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Questions tagged [nonlinear-analysis]

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

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5 views

Positive and negative eigenvalues via Brouwer degree

The problem is: if $B=B(0,1)\subset\mathbb{R}^n$ is the open unit ball and $f$ is a continuous function defined on the closed unit ball $\overline{B}$ with $0\notin f(\overline{B})$, then there exists ...
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23 views

Property of continuous functions defined on a sphere of R^n where n is odd

If $B=B(0,1)$ is the unit open ball in $\mathbb{R}^n$ with $n$ odd and $f:\partial B \rightarrow \partial B$ is continuous, then exists $x\in \partial B $ such that $f(x)=x$ or $f(x)=-x$. Any idea of ...
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29 views

Finding a conjugation given a first integral

In the ODE given by: $x'=X(x)$ , where $X$ is my vectorial field $\in C^1$ in an open subset of $\mathbb R^n$ , If $X$ have $f$ a first integral and $df(p)\neq0$ then there exists a neighborhood $V_{...
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161 views

How to determine linear terms from the nonlinear dataset?

Let us take the parametric curve r($t$) = [$t^2$;$t$], $t$ = [0,1]. Using this equation, I generate 1000 points. Now my goal is to determine the value of $t$ for each point on the curve without using ...
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4answers
90 views

Polynomial growth implies locally Lipschitz?

Let $f:\mathbb{R}^m\to\mathbb{R}^m$ satisfy $\|f(x)\|\le c\|x\|^n$ for some (re-edit:) $n\in\mathbb{N}$ with some constant $c>0$. Is $f$ locally Lipschitz? I see that it is around $x=0$: $\|f(x)-f(...
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7 views

Preservation of Minima with a Non-Linear Monotonic Mapping

I am trying to make a transformation on the set of parameters within the Ising model namely, $a_i$ and $b_{i,j}$. The Hamiltonian is: $H = \sum_{i} a_ix_i + \sum_i \sum_j b_{i,j}x_ix_j $ They need to ...
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25 views

Non-linear Basis Functions for PDE

An idea popped into my head awhile ago while doing a project on non-linear effects of systems of coupled oscillators, but I'm not an expert on this subject so I don't know if it's any good. From ...
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17 views

Hessian Metric and Bregman divergence

I read from a paper that Bregman divergence is an approximation to the Hessian metric when the two points are nearby. What is the definition of Hessian metric? How can we derive this approximation?
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1answer
48 views

Regularity of coefficients in Galerkin method

Let $V\subset H\subset V^*$ be an evolution triple and suppose that $u\in W^{1,2}(0,T;V,H)$, where $$W^{1,2}(0,T;V,H)=\{f\in L^2(0,T,V)\,|\,f'\in L^2(0,T,V^*)\}.$$ Now, let $\{w_1, w_2,...\}$ be a ...
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When a nonlinear equation is regular?

I was solving a nonlinear equation in matlab using the function fsolve. Matlab says "the problem appears regular as measured by the gradient". In the definition of problem appears regular it is ...
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49 views

Solve the nonlinear equation

Suppose that $f:E\to F$(between Banach spaces), is of the form $$f(x)=f(0)+D(x)+N(x).$$ Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\...
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1answer
21 views

Source of non-linear Laplace equation

Consider the following non-linear generalization of the Laplace equation $$\Delta \phi - \frac{\sum_i (\partial_i \phi)^2}{2 \phi} = 0$$ I am looking for spherically symmetric solutions, so I assume ...
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50 views

mapping degree of $f: X \rightarrow Y$ with $\dim X \neq \dim Y$

I am writing about the mapping degree (also called Brouwer degree or topological degree). When calculating the degree for a function $f$, one has to use the determinant of the Jacobian of $f$. This ...
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12 views

Basic example of evolution triple

Let $G$ be a bounded region of $\mathbb{R}^n$ with $n\ge1$. We set $$V=\dot{W}^{m}_{p}(G),\qquad H=L^2(G),$$ with $2\le p<\infty$ and $m\ge 1$. Then "$V\subset H\subset V^*$" is an evolution triple....
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49 views

