Questions tagged [nonlinear-analysis]
For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.
391
questions
0
votes
0answers
8 views
How to add nonlinear function to equation without transformed output
I am trying to adjust an equation to account for a non-linear trend, while preserving the final product without a transformation.
My target, is to estimate a concentration gradient, e.g. If the air ...
0
votes
0answers
8 views
Confusion about the optimal parameter value non linear least squares
The normal equation for the nonlinear least squares is denoted as
$ \boldsymbol{\Delta} \boldsymbol{\beta}=\left(\mathbf{J}^{\mathrm{T}} \mathbf{J}\right)^{-1}\mathbf{J}^{\mathbf{T}} \boldsymbol{\...
0
votes
1answer
27 views
How do I perform Non linear Least Squares on a model with predefined lag structure?
Suppose I have the following formula:
$$y_t = \beta_0\sum_{i=0}^p w(\delta;i)x_{t-i}$$
Where $\displaystyle w(\delta;i)=\frac{\exp(\delta_1 i+ \delta_2 i^2)}{\sum_{i=0}^p \exp(\delta_1 i+ \delta_2 i^2)...
-1
votes
0answers
25 views
solving first order nonlinear equation [closed]
I need to solve this equation n`(t)=co I(t)+ c1 n(t)+ c2 n^2(t) +c3 n^3(t) by hand analysis not computer programs. what other methods than Euler's method?
Thank you
0
votes
0answers
12 views
possible structure for the optimal solution
Consider the following problem
$$\max_{x_{ij} \in [0, 1]} \sum\limits_{i=1}^n\sum\limits_{j=1}^n c_{ij} f_{ij}(x_{ij}) \frac{g_i(x)}{\max(g_1(x), g_2(x), \cdots, g_n(x))}, $$
where $f_{ij}$ is a ...
1
vote
0answers
19 views
What does the non-existence of Lyapunov number mean? [closed]
For discrete dynamical system, $\mathbf{F}:\mathbb{R}^m\rightarrow\mathbb{R}^m$, $k$-th Lyapunov number of the orbit beginning from $\mathbf{x}_0\in \mathbb{R}^m$ is defined as follow.
\begin{align}
\...
1
vote
0answers
12 views
A non-linear convolution recursion relation [closed]
Is it possible to solve the following recursion:
\begin{equation}
T_{2n+1} = L_{2n+1} \sum_{n_1,n_2,n_3 =0}^{n_1+n_2+n_3 = n-1} T_{2n_1+1}T_{2n_2+1}T_{2n_3+1}
\end{equation}
Where $T_1 = 1$ and $L_{2n+...
1
vote
0answers
38 views
Nonlinear ODE involving trigonometric terms
I need to solve the following ODE:
$$y'+A\sin(2y)+B\sin(y)+C\cos(y)+D=0$$
I want to find y' (explicit/unexplicit) rather than y.
I tryied to use the Weierstrass transform, after substituting the ...
1
vote
1answer
33 views
Proof that, if the norm of any normed space is (Fréchet) differentiable, then the derivative is continuous.
I want to prove the following lemma.
Let $X$ be a normed space and its norm $\|.\|$ be Fréchet differentiable on $X \backslash \{0\}$. Then the derivative is continuous there.
I've got this from a ...
1
vote
0answers
32 views
In which points is the supremum norm on $C[0,1]$ and $c_0$, respectively, Gâteaux/Fréchet differentiable?
$f : U \to Y$ where $U\subset X$ is open and $X, Y$ are normed spaces is called Gâteaux differetiable at $u\in U$ if there exists a bounded linear operator $T$ from $U$ to $Y$ such that for $h\to 0$ ...
0
votes
0answers
21 views
How to prove the optimal solution exists for a nonlinear optimal control problem?
I am doing a nonlinear optimal control design project, typically a practical engineering problem.
It is easy to solve that with direct method. However, I am wondering is there any way to prove that ...
1
vote
1answer
26 views
Example for an operator that is strictly monotone but not maximally monotone (or the other way)
While the definition of strictly monotone = nowhere constant operators seems intuitive, I find it hard to picture in which way maximal monotone operators ($\forall (u,u') \in X \times X', \langle u'-v'...
