Questions tagged [nonlinear-analysis]

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

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Multiplication of a time-domain sinusoid to a s-domain (Laplace) signal?

I am confused between the transformations between the time-domain and the frequency domain. I have a signal y(t) which is a sum of multiple sinusoids. I band-pass filter this signal to extract one ...
Ayush Sharma's user avatar
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Singularity of a non- linear second order ODE

I have the encountered a singularity in the equation below . $$ y^{\prime \prime}(x)+\frac{2}{x} y^{\prime}+\left[y-\left(1+\frac{2}{x^2}\right)\right] y(x)=0, \quad 0<x<+\infty, $$ with ...
SR9054505's user avatar
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Seeking name of "trick" involving operators like $A + \tau B$, where $B$ is Lipschitz.

Theorem. On Hilbert space $V$, suppose $T: V \to V$ is nonlinear and that $T = A + \tau B$ where $A$ is linear and strongly monotone, $B$ is nonlinear and Lipschitz, and $\tau > 0$ can be made ...
1Teaches2Learn's user avatar
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Given a posdef matrix $M$, find $x$ such that $x_i = \operatorname{sign}\left(\sum_j M_{ij}x_j\right)$

Let $M$ be a real symmetric positive definite matrix. Can we characterize the sign vectors $x$, that satisfy the condition: $$x_i = \operatorname{sign}\left(\sum_j M_{ij}x_j\right)$$ That is, this ...
a06e's user avatar
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Estimate for a second order non-linear ODE

I am considering the following non-linear ODE \begin{cases} \ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\\\ y(0) = 0\\\\ \dot y(T) = c \end{cases} ...
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How to visualize low-dimensional torus in a high-dimensional system?

I have a system of very high-dimensions (1000s of independent variables), but I could show that the dynamics is attracted to a 1D limit cycle or a 2D torus (with commensurate frequencies, so still ...
Axel Wang's user avatar
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How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?

Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, I heard that $\operatorname{Id}-K$ is a proper map. Proper map ...
boundary's user avatar
1 vote
0 answers
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Updated Gorelik principle

One version of the Gorelik principle is the following: Let $E,X$ be Banach spaces and suppose $U:E\to X$ is a Lipschitz isomorphism (that is, a Lipschitz bijection whose inverse is also Lipschitz). ...
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Inequalities for the solution of $x = (x-a) e^{x+a}$.

Let $a > 0$. The equation $$(x-a) \, e^{x+a} = x $$ must be solved for $x > 0$. Since the solution does not have a closed form, I would like to obtain bounds for the solution. Until now, I was ...
P.S. Dester's user avatar
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Define a compact set $\mathcal{X}$ such that $x(t) \in \mathcal{X}$ for all $t \geq t_0$ [closed]

I have state vector $x \in \mathbb{R}^{\text{n}} $, that behaves following this inequality $ \|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3 $ where $c_1, c_2, c_3$ are positive constants. This ...
SpaceTAKA's user avatar
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Existence and uniqueness of $-\Delta u+u^2=f $

My question is very similar to the one given here: Existence and uniqueness of $-\Delta u + u^3 =f$ If we consider the equation $-\Delta u+u^2=f$ in 3 dimensions with $f\in L^2(\Omega)$ is there ...
micha's user avatar
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Boundary of p(x)-triharmonic operator

Regarding the boundary conditions often associated with this operator, why is the assumption $\Delta^{2} u=0$ commonly made, and what are the implications of this hypothesis for the solution ...
Adnanovic's user avatar
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Solving a super-quadric equation

I have to solve the following scalar non-linear equation. \begin{equation*} \xi^{\frac{2}{\varepsilon}}+(\xi-k)^{\frac{2}{\varepsilon}}=1 \end{equation*} with respect to $\xi\geq 0$. Here $\varepsilon&...
matteogost's user avatar
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What should I prove to show the states lie within a compact set?

I'm trying to prove the local stability of a nonlinear system and got the following inequality. $ \|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3\epsilon_m\cdots $(i) where $c_1, c_2, c_3$ are ...
SpaceTAKA's user avatar
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Nonlinear elliptic equation with Dirichlet conditions without weak solution

I am not familiar with the theory of nonlinear PDEs and wonder if there is a theorem that states sufficient and necessary conditions for the existence of weak solutions to the problem $\begin{cases} -\...
scottish's user avatar
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Second-order Taylor expansion for Operators

Let $u(t)$ and $v(t)$ be functions in $C^{\infty}$. Then let $A(u)$ be an operator. A valid reference mentioned that the second-order Taylor expansion of the operator $A$ is: $$A(u+v) = A(u) + dA(u)[v]...
Redsbefall's user avatar
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Existence of solution to a advection-reaction equation with forcing term

Consider the advection-reaction equation in One-Dimension $\dfrac{\partial u}{\partial t} + \dfrac{\partial u}{\partial x} = u(1-u) + f(x,t); x\in\mathbb{R}, t>0$ with initial condition $u(x,0) = g(...
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Solve for y: $\frac{d^2y}{dx^2} = A\sin(y)+B\cos(y)$

Current Progress I am currently unable to proceed further, any help is welcomed. Looks forward to seeing different approaches to this differential equation. $ {y}'' = A\sin(y) + B\cos(y) \\g(y)=A\sin{(...
zich's user avatar
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Reference proof from Zarantonello's 1960 Article

I cannot find anywhere an online version or reference proof of Zarantonello's 1960 article: E. H. Zarantonello. Solving functional equations by contractive averaging. Math. Research Center Report, 160,...
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Does this inequality guarantee the global stability in this paper?

