4
votes
1answer
40 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: ...
0
votes
0answers
13 views

How the Jacobian is connected to the movement of particle from one domain to another? [on hold]

I am dealing with the proof of Reynold-Transport Theorem. There the Jacobian is used for the changing position of particles from one domain to another. Can anyone help me to understand what does ...
1
vote
0answers
41 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
28 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
27 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
65 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
74 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
71 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
44 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
57 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
30 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
42 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
74 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
511 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
76 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
213 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
0answers
37 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
43 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
52 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
42 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
18 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
63 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 ...
2
votes
0answers
23 views

For all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for all $p \in \Delta_1$

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
43 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
53 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
63 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
48 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
54 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 ...
1
vote
0answers
21 views

$\gamma(t)$ is not asymptotically stable unless $\int_0^T \nabla \cdot f(\gamma(t))dt \leq 0$

Let $f \in C^1(E)$ where E is an open subset of $\mathbb{R^n}$ containing a periodic orbit $\gamma(t)$ of $x'=f(x)$ of period $T$. Then $\gamma(t)$ is not asymptotically stable unless $$\int_0^T ...
0
votes
0answers
22 views

How to compute transfer function from Laplace Transform

My system of interest has the following EOM (V is my input variable): $\ddot{x} = g - k_{1}V(t) + \dot{x}k_2$ Taking the Laplace with initial conditions of zero, I get: $s^2X(s) = \frac{g}{s} - ...
0
votes
0answers
25 views

Hill's problem for moon trajectories.

When we work with the three-body problem, we have a parameter $\mu$ that shows the ratio of the two biggest bodies with $\mu\in(0,1)$. This let's us do practical applications easily. For example we ...
0
votes
1answer
42 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
1
vote
1answer
26 views

Is $ \gamma(t) = \left( A \cos(\sqrt{a} t),B \cos \! \left( \sqrt{b} t \right) \right) $ dense in the rectangle $ [- A,A] \times [- B,B] $?

What conditions must $ a $ and $ b $ satisfy in order for the curve $$ \gamma(t) = \left( A \cos(\sqrt{a} t),B \cos \! \left( \sqrt{b} t \right) \right) $$ to be dense in the rectangle $ [- A,A] ...
0
votes
0answers
22 views

Looking for discrete non linear dynamic system solution hints

I am studying a networking congestion control problem for which I would like to solve the following non linear discrete first order dynamic system (hope I got that correctly, I am no mathematicien ...
1
vote
2answers
35 views

Finding a Lyapunov function for a given system of equations

I've got the following system of equations: $$ \begin{cases} x_1'=-8x_1^3-x_2 \\x_2'=-4x_2-4x_1^3 \end{cases} $$ I'm trying to check, if the equilibrium point in $(0,0)$ is stable or not. I am ...
3
votes
0answers
224 views

Estimating a dynamical system's behavior without using Liapunov theorem

Assume that we have the following dynamical system $$x'=(\epsilon x+2y)(1+z)$$ $$y'=(-x+\epsilon y)(1+z)$$ $$z'=-z^3$$ Then how can I show that any solution that started from the region $z>-1$ ...
5
votes
0answers
177 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...
3
votes
1answer
27 views

Does Hyperbolic + Not Asymptotically Linearly Stable imply Not Asymptotically Stable?

Topic: Stability of Autonomous Non-linear ODEs I'm wondering whether having a hyperbolic critical point that's not asymptotically linearly stable (ALS) in the linearisation of a system implies that ...
1
vote
2answers
63 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
0
votes
1answer
33 views

Intuition or wisdom for stability and instability properties of locally linear system. Boyce, p513, Table 9.3.1

Our instructor requires us to memorize this table for our differential equations exam. So I wonder if anyone has some deeper intuition or observation to help with this? For example, I noticed ...
1
vote
1answer
91 views

Nonlinear first order system of ODEs

While solving some physical problem, I have obtained the following system of differential equations with boundary conditions: $$\left\{\begin{matrix} \frac{d\phi_1}{dz}=\frac{m^2}{\lambda}- ...
2
votes
0answers
56 views

modified ODE has same trajectories as original system and associated flow is defined for all $t \in \mathrm{R}$ [closed]

I really don't know where to start with this problem. Consider the differential equation $\dot{x} = f(x)$ with $f \in C^1(\mathrm{R}^n,\mathrm{R}^n)$. Consider the following modified differential ...
0
votes
1answer
47 views

FermiPasta-Ulam problem

Consider $H(q,p) = \frac{1}{2} \sum\limits_{j=1}^{n+1} {(p_j^2 + (q_{j}-q_{j-1})^2)}$ $H(q,p) $ is the Hamiltonian considered in the FermiPasta-Ulam problem. Consider canonical transformation $Q = ...
6
votes
1answer
139 views

Confluent Heun equation. Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
15
votes
1answer
474 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
0
votes
1answer
34 views

Level sets of a conserved quantity are trajectories of differential equation

If we have a differential equation $\mathbf{\dot{x}}=\mathbf{F}(\mathbf{x})$ and we have conserved quantity $E(\mathbf{x})$, which means $\dot{E}=0$, then I don't understand why level sets of $E$ are ...
9
votes
0answers
109 views

Notions of stability for differential equations

Consider a system of differential equations $$\dot{x} = f(x,u)$$ $$y = h(x,u)$$ where $x(t), u(t)$ are vectors in some $\mathbb{R}^n$. We define the infinity norm of a function in more-or-less in the ...