On solution of a nonlinear differential inequality

I have the following differential inequality: $$f'(x)\geq cf(x)^a,\quad \forall x\in[0,1]$$ where $0<a<1,\, f(x)\geq 0$. I'm taking the following approach to solve the problem: $$f'(x)\geq cf(x)...
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35 views

To show that a 2D nonlinear ODE undergoes a pitchfork bifurcation

Show that a pitchfork bifurcation occurs when $\mu = 0$ for the system $$\dot{x} = \mu x + xy + 3y^2$$ $$\dot{y} = -2y + x^2 + 2xy^2$$. Attempt at a solution If $\dot{x} = 0$ then $$x = \frac{-3y}{\...
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1answer
33 views

To show that $\dot{x} = -y - x +y^2 - x^2,\; \dot{y} = xy$ has no periodic orbits.

I've tried using index theory, but there's a non-isolated fixed point at $(0, 0)$ (the remaining fixed points at $(0, 1)$ and $(-1, 0)$ are saddles). The terms in the equations have even indices and ...
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101 views

Proof check: Show that $x_{n + 1} = f(x_n) = 15 - x_n^2$ is a (Smale) horseshoe

A solution is given here: However, I believe it to be incorrect because $f(K_1) = f(K_2) \neq (-\sqrt{15}, \sqrt{15})$. Attempt at a solution: Referring to the diagram from the question... By the ...
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4answers
54 views

Example of nonexpansive mapping.

I am trying to construct some examples of the nonexpansive mapping $T$ from $R^2$ to $R^2$ such that $T$ should have fixed points more than one. But I could not construct. Can somebody help me? Please....
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26 views

To classify bifurcations by calculating partial derivatives evaluated at the bifurcation point

Question: Given a one dimensional system $\dot{x} = f(x;\; \mu)$ for which a bifurcation of fixed points occurs at $(x^\star, \mu_c)$, where $x^\star$ is a fixed point and $\mu_c$ is the ...
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1answer
51 views

To find the approximate period and approximate equation of the limit cycle for a system with a Hopf bifurcation

Question: The system $$ \dot{x} = 3y + 3x^3 + xy $$ $$\dot{y} = -3x + \mu y + 2xy^2 - y^3$$ undergoes a Hopf bifurcation at $(x, y) = (0, 0)$ as $\mu$ passes through 0. Calculate the approximated ...
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1answer
37 views

Does the fixed point corresponding to a Hopf bifurcation vary in the $(x, y)$ plane as the system parameter $\mu$ changes?

The question is for a 2D system, but for the sake of simplicity, let's consider a 1D system $\dot{x} = \mu + x^2$. Then for $\mu < 0$ the fixed point $x = \pm\sqrt{\mu}$ varies along the $x$-axis ...
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2answers
57 views

Using Hartman-Grobman to determine stability of ODE

Take the system of ODES: $\dot x=(\epsilon x+2y)(x+1)$ $\dot y=(-x+\epsilon y)(x+1)$ Linearise this system and find the eigenvalues of its Jacobian at the origion Answer: Eigenvalues $\lambda_{\pm}...
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1answer
35 views

How to determine whether the vector valued function satisfy the Lipschitz condition?

Given, vector-valued function $\phi(x)=\left[ \begin{array} aa x_{1}^{2}/(x_{1}+b)+x_{1}x_{3}+\frac{6}{\pi}arctan(\frac{x_{1}}{b}-180)+e+c \\ x_{1}x_{3}+x_{2}x_{4}\\ 0\\ 0\\ 0\\ 0\\ 0 \end{array}\...
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1answer
28 views

Uniqueness of values for a transcendental equation $(\sigma\beta)^2(-1+x/\beta+e^{-x/\beta })=y$

I been struggling with a nonlinear equation for a couple of weeks (months). Maybe you can give me a hint. I need to prove the following. Given the equation \begin{equation} \tag{1}\label{eq} (\...
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31 views

