Questions tagged [nonlinear-analysis]
For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.
322
questions
0
votes
1answer
13 views
For $T(t)$ strongly continuous, check that $T(t)x - x = tAx + \int_0^t (t-s) T(s)A^2 x ds$.
I am reading Lemma 2.8 of "Semigroups of Linear Operators and Applications to
Partial Differential Equations" by Pazy:
Let $A$ be the infinitesimal generator of a strongly continuous semigroup $T(t)...
0
votes
0answers
32 views
Show that the cosine map with limit is the logistic map
Firstly, I will note that this question has been asked before 6 years ago but I would like some further explanation of the answer to that question and I'm unable to comment further on that thread.
...
1
vote
2answers
37 views
Stommel model, 1961.
I'm doing some calculations from the article of Stommel $1961$: Thermohaline convection with two stable regimes of flow. At a certain point he write down a system of two nonlinear ODEs which has the ...
2
votes
0answers
36 views
Analytic estimates of limit cycle parameters
Suppose we have a two-dimensional system of differential equations, say, the well-known Van der Pol oscillator:
$$ \dot{x}=y, \dot{y}=\mu (1-x^2)y-x $$
Everyone knows that the study of limit cycles ...
0
votes
0answers
8 views
Explain why the second-order optimality conditions are unable to resolve this case
Consider the Problem
$$\textrm{Minimize} \ \ \{(x_1 -1)^2 +x_2^2; \ \ g_1(x)=2kx_1 -x_2^2 \leq 0\},$$ for the case $k = 1$.
1) Provide an analytical argument to show that $X =(0,0)^{'}$ is an ...
0
votes
1answer
55 views
Bifurcation points
I have this equation $\dot u = u^3 -2u^2 - ku + 2k$, where $k \in \mathbb R$, and dot corresponds to derivatives with respect to $t$. I have some confusion about determining the bifurcation points and ...
2
votes
1answer
28 views
Degree of a function maps $\Omega \subset \mathbb{R}^n \to \mathbb{R}^m$ for $m < n$
I am doing self-reading on some degree theory using "Topological Degree Theory and Applications" by Donal O’Regan, Yeol Je Cho, and Yu-Qing Chen; I am totally new to this field and not quite familiar ...
1
vote
1answer
34 views
What is the best current summary of the limits of (tempered) distribution theory where it comes to multiplication of distributions?
As I understand it, it has been shown that a consistent calculus of (tempered or otherwise) distributions involving multiplication cannot be constructed. Since the problem seems central to the theory ...
0
votes
0answers
22 views
Derivation of internal dynamics of nonlinear system in order to derive Byrnes-Isidori Normal Form
I have a nonlinear system (Ball & Beam) which is described by the following equations of motion:
$$ \ddot{y} + \frac{mg}{a} \sin(θ) -\frac{m}{a}y\dot{θ}^2 = 0 $$
$$ \ddot{θ} + \frac{2m}{b}y\dot{...
0
votes
0answers
4 views
evaluating stability of system in the Equilibrium Point with Lyapunov function
consider nonlinear system:
$\dot x_1=x_2$
$\dot x_2=-x_2-kx_1^3$
evaluating stability of this system in the Equilibrium Point of this system with Lyapunov function V(x)=$fpf^T$.
I find x=0 is ...
1
vote
0answers
91 views
Perturbation at equilibrium point
I am studying one system of nonlinear ODE's s.t. the $(x_1,x_2,x_3)=(1,0,0)$ is an equilibrium point. On this system I have:
$$(1) x_1,x_2,x_3\geq 0;
\\
(2) x_1+x_2+x_3=1 $$
Well, one author, ...
0
votes
0answers
6 views
nonlinear-analysis and limit cycle
Consider the system
$\dot x_1=4x_1^2x_2-f_1(x_1)(x_1^2+2x_2^2-4)$
$\dot x_2=-2x_1^3-f_2(x_2)(x_1^2+2x_2^2-4)$
in that $x_if_i(x_i)\ge0$ and $f_i(0)=0$
Show that almost all system paths lead to ...
1
vote
0answers
39 views
Geometrical methods for studying systems of nonlinear differential equations of high orders, suitable in the computational plan
Recently, I began to study systems of high-order nonlinear differential equations. As an example, I can cite the system of equations from this topic.
