Questions tagged [nonlinear-analysis]
For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.
0
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0answers
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 $V_{...
0
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0answers
161 views
How to determine linear terms from the nonlinear dataset?
Let us take the parametric curve r($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 ...
1
vote
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(...
1
vote
0answers
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 ...
1
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0answers
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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0answers
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?
1
vote
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 ...
1
vote
0answers
22 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
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)(\...
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
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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0answers
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....
2
votes
0answers
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:
$$f'(x)\geq cf(x)...
2
votes
0answers
35 views
To show that a 2D nonlinear ODE undergoes a pitchfork bifurcation
Show that a pitchfork bifurcation occurs when $\mu = 0$ for the system
$$\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}{\...
0
votes
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 ...
0
votes
0answers
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 ...
1
vote
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....
1
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0answers
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 ...
2
votes
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 ...
3
votes
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 ...
0
votes
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}...
1
vote
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}\...
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
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 \...
1
vote
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 ...
3
votes
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:
$$y = \ln\left(\frac{1}{2}c_{1}\left(\tanh^2\left(\frac{1}{2}\sqrt{c_1(c_2+x^2)}\right)-1\...
1
vote
2answers
42 views
$\int_a^b x'(t)dt=x(b)-x(a)$ in Banach space $X$
Let $X$ be a Banach space. Then $\int_a^b x'(t)dt=x(b)-x(a)$ if $x:[a,b]\rightarrow X $ is continuously differentiable.
I have a few problems understanding what I have to show.
First of all, what is ...
0
votes
0answers
12 views
FInd the measure of noncompactness
Find the measure of noncompactness of the set $B\subset C([0,1])$ defined by
$B=\big\{ x\in C([0,1]) \big|x(0)=0,x(1)=1,0\leq x(t)\leq \frac{1}{2},t\in[0,\frac{1}{2}],\frac{1}{2}\leq x(t)\leq 1,t\in[\...
0
votes
1answer
49 views
What is the “bifurcation of a fixed point”?
Isn't the bifurcation of a system, say $\dot{x} = \mu \sin(x) - 2x$, related to the system rather than a fixed point, say $x^{\star} = 0$?.
My understanding is that a bifurcation occurs when there's ...
1
vote
1answer
31 views
Why do every strong extremum is simultaneously the weak extremum?
My Doubt
Here $||f||_{1}=\sup_{x\in[0,1]}|f(x)|+\sup_{x\in[0,1]}|f'(x)|$ where
as $||f||_0=\sup_{x\in[0,1]}|f(x)|$. We can easily prove from
definition that
$$||f||_0=\sup_{x\in[0,1]}|f(x)|...
0
votes
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!
9
votes
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 ...
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 $u_0\in H$ ...
1
vote
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 ...
0
votes
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'\...
2
votes
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^*$?
2
votes
0answers
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$...
2
votes
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{...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
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$?
1
vote
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 ...
0
votes
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=...
1
vote
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 ...
0
votes
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.
3
votes
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 ...
0
votes
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 ...
1
vote
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 $...
