Questions tagged [perturbation-theory]
Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.
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Region of attraction of simple ODE with perturbation
There are a few nice discussions about ROA covering a few subtopics:
Region of attraction of : $x'=-y-x^3,y'=x-y^3$ via Lyapunov Function
Region of attraction and stability via liapunov&#...
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Analyze $y' = y\cdot \cos(x+y)$
Anyone familiar with Mathematica will realize this ODE is the first example of the NDSolve command's documentation, a screenshot of which is shown below
I am wondering if it possible to get some ...
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Method of Dominant Balance with high order system
This question comes from Bender and Orszag's Asymptotic Methods and Perturbation Theory.
I'm practicing applying the method of dominant balance to study behavior as $x\to \infty$ for systems which ...
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Eigenvectors not stable under matrix perturbation
Let $A,B$ be square matrices of the same size but arbitrary components. It is a fact that the eigenvalues of $A$ and $A+\epsilon B$ are close for $\epsilon$ small, but the eigenvectors may be ...
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Numerically unstable matrix decompositions (to simple perturbations e.g. $A+ \varepsilon A$)
So basically the title says it all. I would like to find a decomposition, or something similar, e.g. any transformation that would "notice" a slight perturbation in a given matrix; as in $f(...
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Inverse of almost diagonal matrixes
Consider an $n\times n$ matrix $A$, and it's perturbation matrix $dA$.
Let for simplicity $A=I$ be a matrix with ones on the diagonal and zeros off-diagonal.
Let $dA$ have zeros on the diagonal and ...
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Boundary Layer, leading order, Pertubation Theory, Differential Equations
I have got the following problem, taken from Multiple Scale and singular perturbation methods, Kevorkian & Cole book, page 94, exercise 1.b.:
Find the leading order of the problem:
$\varepsilon ...
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Bounding the roots of the sum of two monic polynomials with real coefficients.
Let $P_1(z)$ and $P_2(z)$ be monic polynomials with real coefficients and roots $\{z_1^{(1)},z_1^{(2)},...\}$ and $\{z_2^{(1)},z_2^{(2)},...\}$, respectively. Are there any results relating the non-...
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Eigenvalue, Eigenvector, of a Tridiagonal Symmetric "nearly" Toeplitz Matrix
I am trying to find the eigenvalues/eigenvectors of a NxN tridiagonal symmetric "nearly" Toeplitz matrix, except that a modification on the top left corner
\begin{pmatrix}
a^2 & -a \\
-a & 1+a^...
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Inverse of perturbed Kac-Murdock-Szegö matrix
A Kac-Murdock-Szegö (KMS) matrix is a matrix of the form $A_{ij}=\rho^{|i-j|}$ for $i,j=1,2,\ldots,n$ and $\rho\neq1$.
The inverse of $A^{-1}$ is well known, see e.g. https://journal.austms.org.au/ojs/...
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$D((A+B)^*)= D(A^*)$ if $B$ is $A$-bounded with $A$-bound $0$
Let $A:D(A) \subseteq H \to H$ and $B:D(B) \subseteq H \to H$ be linear operators on a Hilbert space $H$ such that $A$ is a closed densely defined operator and $B$ is relatively bounded with respect ...
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Taylor approximation to a matrix logarithm of a product of matrix exponentials
Let $\mu_i,\nu_i\in\mathbb{R}^{n\times n}$ for $i\in{\{1,\dots,I\}}$, and let $\sigma\in\mathbb{R}$. Assume that $\log{\prod_{i=1}^I{\exp{(\mu_i)}}}$ exists with real entries.
It must be the case ...
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Two variable perturbation analysis of differential equations
I have following set of equations,
$\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$
$\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - 2 \epsilon_2 y(t) + 2 \epsilon_1 \epsilon_2 (...
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IVP Perturbation With Small Non-Linear Term
EDIT: Sorry to bump this without having anything extra to add, but I still cannot reconcile my solution with what was asked (in (2)). Could someone with expertise in this subject take a look? I ...
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How to bound the change between optimal points when perturbing an objective function?
Let $A,B \in \mathbb{R}^{n \times n}$ be two positive semi-definite matrices and let $a > 0$ be a constant.
Consider the following maximization problem
$$
\max_{x \in \mathbb{R}^n, \gamma}\ x^T \...
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