3
votes
1answer
48 views
Is the sine operator on $L^2[0,1]$ Fréchet differentiable or not and why?
This problem has given me some trouble. Let $F$ be the operator on $L^2[0,1]$ defined by $F(g)(t)=\sin g(t)$. I'm trying to determine whether or not $F$ is (Fréchet) differentiable in that space. I ...
0
votes
0answers
29 views
How to find a quadratic convergence function on fixed point iteration method on root finding?
I've read several references, and it is true that:
A point is called a fixed point if $f(x_0) = x_0$.
It can further be reduced to find root of a non-linear function $f(x) = g(x) - x = 0$
The fixed ...
0
votes
0answers
13 views
Range preserving matrix derivative
Let $M(x) \in \mathbb{R}^{d\times d}$ be a matrix-valued function on $\mathbb{R}^d$
and let $F(x) = M(x)x$.
Suppose $M(x)$ is differentiable at $x$ and the rank of $M$ is constant for an openset $\...
0
votes
0answers
42 views
Solve a Piecewise Non-Linear Function
I have a problem I am having hard time to wrap my head around. It looks relatively simple. So, it is a piecewise non-linear profit function $\pi(x)$as below:
$$
\pi(x)=\begin{cases}mq_2(1+x)-c_A(q_2x)^...
0
votes
0answers
12 views
What is the difference between asymptotic method and perturbation method? Are they the same?
I am confused with "asymptotic method (theory)" and "perturbation method (theory)".
What is the difference between them? Are they the same?
Thank you so much.
HC6
2
votes
1answer
60 views
SPECIAL CASE first order PDE, characteristic curves method
I know how to use the characteristic method but I am in a little trouble with the following PDE:
$$
\partial_xu(x,y)+\partial_yu(x,y)=a(x)u(x,y)u(x_0,y)\\
u(x,y=0)=f(x)
$$
where $x_0$ is a fixed value ...
0
votes
1answer
36 views
How to know when a transcritical bifurcation occurs (Example 3.2.1 Strogaz Nonlinear Dynamics and chaos)
Picture of Question + solution
Hi,
In this question After we expand $\dot{x}$ around $x^*$ = 0 , we get $\dot{x}$ = (1-ab)x + ($\frac{1}{2}$a$b^2$)$x^2$ + O($x^3$).
How do we jump from this to knowing ...
0
votes
0answers
17 views
Study of Strange Attractors
I have a background in Mechanical Engineering and I am currently studying non-linear systems with a focus on 'strange attractors.' What suppplementary reading should I do or what courses should I ...
2
votes
0answers
23 views
If $f(t) = a_0 + a_1 t + a_2 t^2+\mathcal O(t^3)$, what does the condition $|a_1|/|a_2| > 1$ signify geometrically / analytically?
Let $f:\mathbb R \to \mathbb R$ be a function which is twice continuously differentiable in a neighborhood of zero, with Maclaurin expansion $f(t) = a_0 + a_1 t + a_2 t^2+\mathcal O(t^3)$.
Question. ...
1
vote
0answers
24 views
What is the classification of the bifurcation of a tent map?
Considering the tent map where $x_{n+1} = f(x_{n})$ and $f(x)$ is defined as
$$
f(x)=
\begin{cases}
\mu x, & 0 \leq x\leq \frac{1}{2} \\
\mu - \mu x, & \frac{1}{2}\leq ...
0
votes
1answer
32 views
Linearization of a fixed point (dynamical systems)
Just looking at the following piece of math from Strogaz's dynamic / chaos book.
What I don't understand is the last part, where he claims that O($Ƞ^{2}$) is negligible if $f^{'}(x^{*})!=0$.
I guess ...
0
votes
1answer
32 views
Intuition behind regularization of non-coercive variational inequalities
I am looking into the theory of non-coercive variational inequalities and I came accross the following approximation techinique:
Fix an infinite dimensional Hilbert space $H$ and let $K \subset V$ be ...
3
votes
0answers
25 views
Why are continuous partial derivatives up to order two (rather than one) of nonlinear autonomous (2D) systems sufficient for linear approximation?
