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Questions tagged [nonlinear-analysis]

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

3
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1answer
62 views

Solve differential equation $xyy'=x^4+y^4$

How to find general solution to this differential equation (if it exists): $$ xyy'=x^4+y^4 ?$$ I do not know how to even approach it since I never dealt with nonlinear equations. Only thing that I ...
0
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0answers
16 views

How to find explicit solution for recursive sequences with nonlinear terms?

Are there any general known methodologies or algorithms to derive an explicit equation for a nonlinear sequence? As an example, I am struggling to find an explicit expression for the sequence below: $...
0
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1answer
32 views

Riccati and Linear 1st Order ODE Parallel

I have noticed a certain similarity between Riccati 1st Order ODEs and linear 1st Order ODEs. Specifically, the general solution for each is given by any particular solution plus some function of the ...
0
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0answers
11 views

Lookup tables inversion

I'm having troubles solving the following problem. I have 2 2D lookup tables of the form $\lambda_1 = f_1(x, y) \\ \lambda_2 = f_2(x, y)$ where $x\in \{x_1, x_2,...,x_N\}$ and $y\in \{y_1, y_2,...,...
0
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1answer
29 views

Continuity for a nonlinear functional

I'd like to check the continuity for the nonlinear functional $T: (C^{0}([0,1],\Vert \cdot \Vert_{\infty}) \rightarrow (C^{0}([0,1],\Vert \cdot \Vert_{\infty}) $ , with $T(f)(x)=\arctan(f(x))$. I ...
0
votes
1answer
21 views

Determining components of Halley's method

I'm working on a homework problem about Halley's method and I'm not quite sure where to start and how I prove a cubic convergence. Consider the iterative method $$x_{n+1} = x_{n} − \frac{h_n}{Ah_n + ...
0
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1answer
49 views

What algorithm to use to find the non-linear mapping function between 2D shapes generated from biosignals attractors?

I have two biosignals recording the same phenomenon with different methods. Target signal is a reference signal and I would like to find some non-linear mapping from the original signal to the ...
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0answers
16 views

Uniqueness and existence of this system, verifying my answer

I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve. 1)Getting the Jacobian, I obtain $$ J= \begin{bmatrix} 0 & 1\\ -1-2xy & 1-...
-1
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1answer
39 views

Nonlinear equation analysis withe epsilon value [closed]

Consider the nonlinear equation $$\frac{d^2x}{dt^2}+\epsilon\sin(x)=0,~~\epsilon \ll 1\\ x(0)=0,~~\dot{x}(0)=1$$ and find... A. The value of $x_0$ as $\epsilon$ goes to $0$ B. The first order term $...
1
vote
1answer
38 views

Solve an equation involving the error function

Let $0<a<1$ be given. The equation: $$a = 1 - \frac{2\sqrt{x/\pi}}{\mathrm e^x \mathrm{erf}(\sqrt x)}$$ has a unique root $x$, because the right-hand side is increasing in $x$, and goes to $0,...
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0answers
27 views

boundary problem: use main theorem of monotone operators

I am trying to investigate for which $\alpha \in \mathbb R$ the boundary problem $$-u''(x)+\alpha sin(u(x))u'(x)=f(x)$$ $$u(a) = u(b) = 0$$ is weakly solvable using the main theorem of monotone ...
2
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0answers
23 views

How to prove that a linear differential operator generates a semigroup?

I am working on a nonlinear PDE and I should now prove that the linear operators $L:=\partial_x^3+\partial_x^2$, $S(u):=u L$ and $T(u):=L+S(u)$ are generators of a $C_0$ semigroup such that $\|e^{-s A}...
0
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0answers
18 views

Fourier Transform of B-splines linear combination, any application to spectral analysis of non stationary series?

In a framework of non-linear regression one could opt for using B-splines in order to approximate the values $y$ assumed by an unknown function $f(x)$. If we denote with $B_j(x,p)$ the value of the ...
1
vote
1answer
41 views

Transform a non-linear differential equation into a linear equation

Following the work of Yaoji Lu - 1967 (here's a link to the full paper) I got stuck at the step when the author transform a non-linear differential equation into a linear equation (eq. 3.9 pag. 19). ...
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0answers
42 views

How to prove the best polynomial approximation operator is continuous?

For any function $f\in C[0,1]$, it is well known that there exist an unique polynomial $p^{*}\in P_n[0,1]$ such that $||p^{*}-f||_{\infty}\leq ||p-f||_{\infty}$ for any $p\in P_n[0,1]$. In this ...
0
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0answers
8 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 ...
0
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0answers
25 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 ...
1
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0answers
33 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_{...
0
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0answers
484 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 ...
1
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4answers
96 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(...
1
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0answers
8 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 ...
1
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0answers
28 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 ...
0
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0answers
21 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?
1
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1answer
50 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 ...
1
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0answers
25 views

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 ...
1
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0answers
51 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)(\...
1
vote
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 ...
1
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0answers
52 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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0answers
14 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....
2
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0answers
52 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)...
2
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0answers
42 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}{\...
0
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1answer
34 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 ...
0
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0answers
117 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 ...
1
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4answers
80 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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0answers
30 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 ...
2
votes
1answer
55 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 ...
3
votes
1answer
43 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 ...
0
votes
2answers
76 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}...
1
vote
1answer
40 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}\...
0
votes
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} (\...
0
votes
0answers
33 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 \...
1
vote
2answers
133 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 ...
3
votes
1answer
69 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\...
1
vote
2answers
43 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 ...
0
votes
0answers
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[\...
0
votes
1answer
67 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 ...
1
vote
1answer
37 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)|...
0
votes
2answers
98 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!
9
votes
1answer
168 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 ...
0
votes
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 $u_0\in H$ ...