A question on homotopy invariance of the topological degree

Let $\Omega =B_1(0)=\{z\in\mathbb{C}\cong\mathbb{R}^2 : \lvert z\rvert<1\}, y=0,$ and define $h(t,z)=\begin{cases} \lvert z\rvert, & t=0, z\in\overline\Omega \\ \lvert z\rvert \...
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2answers
126 views

Show that the origin of the following system is stable (non-linear vs linear stability)

$$\dot{x} = -2y -x^3 + x^2 y^2$$ $$\dot{y} = x - x^2 y$$ Jacobian evaluation at $(0, 0)$: $$\begin{pmatrix}0 & -2\\ 1 & 0\end{pmatrix}$$ Clearly the determinant is greater than zero but the ...
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1answer
67 views

Can $y'' = e^y$ be converted into a linear ODE through successive variable substitutions?

Consider the following second-order nonlinear ODE: $$y'' = e^y$$ The solution to this ODE is known precisely: $$y = \ln\left(\frac{1}{2}c_{1}\left(\tanh^2\left(\frac{1}{2}\sqrt{c_1(c_2+x^2)}\right)-1\...
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2answers
42 views

$\int_a^b x'(t)dt=x(b)-x(a)$ in Banach space $X$

Let $X$ be a Banach space. Then $\int_a^b x'(t)dt=x(b)-x(a)$ if $x:[a,b]\rightarrow X $ is continuously differentiable. I have a few problems understanding what I have to show. First of all, what is ...
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12 views

FInd the measure of noncompactness

Find the measure of noncompactness of the set $B\subset C([0,1])$ defined by $B=\big\{ x\in C([0,1]) \big|x(0)=0,x(1)=1,0\leq x(t)\leq \frac{1}{2},t\in[0,\frac{1}{2}],\frac{1}{2}\leq x(t)\leq 1,t\in[\...
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1answer
49 views

What is the “bifurcation of a fixed point”?

Isn't the bifurcation of a system, say $\dot{x} = \mu \sin(x) - 2x$, related to the system rather than a fixed point, say $x^{\star} = 0$?. My understanding is that a bifurcation occurs when there's ...
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1answer
31 views

Why do every strong extremum is simultaneously the weak extremum?

My Doubt Here $||f||_{1}=\sup_{x\in[0,1]}|f(x)|+\sup_{x\in[0,1]}|f'(x)|$ where as $||f||_0=\sup_{x\in[0,1]}|f(x)|$. We can easily prove from definition that $$||f||_0=\sup_{x\in[0,1]}|f(x)|...
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2answers
91 views

Nonlinear Ordinary Differential Equations

Does anyone knows if the following equation $$ x'=\frac{t}{x^4}$$ is a nonlinear ordinary differential equation ? Because usually the nonlinear ODE is of the form $$x'=tx^4$$ Thanks in advance!
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1answer
159 views

Statements on the behavior of solutions to $y' = \sin(xy)$ for large $y(0)$

Consider the following initial-value problem involving a nonlinear first-order ODE: $y' = \sin(xy), \quad y(0) = y_0$. For large enough values of $y_0$, the solutions to this equation appear to ...
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0answers
18 views

A question about the assumptions of Galerkin's Method

Let $V, H$ be real, separable Hilbert spaces and suppose that $V$ is dense in $H$. Since $V$ is separable, there exists a countable basis, namely $\{w_1, w_2,...\}$. Suppose we are given $u_0\in H$ ...
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0answers
19 views

Solution Uniqueness of a Nonlinear Degenerate Problem

I'm studying the following problem: Let $X$ be a reflexive and separable Banach space, $H$ a separable Hilbert space (both real) such that $X\subseteq H$ is dense and continuous; $A: X\to X'$ not ...
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2answers
42 views

Weak closure of intersection in reflexive Banach space

Let $X$ be a reflexive Banach space. Let $\mathcal{S}$ be a family of finite-dimensional subspaces of $X$. Consider a bounded sequance $(x_Y)_{Y\in \mathcal{S}}\subset X$. Define $$C_Y=\bigcup_{Y'\...
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1answer
30 views

Is adjoint operator a projection?