In Boyce and Diprima's ODE's and BVP's (10th edition page 522), it says that for the nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y) \qquad\qquad\qquad (10),$$
"The system ...
1
vote
0answers
21 views
Linear subspace of a normed linear space is chebyshev.
We know that for a normed space $X$ and for a subset $G$ of $X$, where $x\in X$ and $g_0 \in G$, a best approximation of $x$ is $\|x-g_0\|=\inf_{g \in G} \|x-g\|$. Furthermore $G$ is a Chebyshev ...
0
votes
1answer
19 views
Prove that the element $(0,1)$ has infinite many best approximations in the linear subspace $B=\{ (x,x)| x\in\mathbb{R} \}$
I have tried this problem as:
By using the definition
For a subspace $S$ of a normed linear space $X$, for all $x$ belongs to $X,g\in S$ is a best approximation of $x$ if $\|x-g\|=\inf\{\|x-g'\|:g'\...
1
vote
1answer
22 views
On continuity of the Gateaux derivative of p-Laplacian operator
Let $\Omega\subset \mathbb{R}^n, N\geq3$, be an open set. For $p\in(1,+\infty)$, define a functional $J:W_0^{1,p}(\Omega)\rightarrow\mathbb{R}$ by
$J(u)=\int_\Omega |\nabla u|^p\,dx.$
Then $J$ is ...
0
votes
1answer
22 views
Obtain non linear solution using neural network
The function f(x)=theta·x where theta is a row vector and x is a column vector, is a linear function. How can I obtain a non-linear function g(x) using a multi layer network, that also takes in x as ...
1
vote
1answer
32 views
Relative homology of sublevels is the homology of the attached $k$-cell
Let $M$ be a smooth manifold of dimension $N$, $f: M \to \mathbb R$ smooth and let $p$ be a nondegenerate critical point, $f(p) = c$. Suppose that it is the only nondegenerate critical point in $f^{-1}...
0
votes
0answers
37 views
2 equivalent way of applying activation functions in RNN
I don't understand why in RNN the 2 following ways of applying the activation functions are equivalent:
First way:
$$
h_t = W\sigma(h_{t-1}) + U x_t + b
$$
Second way:
$$
g_t = \sigma(Wg_{t-1} + U x_t ...
7
votes
1answer
121 views
What is the most general Carathéodory-type global existence theorem?
I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$
$$
\begin{equation}
\left\{ \begin{aligned}
x'(t) &= f(t, x(t)), \qquad t \...
0
votes
0answers
10 views
Motivation for Pseudomonotone Operators
I just got introduced to pseudo monotone operators $A:X\rightarrow X'$ in Banach spaces, i.e.
$$\forall u_k\rightharpoonup u\subset X \text{ with } \lim\sup \langle Au_k,u_k-u\rangle\leq 0$$
$$\...
0
votes
0answers
42 views
Is there an analytic solution for the following non-linear equation $x'(t) = a(t) + b\, x(t) + c\,x^3(t)$?
Here $b, c$ are constants. Of course there exists locally a solution by the Picard-Lindelöf theorem but I'm looking for an explicit expression.
In fact, this question comes from here but I changed the ...
1
vote
2answers
86 views
Is there by chance an analytic solution of the following non-linear ODE: $x'(t) = a\, x(t) + b\, x^3 (t)$?
I'm in fact interested in a PDE for which I try to get some intuition (roughly, I interpret my PDE as a function of time with value in a space of functions of space variables). Does someone by luck ...
2
votes
0answers
50 views
Static meniscus - analytical solution [closed]
I have been trying to understand how to find the solution to the following ODE
$$\frac{d}{dx} \frac{1}{[1+(dh/dx)^2 ]^{1/2} }=h$$ where $h=f(x)$.
It would be great if someone could help me understand ...
0
votes
1answer
20 views
Onto property of the function
How can we prove the onto property of this non-homogeneous function
$$f(x)= (|x|^{p-2} + |x|^{q-2}) x \quad \text{for} \quad x \in \mathbb{R} \quad \text{where} \quad p, q >1.$$
Any ideas (without ...
