On the many different ways to turn non-linear systems of equations into linear ones.
Questions tagged [linearization]
318 questions
12
votes
2 answers
Lyapunov stability question from Arnold's trivium
V.I. Arnold put the following question in his Mathematical Trivium:
Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?
It puzzled me for a while, since my experience doesn't include such a…
H. M. Šiljak
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11
votes
1 answer
What is the 'linearization' of a PDE?
Specifically I am looking at the proof of Lemma 4.1 on page 9 here, where the graphical form of curve shortening flow is given, and then its 'linearization'. I am struggling to find any online resources that explain what this means, and what the…
jl2
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10
votes
0 answers
Noether's theorem in the critical heat equation
I am watching a serie of lectures on "Blow up solution for the energy critical heat equation" from Monica Musso on YouTube and at some point she states a result I do not understand. Let met recall the setting.
We are studying the following…
Falcon
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7
votes
0 answers
Linearization of scalar curvature: $DR|_g(h)=-\Delta_g(\mathrm{tr}_g h)+\mathrm{div}_g(\mathrm{div}_g h)-\langle\mathrm{Ric}_g,h\rangle_g$
I'm working on an exercise from Geometric Relativity by Dan A. Lee, but things didn't go well:
Following Lee's hint, I was trying to use Exercise 1.12 and view $\color{red}{g}$ as the background metric ($\bar g$ in Exercise 1.12). In Exercise…
Wombat
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6
votes
3 answers
How to write boolean expressions as linear equations
I want to convert a set of boolean expressions to linear equations. In some cases, this is easy. For example, suppose $a, b, c$ $\in$ {0,1}. Then if the boolean expression is: $a$ $\ne$ b, I could use the linear equation $a + b = 1$.
To give a more…
user66360
6
votes
1 answer
What is the significance of the linearization of a non-linear PDE?
This may be too general a question so please let me know if I need to make it more specific.
I am a first year graduate student in PDEs, and as such have not had much exposure to non-linear PDEs. I am starting to look at research papers, in many of…
jl2
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5
votes
0 answers
Linearization of Gradient Flow
As someone who has only "theoretical" knowledge in Riemannian geometry, I have a hard time trying to wrap my head around how to actually compute the so called "linearization" of a gradient flow on a manifold.
The setting is: We have a symplectic…
Nuke_Gunray
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5
votes
0 answers
Example for Carleman Linearization resulting in a linear system
The Carleman linearization came to my attention due to this article. I tried to understand this method but so far i wasn't succesful i tried to understand page 39 of this presentation however the example didn't make sense for me.
Could someone…
worldsmithhelper
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5
votes
1 answer
Understanding a proof about Riemannian metrics in three dimensions always being diagonalizable
I've recently been working through Deturck's and Yang's Existence of elastic deformations with prescribed principal strains. First and formost, I'm interested in it's proof that Riemannian metrics can in three dimensions always be diagonalized, that…
moran
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4
votes
0 answers
How to define linearization of a dynamical system on a manifold with affine connection?
In Euclidean space, if I have a smooth dynamical system $\dot{x}=F(x)$, it's linearization about a solution $x(t)$ is $\dot{v}(t) = DF(x(t))v(t)$, where $DF(x)$ is the Jacobian matrix of $F$ at $x$.
I am curious on how linearization works on smooth…
Spencer Kraisler
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4
votes
1 answer
Linearize optimization problem with absolute value
Is there any method to linearize the following optimization problem?
\begin{align}
\min_{x,y} &~~ c~[x; y] \\
\text{s.t.} &~~ \sum x\leq \alpha_1 \\
&~~ \sum |y|\leq \alpha_2 \\
&~~ \sum y= 0 \\
&~~ x+|y| \leq 1 \\
&~~ (x,y)\in \{0,1\} \times…
Reda
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4
votes
1 answer
Linearization of a PDE
I have been struggling with some linearization argument of the following paper: "M. Weinstein: Modulational stability of ground states of NLS". In order to give a bit of context to my question, let us consider the NLS equation $$…
Sharik
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4
votes
2 answers
Newtons method for finding reciprocal
Define a function 1 which is $f_1(x)=a-1/x$
and function 2 which is $f_2(x)=1-ax $
If I set both to zero I am looking for when $x=1/a$ as the root using Newtons method.
When I do this I get two different answers however and they should surely both…
user129299
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4
votes
1 answer
Stationary points and linearisation of non-linear system
So, the problem is:
Find and discuss the behavior of the stationary points of the system :
$$ x'=-y+x\cdot (x^2+y^2)\cdot \sin\sqrt{x^2+y^2} =f(x,y)$$
$$ y'=x+y\cdot (x^2+y^2)\cdot \sin\sqrt{x^2+y^2}=g(x,y)$$
So in the beggining I Linearised the…
Paris K. Patsogiannis
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3
votes
1 answer
Feedback linearization of a controllability form system
Given the system:
$$\begin{cases}\dot{x_1}= x_2\\\dot{x_2} = -10x_1+1.8{x_1^2}-0.25x_2 +u,\end{cases}$$
where
$$u=-1.8{x_1^2}+v,$$
I get the system:
$$\begin{cases}\dot{x_1}= x_2,\\\dot{x_2} = -10x_1-0.25x_2 +v,\end{cases}$$
with the…
mdeli
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