All Questions
Tagged with perturbation-theory dynamical-systems
42
questions
2
votes
1
answer
61
views
Approximating solution to vector recurrence relation with element-wise exponential
$$
\mbox{Let}\ A \in \mathbb{R}^{n \times n}\ \mbox{and}\ \gamma_{1}, \ldots, \gamma_{n} \in \mathbb{R}\ \mbox{with}\ \gamma_{i} > 0,\ \forall\ i.
$$
$$
\mbox{Define}\ \operatorname{f}: \mathbb{R}^{...
- 33
4
votes
2
answers
155
views
Periodic perturbation of ODE
Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.5, the author mentioned that the following is well-known in the theory of ODEs.
If $x_0$ is an ...
- 351
2
votes
0
answers
52
views
Leading order perturbation to the solution of a dynamical system
I was reading the paper 'A Proposal on Machine Learning via Dynamical Systems', where I came across the following steps:
Consider a system-
$$\frac{dz}{dt} = f(A(t),z),$$ with $z(0) = x.$
So, the ...
- 21
3
votes
2
answers
107
views
Birth-death : Always more than 1 bifurcation?
Say I have a (smooth) function $f : \mathbb{R}^n \to \mathbb{R}$, and a critical point $x$ (ie, $f'(x) = 0$). I call this point degenerate if $\det \text{Hess}_x f = 0$ (so, equivalently, if the ...
- 2,311
1
vote
0
answers
24
views
Perturbations of an integrable system with no resonant tori
I think the following is probably a trivial application of KAM theory, which I know little about, so I am hoping someone can give me an answer pointing me in the right direction.
Suppose I have a ...
- 391
0
votes
1
answer
111
views
Deriving a Hopf Bifurcation – Perturbation Method vs. Jacobian Matrix
When deriving a Hopf bifurcation of a dynamical system, the usual process is:
Find a fixed point $(x_0, y_0)$
Perturb the system about the fixed point $(x_0+\tilde{x}, y_0+\tilde{y})$
Linearize, ...
5
votes
2
answers
463
views
Making the "Two-Timing" method rigorous (Perturbation theory)
I'm reading about the method of two-timing in section 7.6 of Nonlinear Dynamics and Chaos by Strogatz, and I have some questions about how to make this concept rigorous. In this section the book ...
- 1,054
0
votes
1
answer
86
views
Prove all solutions to $ x^{\prime}=A(t) x, \quad x(0)=x_0 $ are $T$-periodic.
Suppose $A(-t)=-A(t)$ and $A(t+T)=A(t)$. Prove all solutions to
$$
x^{\prime}=A(t) x, \quad x(0)=x_0
$$
are $2T$-periodic.
My try
$t_0=0$ here. $A$ is an $n \times n$ matrix.
Method 1:
We know that ...
- 9,098
3
votes
1
answer
302
views
Examples of dynamical systems that have structural stability
I am looking for simple examples of structural stability, I read the definition of structural stability but couldn't figure out a concrete example of a system, its perturbated version and its ...
- 165
1
vote
0
answers
62
views
Decaying solution and periodic-$1$ solution to $y''(x)=y(x)-y(x)^2$
I came across a problem in the dynamic systems theory, which is very similar to the following simple example.
Consider the simple model equation
$$
y''(x)=y(x)-y(x)^2. \tag{*}$$
Let us denote by $y_\...
- 4,865
1
vote
2
answers
82
views
unique equilibrium in a perturbed system
In today's lecture, I see that if $x_*$ is an asymptotic stable and hyperbolic equilibrium of the $\dot{x}=a(x), \, x\in\mathbb{R}^n$. But then prof said that "it's obvious" if we give a ...
user707954
1
vote
1
answer
280
views
WKB for non-homogeneous ODE
Consider the ODE $$\epsilon^2 y'' + \epsilon x y' - y = -1, \; y(0) = 0, \; y(1) = 3$$
I've seen the WKB method applied to homogeneous (linear) ODEs, but here we have the $-1$ term. I could perhaps do ...
- 301
0
votes
1
answer
193
views
Why is $\sin^3(\tau + \phi)$ not a Secular term in the context of the van der Pol oscillator
On page $223$ of Strogatz is the example of Two-Timing applied to the van der Pol oscillator $$x'' + x + \epsilon (x^2 - 1)x' = 0$$ Introducing the perturbation $x = x_0(\tau, T) + \epsilon x_1(\tau, ...
- 301
1
vote
0
answers
54
views
Existence of equilibrium point under asymptotically vanishing disturbance
Consider an autonomous dynamical system $\dot{x}=f(x)$ and suppose that we know that the solutions converge to a limiting set of equilibrium points $\Omega$ in $\mathbb{R}^n$. Consider now the ...
- 153
1
vote
0
answers
43
views
Bounded solution of an ODE (with perturbation)
Let $\epsilon >0$ and $f\in \mathcal{C}^1(\mathbb{R}^{d+1})$, and we consider the following Backward Foward ODE :
$$\begin{cases}
\dot z^{\epsilon}(t)=f(z^{\epsilon}(t),t), & t\in [0,2]\\
z^{\...
- 1,415