1
vote
0answers
23 views
Rewriting a nonlinear transformation in multiple dimensions
Let $x,y\in\mathbb{R}^n$, $n>0$. Suppose we have an arbitrary nonlinear mapping $T(x):\mathbb{R}^n\to\mathbb{R}^n$, that basically performs a nonlinear coordinate transformation for which we assume ...
1
vote
0answers
43 views
What is this hybrid symbolic-numeric algorithm for solving non-linear ODEs called?
I'm an engineer so please forgive my lack of knowledge in exact mathematical terminology:
For linear differential equations, it is possible to use the principle of superposition or convolution ...
0
votes
1answer
48 views
Non-linear system of differential equations
I am trying to solve this differential equation which popped up in an engineering problem.
\begin{align}
&a\dot{V}(t) + b P(t) &= x(t)\\
&V(t)P(t) &= y(t)
\end{align}
The values $a$ ...
-1
votes
1answer
42 views
Riccati equation solution as a ratio of Bessel functions
I'm trying to find a general solution to this Riccati equation:
$f'(x) = -a f(x)^2 + g(x)$
where $a$ is a constant, $g$ is a smooth real-valued function of the real independent variable $x$, and $f$ ...
1
vote
0answers
30 views
On a compact imbedding problem
Denote $$\mathcal{H}=\{u\in H^1(\mathbb{R}^3):\int_{\mathbb{R}^3} V(x)|u(x)|^2\,dx<\infty\},$$ where $V(x)\in L_{loc}^\infty (\mathbb{R}^3)$, $V(x)\geq0$ and $\lim_{|x|\rightarrow \infty} V(x)=\...
2
votes
1answer
69 views
Numerical method to solve a non-linear ODE
I want to solve numerically the following non-linear ordinary differential equation:
$$f''(x)=A(1+f'(x)^2)^{3/2}-\frac{f'(x)}{x}(1+f'(x)^2)$$
where $A$ is a constant and $x\in[0,R]$. The ODE is also ...
2
votes
0answers
33 views
What is the so-called “bootstrap” argument in Mathematics and its application to nonlinear Schrodinger system.
We have the following nonlinear Schrodinger equations ($n\leq3$):
$$\begin{cases}
\Delta u_1 -u_1+\mu_1u_1^3+\beta u_1u_2^2=0\\
\Delta u_2 -\lambda u_2+\mu_2u_2^3+\beta u_1^2u_2=0\\
u_1,\,u_2\in H^1(\...
1
vote
0answers
55 views
$I-f$ is a nonlinear homeomorphism on an infinite dimensional Banach space. $P$ is a linear projection. Is $I-Pf$ a homeomorphism?
$f:X\to X$ is a nonlinear operator where $X$ is an infinite dimensional Banach space, $I-f$ is a homeomorphism, and $P:X\to X$ is a linear projection. Additionally, we may assume that $I-Pf$ is ...
0
votes
0answers
16 views
Are there any other techniques besides linearization to approximate nonlinear systems?
It seems like anytime one proposes some approach to “solve” nonlinear equations/systems it always involves converting the problem to a linear system, which is of course tractable.
Are there other ...
1
vote
0answers
23 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^{\...
0
votes
0answers
38 views
Extension of an ODE to dynamical system with certain properties
We have a gradient ODE:
$\frac{dx}{dt}=\frac{df}{dx}$
where $f=-x^2$
I want the condition to be met on a given system:
$x''+x'=0$
I.e. initial ODE turns into a dynamic system:
$\begin{cases} \frac{dx}{...
-1
votes
2answers
32 views
How to do a 3 constant reciprocal (multiplicative inverse) regression
I am trying to fit a set of data to a curve such as:
$y=\frac{m}{x-a}+b$
Without the constant $a$, it is easy to define $z=\frac{1}{x}$ and convert it to a linear model. But I have not been able to ...
4
votes
1answer
53 views
Non-linear function with properties related to Gaussian distribution
Let $\mathcal{A}_n$ be the set of functions $f : \mathbb{R} \to \mathbb{R}^n$ such that:
$f$ is continuous and $f$ is differentiable almost everywhere
if $X \sim \mathcal{N}(0, 1)$, then:
$E[f(X)] = ...