Phase portrait of n-dimensional state-space ...
1
vote
1answer
24 views
proving the stability of the equilibrium point with a Lyapunov function
The below nonlinear system is considered as a pendulum with a nonlinear damping coefficient:
$$
\ddot y+(a+b\cos(y))\dot y+c\sin(y)=0, \qquad a\geq b\geq 0
$$
Use the energy of the whole system as a ...
0
votes
0answers
26 views
Energy estimate for nonlinear term
Let $ \Omega$ be open and bounded. Consider that $u$ is smooth solution of
\begin{equation}
\frac{\partial u}{\partial t}=(u\cdot \nabla)u~~~in~\Omega\times[0,T]\\
u=0~on~~~\partial \Omega\times[0,T]\\...
0
votes
0answers
69 views
First order, non-linear ODE
I am solving the following equation:
$$f(x)^2+\frac{f(x)^2}{f'(x)^2}=h(x)$$
for a known function $h(x)\geq 0, x\in \mathbb{R}$.
I'm asking for help with:
Identifying this ODE in terms of well-...
0
votes
0answers
15 views
Show that if this nonbinding constraint is deleted, it is possible that $\bar{x}$ is not even a local minimum
Hello guys I am looking for some help for this nonlinear problem
Let $\bar{x}$ be an optimal solution to the problem of minimizing $f(x)$ subject to $g_{i}(x)\leq0, i=1,...,m$ and $h_{i}(x)=0, i=1,.....
0
votes
0answers
12 views
what is the Quanitity to measure the boundedness of non-linear recurrence relation
Consider the nonlinear first-order recurrence
${x_{n}=f(x_{n-1})}$
How can I find out if the recurrence stays bounded for an initial value of ${x_0}$.
looking at the logistic map as an example, ...
0
votes
0answers
29 views
Corner Solution To A Recursive, Strictly Concave Function?
I was reading through Dynamic Programming by Richard Bellman today, and I got to exercise 7 in chapter one. You are asked to prove a theorem, but I feel like the theorem itself is... well, not quite ...
0
votes
2answers
68 views
Non linear system -control problem
How can i solve this problem in MATLAB?
I have this non linear system:
$\frac{dx_1}{dt}=5sin(6t)-x_1$
$\frac{dx_2}{dt}=3x_1x_2-2x_2+1$
$y=x_2$
Initial condition are the following:
$x(0)=[-2,-1....
2
votes
1answer
35 views
(Book recommendation please!) Is there any difference between $L^p_tL^q_x(R \times R^n)$ and $L^p(R \to L^q(R^n))$?
In real analysis , we usually consider the space $L^p_tL^q_x$ consists of the measurable functions on $R^{n+1}$
$$f:R \,\,\times \,\, R^n \to C$$
$$(t,x)\to f(t,x)$$
With $$\int\int |f(t,x)|^q \, dx^{...
0
votes
0answers
21 views
finding equilibrium points
For all positive values of a, b and c, determine the type of equilibrium of the system
$$
\dot{x_1}=a-bx_1+x_1^2x_2\\
\dot{x_2}=cx_1-x_1^2x_2
$$
I find equilibrium of the system $x_1=\frac{-a}{-b+c}$ ...
0
votes
1answer
16 views
Regarding a Nonlinear Operator on a Cone $K$
We say that a nonlinear operator $N: E \rightarrow E $ on a Banach space $E$ is cone preserving (i.e. positive) if it maps the cone $K \subseteq E$ into itself, that is $N(K) \subseteq K$.
My ...
0
votes
0answers
13 views
what is the geometric representation of Lyapunov stability
Take a look at the theorems below, in a dynamical systems linear or nonlinear, we construct a function usually the energy-function of a physical system, compute its derivative and apply the theorem to ...
0
votes
0answers
10 views
Newton's method on Hilbert space problems / References
Let $S:H \rightarrow K$, a nonlinear functional. Where $H$ and $K$ are two Hilbert spaces.
I have the existance of the zero of $S$ i.e $$\exists x^* \in H ,\, \, \, S(x^*)=0_{K}.$$
So in order to ...
