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 ...
2
votes
1answer
43 views

Link between the two definitions of a “hyperbolic point”

The common definition of a hyperbolic point for a flow of a vector field $f$ is a fixed point in which the eigenvalues of the Jacobian matrix of $f$ all have non-zero real parts ...
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. ...
0
votes
0answers
49 views

what is the meaning/characteristics of the component-wise product of right and left eigenvectors.

I have a generic, but seemingly simple question : what is the meaning/characteristics of the component-wise product of right and left eigenvectors (for the same eigenvalue of course) ? let's call ...
1
vote
0answers
60 views

Eigenvalues 'on' Unit Circle - Stability Characteristics?

If the eigenvalues of the system matrix contains most of the Eigenvalues exactly "on" the boundary of the unit circle, with a very few actually inside - What does it imply about the stability ...
0
votes
1answer
53 views

Help With Eigenvectors and Dynamical Systems

I have the following system of differential equations: $ \frac{d}{dt} \left[ \begin{array}{c} A(t)\\ N(t)\\\end{array} \right] = \left[ \begin{array}{c c} -(a+b) & 0\\ a & -(a+b)\\ ...
2
votes
1answer
104 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
4
votes
1answer
112 views

Trying to prove a matrix is always convergent.

I have a matrix $Z$ of the form $Z = \left[Q^{-1}-Q^{-1}A^T\left(AQ^{-1}A^T\right)^{-1}AQ^{-1}\right]\Phi$ where, $\Phi$ is a diagonal matrix of real non-negative values. $\Theta$ (not ...
5
votes
1answer
826 views

about Jacobian and eigenvalues

I am studying the dynamical system on a discrete standard map $$x_{n+1} = f(x_n, y_n)$$ $$y_{n+1} = g(x_n, y_n)$$ First of all, could anyone explain the difference between the stationary point and ...
0
votes
1answer
83 views

Help to find eigenvalues

In this paper the authors have the dynamical system $$\begin{align} T_f \dot{y}_f & = -y_f + (1-\alpha(v))\varphi(z,d) &(1)\\ T_r \dot{y}_r & = -y_r + \alpha(v) \varphi(z,d) &(2)\\ ...
2
votes
1answer
132 views

Eigenvalues of diff-system(can't understand)

In this paper the authors have the dynamical system $$\begin{align} T_f \dot{y}_f & = -y_f + (1-\alpha(v))\varphi(z,d) \\ T_r \dot{y}_r & = -y_r + \alpha(v) \varphi(z,d) \\ \dot{z} ...
0
votes
1answer
194 views

does the following dynamic system converge to a steady state?

This is an economics problem, but I'm pretty sure this kind of thing comes up elsewhere. I've used dynamic programming to find the optimal path of a system (law of motion), which is: ...
2
votes
1answer
1k views

Zero State, Stable Equilibrium, Dynamic System

Could someone please help? The question reads: For which real numbers $k$ is the zero state a stable equilibrium of the dynamic system $x_{t+1} = Ax_t$? $A = \begin{bmatrix} 0.1 &k \\ 0.3 & ...