2
votes
0answers
28 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
34 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
72 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
2answers
421 views

Understanding this ODE

(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
72 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
207 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
35 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 ...
1
vote
1answer
40 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
49 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
39 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
73 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
61 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
15 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
17 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
20 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
40 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
48 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
60 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
47 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
20 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
21 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
39 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
25 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
33 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
207 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
172 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
58 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
31 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
89 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
133 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
465 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
33 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 ...
8
votes
0answers
103 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 ...
2
votes
1answer
103 views

Properties of this Schrödinger equation / Sturm-Liouville problem.

Given the ODE $$\Psi''(\theta) + \eta \cos(\theta) \Psi(\theta) + \xi \cos^2(\theta) \Psi(\theta)= \lambda \Psi(\theta),$$ where $\theta \in [-\pi,\pi]$, $\Psi(-\pi)= \Psi(\pi)$ and $||\Psi ||_{L^2} = ...
1
vote
1answer
38 views

bifurcation with more than parameter

Problem: Consider the scalar differential equation depending on the parameters $\alpha_1, \alpha_2$ ∈ $\Re$ $x˙ = \alpha_1 + \alpha_2 x − x^2$. Find a change of coordinates $y = \phi(x)$ such that ...
0
votes
1answer
48 views

near identity change of coordinates

Problem: Consider the scalar differential equation $$x' = \frac{4x – 24x^2 – 16x^3}{1 – 12x – 12x^2}.$$ which has a fixed point at $x^* = 0 $. For $x$ close to $x^* = 0 $ find a near identity ...
0
votes
1answer
49 views

Recommend resources on dynamical systems and singularities

I'm looking for resources on bifurcation theory and systems of non-linear differential equations, but am very particular about the way it is taught/explained. I would like the approach to be based on ...
2
votes
2answers
81 views

the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
3
votes
1answer
57 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
2
votes
1answer
86 views

Show that system is Transcritical bifurcation

In what ways can you show that transcritical bifurcation occurs? For example take the system $$\dfrac{dx}{dt}=xr+2x^2 $$
0
votes
1answer
48 views

Dynamical system with periodic orbit

We are given dynamical system $\phi $ in $R^2$, and know that it has periodic orbit (means $\phi(T,x_0)=x_0$ for some $T>0$ and $x_0 \in R$). We are asked to prove that the system has stationar ...
1
vote
2answers
74 views

A system of nonlinear differential equations

We have the following system in $\mathbb{R}^{2}$ $$\dot{y}_1=2-y_1y_2-y_2^2$$ $$\dot{y}_2=2-y_1^2-y_1y_2$$ i) Calculate the equilibrium points en determine their stability. ii) Draw the Phase ...
-1
votes
1answer
120 views

Find values of the parameters in Predator prey model [closed]

$$r' = F_1(r, f) = r − cr^2 − drf$$ $$f' = F_2(r, f) = −f/4 + erf + gf^2$$ Consider the case where $g = 0$. For what values of the parameters, $c, d$ and $e$, which are all assumed to be positive, ...