0
votes
0answers
25 views
Does this simple system of nonlinear equations have a solution?
Sorry if this question is too simple; I just got a bit confused about it. Does this simple system of equations (a couple linear and one nonlinear) have any solution? From some computer simulation, I ...
0
votes
1answer
23 views
Non-Linear BVP - ODE - using finite differences
I have the following ODE
$EI\frac{{{d^2}y}}{{d{x^2}}} = [\frac{{ - qLx}}{2} + \frac{{q{x^2}}}{2}] \cdot {[1 + {(\frac{{dy}}{{dx}})^2}]^{\frac{3}{2}}}$
$\begin{array}{l}
y(0) = 0\\
y(L) = 0
\end{...
1
vote
1answer
43 views
Brent's Method convergence criteria
I am using Brent's method to solve the BEM equations for a wind turbine model. I have run into a scenario where Brent's method has converged i.e., abs(m) is below set tolerance of 1e-8 but the value ...
0
votes
0answers
17 views
Argument variable vs. indexed variable in Fourier transform.
Why is the transformed variable in some fourier transforms written as an index whereas sometimes it is written as an argument ? For example,
$$G_{\bf k}({\bf{v}},t)= \frac{1}{(2\pi)^3}\int g({\bf ...
1
vote
0answers
28 views
Behaviour of Nemytskii operator wtr weak convergence
Let $N_f$ denote the Nemytskii operator and suppose that $u_n\rightharpoonup u$ in $H^1(\mathbb{R}^n)$. It is true that $$ N_f(u_n)\rightharpoonup N_f(u)?$$
And if it is true, in what space we have ...
2
votes
0answers
25 views
Is every infinite-dimensional Banach space coarsely equivalent to its subspaces of codimension 1?
Gowers in his solution to the Banach hyperplane problem constructed a Banach space which is not linearly isomorphic to any of its codimension one subspaces.
However, how about much weaker notion of `...
0
votes
0answers
28 views
Calculation of Symmetry generator of cylindrical KdV equation
I have calculated the generators of the cylindrical $KdV$ equation $$u_t+(u/2t)+uu_x+u_{xxx}=0,$$
but I got three generators, $$X_1=\partial_x,\\ X_2=2t^{1/2}\partial_x+\left(1/2t^{1/2}\right)\...
0
votes
0answers
23 views
Backstepping control of second order nonlinear system
$\dot{x_{1}}=x_{2}^2-3\sin(x_{1})x_{2}$
$\dot{x_{2}}=x_{1}^3-3x_{2}\cos(x_{1})+u^{1/2}$
Question: Using the backstepping method and Lyapunov function, design the controller $u$ that will make the ...
2
votes
2answers
38 views
Mathematics of apportionment of representation in a legislative body
It seems to be eclipsed by coronavirus, but today is the U.S. Census day for this coming decade.
If you are American, where you are living today, where your children are living today, is where they ...
0
votes
0answers
8 views
Best method of converting $4$ given differential equations into linear system control form
The equations are :
$$\begin{cases} \frac{d(ADH)}{dt}= \frac{ADHS- ADH\times DAD}{PV} \\
\frac{dR}{dt}=\frac{RS- 0.135\times R}{PV}\\
\frac{dA}{dt}=\frac{AS - 4.04\times A}{PV} \\
\frac{d(ALD)}{dt}...
1
vote
0answers
28 views
Is this a correct control input for this nonlinear system
Take a look at this system
$$
\begin{align}
\dot{x}_1 &= \cos x_2 + (x_2+1)x_3 \tag{1}\\
\dot{x}_2 &= x^3_1+x_3 \tag{2}\\
\dot{x}_3 &= x^2_1+u \tag{3}\\
y&=x_1 \tag{4}
\end{align}
$$
...
0
votes
0answers
23 views
The relationship between the function and the conjugate
Let $f:E\longrightarrow [-\infty,\infty]$ defined by
$$ f^*(y)=max_{x\in {E}}\{<y,x>-f(x)\}$$
is called the conjugate function of $f$.
where $y\in{E^*}$ and $E,E^*$ are finite-dimensional ...
