Questions tagged [nonlinear-analysis]
For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.
3
votes
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
votes
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
votes
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
votes
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
votes
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
$$x_{n+1} = x_{n} − \frac{h_n}{Ah_n + ...
0
votes
1answer
49 views
What algorithm to use to find the non-linear mapping function between 2D shapes generated from biosignals attractors?
I have two biosignals recording the same phenomenon with different methods.
Target signal is a reference signal and I would like to find some non-linear mapping from the original signal to the ...
0
votes
0answers
16 views
Uniqueness and existence of this system, verifying my answer
I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve.
1)Getting the Jacobian, I obtain
$$ J= \begin{bmatrix}
0 & 1\\
-1-2xy & 1-...
-1
votes
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,...
0
votes
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
votes
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
votes
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).
...
1
vote
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
votes
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
votes
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
vote
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 $V_{...
0
votes
0answers
484 views
How to determine linear terms from the nonlinear dataset?
Let us take the parametric curve r($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
vote
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
vote
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
vote
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
votes
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
vote
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
vote
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
vote
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
vote
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 ...
0
votes
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
votes
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:
$$f'(x)\geq cf(x)...
2
votes
0answers
42 views
To show that a 2D nonlinear ODE undergoes a pitchfork bifurcation
Show that a pitchfork bifurcation occurs when $\mu = 0$ for the system
$$\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
votes
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
votes
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
vote
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....
1
vote
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 $u_0\in H$ ...