Let $X$ be a reflexive Banach space and $Y$ its finite dimensional subspace. Let $i_Y\in \mathcal{L}(Y,X)$ be the embedding operator. Why $i^*_Y\in \mathcal{L}(X^*,Y^*)$ is the projection on $Y^*$?
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37 views

The best approach to tackle the following class of mixed integer nonlinear programming models

I would be thankful if anyone can help me to find the best approach to solve the following MINLP model. In this model, $p_{js}$, $p_{js}$ and $Q$ are parameters. $x_{js}$ is a binary variable and $y_i$...
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0answers
26 views

On definition of coercivity over an open domain

By far I have seen, in convex analysis and optimization literature, a coercive function $f: \mathbb{R}^n \to \mathbb R$ is defined as \begin{align*} f(x) \to +\infty \text{ as } \|x\| \to \infty. \end{...
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1answer
67 views

Approach to solving underdetermined nonlinear system of equations

I've gotten into a problem I haven't really worked with before in my numerics classes. I have a system of four nonlinear equations with six parameters. Newtons method, Boydens method etc. all ...
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1answer
42 views

Generalizing the estimate of norm of solution of an ODE to Banach spaces

I am reading Hale's Oscillations in Nonlinear Systems, and I want to generalize some results to Banach spaces. However, I cannot find a way to generalize the following part (which is in the ...
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18 views

Why is $H$ dense in $V^*$ in Gelfand triple?

Let $"V\subset H\subset V^*"$ be an evolution triple. From the definition, we have that $V$ is dense in $H$. Why $H$ is dense in $V^*$? Is it because of the reflexivity of $V$?
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1answer
48 views

Method of Lagrange multipliers for constrained minimum of functional

I have this problem where I have to find the steepest descent direction of a functional. Basically, it comes down to solving a Poisson PDE subject to a certain constraint ($||v|| = 1$). The image ...
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0answers
26 views

Sensitivity (perturbation analysis) of a system with nonlinear equation

suppose we have a system with 3 unknowns ${\theta, \phi,\rho}$ ;and 3 equations.where we have a constant ${\eta}$, and 3 different ${\delta}$ for each equation. The system is as below(each${\delta_i,i=...
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1answer
78 views

Problems to understand Lyapunov stability - Nonlinear Control

I'm learning nonlinear control and I have already learn how to do phase plots. It was not a big deal. Just using ode45 in Octave/Matlab. But when I going to learn something, I only focus on practical ...
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1answer
32 views

Analytical solution for specific - First Order Nonlinear Differential Equation

I would like to understand how to solve $-K_1\frac{dx(t)}{dt} = x(t)^2K_2 + x(t)^{0.6}K_3 + K_4$ analytically.
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3answers
79 views

Show that $x^3 - 2 x^2 +\log(1+x)(x(3x+4) -2(1+x)^2 \log(1+x))$ is positive

I want to show that (the following just gets rid off large brackets) $$x^3 - 2 x^2 +x(3x+4)\log(1+x)-2(1+x)^2 \log^2(1+x)>0, \ \ \mbox{for}\ \ x\in(0,\infty).$$ My attempt: Transfer all negative ...
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0answers
37 views

Numerical Differentiation of non-linear functions

I am trying to write a code to solve a differential equation using a non-standard method. The complications I am facing boils down to numerically evaluating a non-linear function (such as 1/x). Due to ...
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0answers
16 views

Vector space dimension after nonlinearly mapping spanning vectors

Let $\mathbb{R}^d \supset V$ be a vector space spanned by elements $v_1, v_2, \dots, v_n$, not necessarily linearly independent. Let $v_i = \sum_{j=1}^dv_{ij}\mathbf{e}_j$ be the representations of $...