All Questions
Tagged with complex-systems homework-and-exercises
16 questions
4
votes
3
answers
1k
views
How to do linear stability analysis on this system of ODEs?
I was trying to do linear stability analysis of spring pendulum. I arrived at the differential equations which describe the system. But I am unable to proceed to linear stability analysis. Is it ...
- 416
3
votes
4
answers
250
views
Asymptotic frequency of nonlinear oscillator $\ddot x = -x-{\dot x}^3$ (speed cubed)
A particle oscillates according to the equation
$\ddot x = -x-{\dot x}^3.$ The positive positions of the particle when it changes direction, $\dot x = 0$, are $x_1,x_2,\ldots$.
I want to show that
$$\...
- 131
3
votes
1
answer
291
views
Recursion relations and stability analysis
I have a recursion relation in the form of the following two equations:
$X_{t+1} = X_t + V_{t+1} \\
V_{t+1} = wV_t + cy(g-X_t)$
I want to write these two equations into a matrix form so that I can ...
- 175
3
votes
1
answer
79
views
Stability of Chemical Reactions [closed]
Given the following reactions:
$$A + X \xrightarrow{k_{1}} 2 X$$
$$Y + X \xrightarrow{k_{2}} 2 Y$$
$$Y \xrightarrow{k_{3}} B$$
I was able to write the following rate equations for the concentrations:
$...
- 653
2
votes
0
answers
29
views
Find closed orbit problem (Strogatz 8.3.2) [closed]
I'm having trouble solving this excercise from Strogatz
Consider the following system for a chemical oscilator:
$$
\dot x= a -x+x^2y
$$
$$
\dot y=b -x^2y
$$
Where $a,b>0$ are parameters and $x,y\...
- 97
2
votes
0
answers
135
views
Are the following action-angle variables correct for the Hamiltonian $H = \sqrt{p}f(q)$? [closed]
Suppose I have a Hamiltonian $H=H(p,q)=H(p,q+1)$ defined on a cylinder $\mathbb{T} \times \mathbb{R}^{+}$, such that $$H(p,q) = \sqrt{p}f(q)$$ where $f(q)=f(q+1)>0$ is a periodic function of $q$, ...
- 331
1
vote
2
answers
151
views
Phase diagram method
I was trying to find the famous attractor solution of the inflaton field which follows the equation
$$\frac{d\dot{\phi}}{d\phi}=-\frac{\sqrt{12\pi}(\dot{\phi}^2+m^2\phi^2)^{1/2}\dot{\phi}+m^2\phi}{\...
- 101
1
vote
1
answer
183
views
How to find the value of the parameter $a$ in this transfer function?
I am given a transfer function of a second-order system as:
$$G(s)=\frac{a}{s^{2}+4s+a}$$
and I need to find the value of the parameter $a$ that will make the damping coefficient $\zeta=.7$. I am not ...
- 1,158
1
vote
0
answers
52
views
Potentials that prevent the phase flow of the system [closed]
I am trying to solve a question that my professor gave.
When a particle moves in one dimension $x$ in a potential $U(x)$ , the resulting motion over a very short time interval is specified by Newton’...
- 9
1
vote
0
answers
44
views
Argument of a continuos system frecuency response
Yo there!, I have seen a question in a book that I solved, the way that is defined is what make me curious.
The idea is to prove that the steady state of stable system with transfer function $P(s)$ ...
- 183
0
votes
1
answer
369
views
Lyapunov Exponent of the Logistic map [closed]
My dynamical system professor (and the textbooks we use) all claim that the Lyapunov exponent for the Logistic map with $r=4$ ($x_{n+1} = 4x_n(1-x_n)$) is $\log(2)$. Would someone be able to sketch ...
- 19
0
votes
1
answer
451
views
Phase portrait Hamiltonian system
I have the Hamiltonian:
$$H(x,p)=\frac{p^2}{2m}+\frac{x^4}{4}$$
When I find the critical points associatted with the system $\dot{x}=\frac{p}{m}$ and $\dot{p}=-kx^3$, I find only one critical point ...
- 219
0
votes
1
answer
385
views
Third eigenvalue of Lorentz equations
I was reading and working on the Strogatz's book on nonlinear dynamics and chaos on my own. I was trying to solve problem 9.2.1. The thing is that, I do not understand how can I solve part c) of that ...
- 49
0
votes
1
answer
650
views
What is the Laplace transform of a Linear Time-Varying system?
The Problem
I have the following damped mass-spring system in the form of a Linear Time-Varying (LTV) system:
$$\mathbf{M}(t)\mathbf{\ddot{x}}(t) + \mathbf{C\dot{x}}(t) + \mathbf{Kx}(t) = \mathbf{f}(t)...
- 37
0
votes
1
answer
780
views
How to find the value of the parameter a in this transfer function? [duplicate]
Possible Duplicate:
How to find the value of the parameter $a$ in this transfer function?
I am given a transfer function of a second-order system as:
$$G(s)=\frac{a}{s^{2}+4s+a}$$
and I need to ...
- 1,158
-3
votes
1
answer
123
views
Lyapunov is wrong - got unstable on a stable system [closed]
I'm angry with the Lyapunov stability criteria. Consider this system:
Here, $u$ is the input and $x_1$, $x_2$ are my state variables. Now, solve for the transference of the system, defining my output ...
- 337