1
vote
1answer
35 views
Showing that the sequence $\left\{x_\lambda\right\}_{\lambda > 0}$ is bounded.
(Part of the following can be found at page 36 of "Non-linear Differential Equations of Monotone Types in Banach Spaces" of Viorel Barbu.)
Note for the following, that $A$ is a coercive and maximal ...
0
votes
2answers
60 views
Solve $y'=a(x)y^2+b(x)$
PROBLEM
Solve $$y'=a(x)y^2+b(x)$$
$a(x) = \frac{a_1}{m-M x} , b(x) = \frac{b_1-(m-M x)b_2}{m-M x} $ and $a_1,m,M,b_1,b_2 $ are constants
ADDITIONAL INFORMATION
I find on net it's probably Riccati ...
0
votes
0answers
22 views
refering something as “non-linear” when there is no underlying linear structure
Can I talk about a non-linear shape functional. I understand a shape functional $J$ as some mapping that takes a shape and returns a real (or complex) value.
I would like to talk about a non-linear ...
0
votes
0answers
28 views
Nonlinear function of two Gaussians - Stein's Lemma
Let $g,h$ be independent standard normal variables ($\cal{N}(0,1)$). Fix $\sigma>0$ and pick $f:\mathbb{R}\rightarrow \mathbb{R}$. Under what conditions on $f$, we have that
$$
\mathbb{E}[f(g+\...
0
votes
2answers
91 views
Non-Linear Differential Equation with quadractic terms
I have been doing some exercises about solving differential equations, but I am not be able to solve this one:
Find the implicit solutions of the following DE
$$\dfrac{xx'}{\sqrt{x^2+r(x')^2}}=c,$$
...
0
votes
0answers
16 views
ODE with reverse independent variables like $u_x=u(x)+2u(-x)$
The problem is probably just for fun or maybe somehow related to so-called nonlocal ODE/PDE theory.
Two facts:
the solution to $u_x(x)=u(x)+u(-x)$ is only the zero solution.
the solution to $u_x(x)=...
0
votes
0answers
29 views
Non-linear maps between Fréchet spaces - is this property well know?
Let $E$ and $F$ be Fréchet spaces and $f:E\rightarrow F$ a (not necessarily linear) map. We call $f$ bounded, if it maps bounded sets of $E$ into bounded sets of $F$. If $f$ is linear, then ...
1
vote
0answers
23 views
Possible phase difference in equation of sine functions
Given the equation
$
A_1\sin(\omega t + \theta_1) + A_2\sin(\omega t + \theta_2) + A_3\sin(\omega t + \theta_3) = 0
$
which holds for all $t$.
It seem apparent that this implies that the phase ...
-4
votes
1answer
43 views
Is $T$ a nonlinear map? [closed]
Define $T:\Bbb R^2\to \Bbb R^2$ with $(x,y)\in\Bbb R^2$ and $(e^x,e^y)\in \Bbb R^2$ s.t. $\forall x,y$ $(x,y)\mapsto (e^x,e^y).$ Define the origin to be $(0,0)$ before the map and after the map. This ...
2
votes
0answers
21 views
Third order linear differential equation and Painlevé II solution
Consider the solution to the Painlevé II equation on $\mathbb{R}$
$$q''=2q^3+rq$$
with the boundary condition $q(r)\sim_{r\to +\infty} \mathrm{Ai}(r)$ and consider the function $f$ such that for $r\...
0
votes
0answers
23 views
Nonlinear Regression with machine learning methods
Consider the following example:
I have a big set of test data with input $x$ (of dimension 10) and output $y$. The plot $y$ vs $x$ shows that $y$ depends nonlinear on $x$.
I want to construct a ...
1
vote
0answers
33 views
Span of a Nonlinear Function
Fix a vector $x\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. $\phi$ applies entrywise on vector inputs. Under what conditions we have the equality
$$\mathbb{R}^d=\...
0
votes
1answer
27 views
Continuity of invertible operator.
Let $F$ be a bounded linear operator between the Hilbert spaces $H_1$ and $H_2$. Let $F$ satisfies $$\|z- a\|\leq \|F(z)-F(a)\|^t, \ t>0, \ z \in H_1.$$ Please answer the following two questions:
(...
