All Questions
Tagged with nonlinear-dynamics stability-theory
44
questions
2
votes
1
answer
37
views
Stability of Hamiltonian system on degenerate critical point.
I'm trying to find information on the stability of the following ODE:
$$ x'' = x^4-x^2.$$
We know that it has a Hamiltonian $H(x,y) = \dfrac{y^2}{2} - (\dfrac{x^5}{5} - \dfrac{x^3}{3})$. The orbits ...
- 327
4
votes
1
answer
171
views
Is a system, that is globally asymptotically stable for any constant input also input-to-state stable? [closed]
I am referring to the ISS definition by Sontag of
${\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}$
I understand that 0-GAS is a necessary condition for ISS. But is GAS for all ...
- 51
0
votes
0
answers
76
views
Singularity of a non- linear second order ODE
I have the encountered a singularity in the equation below .
$$
y^{\prime \prime}(x)+\frac{2}{x} y^{\prime}+\left[y-\left(1+\frac{2}{x^2}\right)\right] y(x)=0, \quad 0<x<+\infty,
$$
with ...
2
votes
0
answers
60
views
Radially bounded Lyapunov function and global stability
I came accross this link about the necessity of the Lyapunov function being radially unbounded.
My understanding is that this condition is unnecessary if the time derivative along solution ...
- 35
4
votes
1
answer
146
views
Exponential Stability and Lasalle's Invariance Theorem
It is well known that a system $\dot{x}=f(x)$ with $x \in \mathbb{R}^n$ is exponentially stable if there exists a Lyapunov function $V(x)$ which satisfies
\begin{align}
k_1\Vert x \Vert \leq V(x) &...
- 380
0
votes
0
answers
52
views
from local stability to global stability
Suppose I have the system $x'=F(x)$ with $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$. I denote by $J(x)$ the Jacobian matrix, that is, $J_{ij}(x)=\partial F_i/\partial x_j (x)$.
Suppose I know that for ...
- 35
0
votes
0
answers
45
views
Stability of normal state in chemostat model
The chemostat model proposed by monod was given by,
$$
\begin{align}
\frac{dx}{dt}&=[K(c)-D]x\\
\frac{dc}{dt}&=D[c_0-c]-\frac1yK(c)x
\end{align}
$$
where $x(t)$ is the population of micro-...
- 701
0
votes
0
answers
57
views
The relationship between real negative eigenvalues and convergence rate for ODE.
Let $\pmb{\delta}=\pmb{\delta}^\triangle$ be an equilibrium point for the following ODE ,
\begin{align*}
\frac{\partial \pmb{\delta}(t)}{\partial t}=\pmb{F}(\pmb{\delta}) \ with \ \
\pmb{F}(\...
2
votes
0
answers
54
views
Time Derivative of Dynamical System
Suppose I have a dynamical system of the form
$$
\frac{dx}{dt} = f(x)
$$
Most of the frameworks I am familiar with for analyzing such systems revolve around finding the fixed points $x^*$ where $f(x^*)...
2
votes
1
answer
143
views
Any theorems for Input-output or input-state stability for non-asymptotically stable nonlinear systems?
Update for clarification:
Assume $\dot{x_1}=f(x_1 , x_2)+ au$
where $x_1$ is asymptotically stable for all bounded values of $x_2$. If $x_2$ is kept bounded, will input-output stability theorem apply ...
- 21
1
vote
0
answers
61
views
What can we say about the stability of this equilibrium with purely imaginary eigenvalues?
Equations
I have the following system of Lotka-Volterra equations:
$x'(t)=ax(t)-bx(t)y(t)$
$y'(t)=-gy(t) + dx(t)y(t)$.
This system has a non-trivial equilibrium of:
$(x^*,y^*)=(\frac{g}{d},\frac{a}{b})...
- 261
6
votes
1
answer
89
views
An ODE confusion
I was thinking about a ODE problem recently when I was reading about dynamical system. In school we used to solve the ODE problem $\frac{dx}{dt}=\sqrt{1-x^2}, x=0, t=0$ as $x=\sin(t),$ which will have ...
- 9,098
0
votes
1
answer
276
views
Runge-kutta fourth order for 3 coupled second order equations.
Someone, please help me by checking whether the steps of the application of RK4 in my calculation is correct or not. I could not find any paper/books/write with similar problems or examples. ...
1
vote
0
answers
110
views
Long term behavior of a nonlinear dynamic system
Consider the following system of first order nonlinear difference equation:
$$
\begin{cases}
x_{k+1} = x_k + \alpha(1-x_ky_k^2) \\
y_{k+1} = y_k + \alpha(1-x_k^2y_k) \\
\end{cases}
$$
with a given ...
- 997
1
vote
0
answers
118
views
long term behavior of solution to a first order nonlinear differential equation
Consider the following system of first order nonlinear autonomous ODEs (derivatives are taken with respect to $t$):
$$
\begin{cases}
\dot{x} = -2xy^2+1 \\
\dot{y} = -2x^2y+1 \\
x(0) = x_0,\;y(0)=y_0
\...
- 997