^n$ is continuously differentiable and satisfies $$ f(x) \cdot x < 0 \quad \quad \text{for all } x \in \partial B_R(0) ...
2
votes
1answer
33 views

What to do when regular approaches fail on linear, non-homogeneous ODES.

In my research problem, I have come across the following form of a time varying, non-homogeneous ordinary differential equation. $$\dot x + \frac{k_1}{t} x = k_2t^{3n}\sin(bt) + k_3 t^n \sin(bt) - ...
0
votes
0answers
30 views

Why can't the general solution of separable first order ODE cross the stationary solution?

For example, if we have the following Cauchy problem: $y'=y^2-4, y(0)=0$ In class, our professor told us that $y=-2,2$ are the two stationary solutions, but how could it be, since our initial point ...
1
vote
3answers
84 views

Is there a closed form solution for this differential equation?

I was trying to solve the following ODE, but I cannot find an easy way anywhere. I also tried using Mathematica, but it also does not provide me with a solution. $\frac{dx}{dt}=-k_1 x+(1-x)k_2 ...
0
votes
0answers
30 views

Using the Lyapunov-Perron method to find the local stable/unstable manifolds

Hello Stack Exchange community. I am currently having an issue finding the local stable/unstable manifolds of this system. After going at it for a few hours I believe the person who wrote this ...
1
vote
2answers
24 views

How to find an orbit in implicit form for a first order non-linear system of differential equations?

How to find an orbit in implicit form for a first order non-linear system of differential equations? Say $x'= x - xy$, $y'= y - 2xy$ is our system. How do we find an orbit of it in an implicit form?
6
votes
1answer
101 views

Stability analysis for ODEs with non constant inputs

For a project, I have to deal with systems of ODE's with non constant input such as: $$\begin{cases}\dot x =I(t)x+x^2\\ \dot y=x\end{cases}$$ where I(t) is a random input (for example). In any case, ...
1
vote
0answers
48 views

references about homoclinic or heteroclinic orbit

Could you recommend me some references about the connecting orbit manifolds of differential equation? And what about the methods of studying the existence of heteroclinic or homoclinic orbit in ...
0
votes
0answers
49 views

Stability of critical points on dynamical system

Consider a generic dynamical system \begin{cases}\dot{x} &= f(x,y)\\ \dot{y} &=g(x,y) \end{cases} with a critical point in $P=(x_0,y_0)$ (there may be other critical points somewhere else). ...
1
vote
1answer
52 views

stability of linearly perturbed linear nonautonomous system

I have a linear time-varying linearly perturbed ODE of the form: $$ \dot{x} = [A(t)+B(t)]x $$ where $A(t)$ is a bounded lower-triangular matrix with negative functions on the main diagonal, i.e. ...
0
votes
1answer
24 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...
0
votes
0answers
19 views

$\Phi(\cdot ,x):I_x \rightarrow M$ injective?

I don't understand we the map $\Phi(\cdot ,x):I_x \rightarrow M$ from the excerpt from below of the lecture notes of my professor has to be injectiv. (Here $M$ denotes the domain of the function on ...
1
vote
0answers
62 views

Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
1
vote
1answer
37 views

Global existence of a dynamical problem.

Prove that all the solutions to the system $$ \begin{cases} \dot x= e^{-y^2}\sin(x^n+y^n), \\ \dot y= x^n\sin(x^n+y^n), \end{cases} $$ where $n$ is a fixed natural number, are defined on ...
0
votes
0answers
18 views

Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function
0
votes
2answers
47 views

How to find the derivative of the flow of an autonomous differential equation with respect to $x$

Ok, may be this is a silly question but consider the following. Let $\dot x=f(x)$ be an autonomous differential equation with $f$ having enough smoothness (Say $C^2$). Let $\xi:\mathbb ...
3
votes
0answers
41 views

Let $\eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x)) dt$ then $\eta$ is a $C^1$ function

Consider the following problem. Suppose that $a>0, r >0$ and $\xi:\mathbb R \to [o,\infty)$ is a $C^2$ which vanishes in the complement of the interval $(-r,r)$. Also suppose that ...
0
votes
1answer
41 views

Every solution of the system is attracted to the center manifold

I am trying to solve the following problem. Determine a center manifold for the rest point at the origin of the system \begin{align} \dot x &=-xy \\ \dot y&= -y+x^2-2y^2 \end{align} a) ...
5
votes
1answer
81 views

Stability of nonlinear system given by $\dot{x} = f_1(x) + f_2(x)$

I have a nonlinear system $\dot{x} = f_1(x) + f_2(x)$ defined in a domain $U \subset \mathbb{R}^n$. I know that $x_0$ is an asymptotically stable and the only equilibrium point of the two systems ...
4
votes
1answer
57 views

The system $\dot{x}=x^2$, $\dot y=-y$, has infinitely many (local) center manifolds

Consider the system, \begin{align} \dot{x}&=x^2 \\ \dot y&=-y \end{align} I am trying to show that this system has infinitely many local center manifolds. Here is what I have done so far: ...
1
vote
0answers
43 views

Uniqueness of solution for a system of differential equations

A friend of mine working on Auction Theory needs to establish uniqueness of solution (up to initial and boundary conditions) of a system of differential equations of the form $$ ...
0
votes
0answers
30 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
0
votes
0answers
28 views

Diagonalize Complex ODE

I'm trying to solve for the dynamics of one coordinate of a coupled system of linear differential equations with complex coefficients. Physically, a number of single-pole harmonic oscillators with ...
4
votes
1answer
67 views

Notational issues on differential equations

I am studying dynamical systems and I have some trouble in understanding the notation used for differential equations. For example when I read $$\overset{..}{x}=F(x),$$ how should I interpret ...
2
votes
0answers
85 views

Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
4
votes
1answer
77 views

Interpretation of generalized eigenvector in orbits

First of all, this is my fourth question about dynamical systems in a week, sorry for that. Considering a linear bidimensional dynamical (autonomous) system, the orbits can be plotted in the phase ...
1
vote
0answers
48 views

Central manifold theorem => Stable/unstable manifold?

