Questions tagged [nonlinear-analysis]

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

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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)...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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]\\...
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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-...
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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,.....
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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, ...
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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 ...
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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....
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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^{...
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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 $x_1=\frac{-a}{-b+c}$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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 `...
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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)\...
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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 ...
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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 ...
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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}...
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28 views

Is this a correct control input for this nonlinear system

Take a look at this system $$ \begin{align} \dot{x}_1 &= \cos x_2 + (x_2+1)x_3 \tag{1}\\ \dot{x}_2 &= x^3_1+x_3 \tag{2}\\ \dot{x}_3 &= x^2_1+u \tag{3}\\ y&=x_1 \tag{4} \end{align} $$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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+\...
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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 $$\dfrac{xx'}{\sqrt{x^2+r(x')^2}}=c,$$ ...
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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)=...
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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 ...
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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 ...
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1answer
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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 ...
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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\...
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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 ...
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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=\...
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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: (...

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