Questions tagged [nonlinear-analysis]

For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.

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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 ...
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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{\...
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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)...
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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
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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 ...
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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} \...
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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+...
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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 ...
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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 ...
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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 $h\to 0$ ...
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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 ...
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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'...
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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 ...
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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 ...
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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 $\...
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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)^...
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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
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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'\...
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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 ...
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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 ...
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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}...
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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 ...
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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 \...
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0answers
10 views

Motivation for Pseudomonotone Operators

I just got introduced to pseudo monotone operators $A:X\rightarrow X'$ in Banach spaces, i.e. $$\forall u_k\rightharpoonup u\subset X \text{ with } \lim\sup \langle Au_k,u_k-u\rangle\leq 0$$ $$\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $a$ ...
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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 $f$ ...
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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
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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 ...
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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(\...
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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 ...
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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 ...
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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^{\...
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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}{...
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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
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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)] = ...

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