I'm a bit confused why we always separate the stable/unstable manifold theorem and the central manifold theorem. The stable/unstable manifold theorem applies to a hyperbolic point ...
1
vote
0answers
64 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
2
votes
0answers
31 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
2
votes
1answer
61 views

Prerequisite of Dynamical system and applied PDE

For the further research interest, I want to focus on the application of Dynamical systems and PDE in the field of robotics and neuroscience, particularly from a mathematical points of view. ...
0
votes
1answer
77 views

Solving for Center Manifold with Parameter

I have a system of ODEs given by $$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$ $$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$ where $\beta $ is a parameter. How should I ...
9
votes
3answers
518 views

Eigenvalue problem for ODE with singular coefficients, $-(1-x^2) y'' + py'+qy=\lambda y$

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
5
votes
2answers
78 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
8
votes
3answers
222 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
2
votes
1answer
43 views

Stability properties of discretization of ODE

I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the ...
2
votes
1answer
57 views

Differential Equation Examples for different type of critical point

For a linear system $X'=AX$, there are only limited types of critical points according to the eigen values of $A$. When I want to considering non-linear dynamical system in $\mathbb{R}^2$ and ...
0
votes
2answers
58 views

Extension of Poincaré-Bendixson Theorem to $\mathbb{R}^3$

Hartman mentioned in his ODE book (chapter 7) that Poincaré-Bendixson Theorem is limited to $\mathbb{R}^2$ or $2$-manifold because of Jordan Curve Theorem. Since there is generalization for ...
0
votes
1answer
46 views

Recommendation for dynamical system with complex behaviors

I want to learn the behaviors of dynamical systems, especially the in form of $X'=f(X)$ and $X'=f(t,X)$ in $\mathbb{R}^3$. I know Lorentz system is such a system(typically ...
1
vote
1answer
75 views

Delayed System Help

It is well-known that a small delay may or may not cause stable equilibrium to become unstable. Can anyone help that if for $\tau=0$ the equilibrium solution is unstable and if $\tau>0$ is there a ...
0
votes
0answers
20 views

if $X$ is a vector fild in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? [duplicate]

if $X$ is a vector fild in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? and in dimension $n$? I know there is singularity when $n=2$.
1
vote
1answer
65 views

A phase diagram outlining

I'm trying to solve this differential equation $$x^{ \prime}=f(x)-nx-y$$ $$y^{\prime}=\frac{(f^{\prime}(x)-r)y}{\alpha}$$ where $f:[0,+\infty[\rightarrow \mathbb{R}_{+}$ is an increasing and concave ...
2
votes
0answers
16 views

If $p$ is a regular point $X$ such that $p \in \omega(p)$ then $\omega(p)$ is periodic orbit. [closed]

Let $X$ be a field in $\mathbb{R}^3$, $C^1$ class. If $p$ is a regular point $X$ such that $p \in \omega(p)$ then $\omega(p)$ is periodic orbit.
1
vote
0answers
19 views

Maximum intervals of a solution and singularities [closed]

Let $X$ be a vector field of $C^1$ calsse in $\Delta \subseteq \mathbb{R}^n$. Prove that if $\varphi(t)$ is a trajectory of $X$ defined maximum range $(\omega_-,\omega_+)$ with: $$\lim_{t \rightarrow ...
1
vote
0answers
29 views

For every topological conjugation $h: \Delta_1 \rightarrow \Delta_2$ we have $h(\omega(p))=\omega(h(p))$, for all $p \in \Delta_1$ [on hold]

Let $X_1$ and $X_2$ fields in $\Delta_1,\Delta_2$ subset open in $\mathbb{R}^n$. Then, for all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for ...
1
vote
1answer
48 views

Prove the formula for the Lie derivative of a differential form

If $X$ is a vector field then by $\mathcal F^t_X$ I will denote it's flow. If $\alpha \in \Lambda^k$ then by definition $$ \mathcal L_X \alpha = \frac{d}{dt}(\mathcal F^t_X)^*\alpha \, ...
2
votes
1answer
55 views

Problem with a pushforward of vector field formula (Michael Taylor, “Partial Differential Equations”)

Let $X$ denote a vector field and let $\mathcal F^t_X$ denote its flow. If $X$ and $Y$ are two vector fields we denote by $\mathcal F^t_{X\#}Y$ the vector field satisfying $$ \mathcal ...
1
vote
1answer
74 views

Compact $\omega$-limit set $\Rightarrow$ connected

Consider the flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ and $L_{\omega}(x)$ the $\omega$-limit set of a point $x \in \mathbb{R}^n$. How can I show that if $L_{\omega}(x)$ is ...
1
vote
1answer
51 views

“Reparametrizing” a differential system of the first order (Vinograd theorem?)

Consider a continuous function $f:\Omega\subset\mathbb R^n\longrightarrow \mathbb R^n$ such that for every $x\in\Omega$ the Cauchy problem: $$(\ast)\left\{\begin{array} {ll} y'=f(y)\\ y(0)=x ...
0
votes
1answer
55 views

Dynamical systems, conjugacy

Consider a family of dynamical systems generated by equations: $y'=ax+b, \ \ a,b \in \mathbb{R}$. Is it true that in this family: 1) There are 4 types of phase portraits up to topological ...