I'm reading a very informative paper. But I met some formulations hard to understand. In the stability proof section (Sec. V, Theorem 5.2), they define a Lyapunov function as $V(s) = \frac{1}{2}m\|s\|^...
SpaceTAKA's user avatar
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How is it OK to safely neglect one-step difference of the system state in this paper?

I’m reading a very informative paper. But I met some formulations hard to understand. In assumption 2, they have the inequality of one step difference, $$\Delta u_k \leq \sigma(B_o^{-1})\big(L_a\Delta ...
SpaceTAKA's user avatar
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How is it possible to have discrete-mixed continuous Lyapunov function in this paper?

I'm reading a very informative paper. But I met some formulations hard to accept. In the stability proof section (Sec. V, Theorem 5.2), they define a Lyapunov function as $V(s) = \frac{1}{2}m\|s\|^2$ ...
SpaceTAKA's user avatar
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1 answer
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Spectral Radius at the Unique and Globally Attractive Fixed Point of a Specific Type of Mapping

Consider a mapping $f: \mathbb{R}_+^n \rightarrow \mathbb{R}_{++}^n$. If it is monotonic and strictly subhomogeous, then it is contractive under the Thompson’s metric (See Lemma 2.1.7). Here, ...
maphado fan's user avatar
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Solution for a system of non linear equation

$ x = \phi(x + t)$ is the system of non-linear equation which I need to solve for $x$. Here $x$, $t$, $\phi(.)$ are vectors all having dimension $n$. How to solve this system using Newton's method. ...
Argha Chakraborty's user avatar
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A quasilinear ode

I found the differential equation from a textbook (which is about differential games): $$ \phi'(x)= \frac{1-r-2x}{2(x-x^2)/\phi(x)-M-2} $$ with initial condition $\phi(0)=0$. The textbook just says ...
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Study of the properties of a non-local ODE

I am studying the following non-local ODE $$\dot p(x) \nu_{\varepsilon, \alpha}(x) + \int_{x}^{2x_0}\frac{\dot p(s)}{s + \varepsilon} ds = c \quad \text{for } x \in [0,2x_0].$$ The number $x_0$ can ...
Falcon's user avatar
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Looking for a Functional Form Satisfying Certain NonLinearites

I'm working with a functional form $\alpha = F(x,y,t)$ where t represents time and $x,y $ other factors affecting $\alpha$. I want to find a functional form for F (that also has some parameters) such ...
esos's user avatar
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29 votes
4 answers
925 views

Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$

I am looking for a function over the real line, $g$, with $g*g = g^2$ (or a proof that such a function doesn't exist on some space like $L_1 \cap L_2$ or $L_1 \cap L_\infty$). This relation can't hold ...
BigMathGuy's user avatar
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Can you reverse time and space in differential equations?

Context: In my recent answer I proved that a certain map $T$ is linear, and smooth. In that answer I work with a Cauchy foliation $\Omega_s(x,y,z)$ of a Lorentzian manifold. In this question I will ...
John Zimmerman's user avatar
1 vote
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21 views

Is it correct to exclude the nonlinear terms from the finite strains in linear elasticity, if their differentiation is needed in equilibrium equation?

It is clear that infinitesimal strains are obtained from finite strains (e.g. Green-Langragian or Eulerian-Almansi tensor) by removing nonlinear terms which smallness order is greater than that of ...
Fidel Pestrukhine's user avatar
1 vote
0 answers
42 views

Nonlinear ODE for inverse CDF

I have a CDF function of $F(x)=p = \int_{0}^{x} \mu e^{-\mu v}e^{-e^{\beta v}\lambda \alpha} dv$. Using the theorem of calculus, taking derivative of the CDF with respect to $x$ will result to $$ \...
AHMAD FAIZ BIN MOHD AZHAR MSC2's user avatar
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1 answer
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Hausdorff separation for the definition of Mackey topology

I am reading the definition of the Mackey topology relative to a dual system $(X,Y)$ and the author (Edwards-Functional Analysis) imposes the condition that the dual system must be separated in $Y$. ...
Neutral Element's user avatar
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Find parameters of a gaussian transformation

I have a system of equations where the relationship between input and output is derived from a pixel lattice: \begin{equation} x_i(k+1) = \sum_j \alpha e^{ \frac{\left(dr_{ji}^2 + dc_{ji}^2 \right)}{2 ...
user3284182's user avatar
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0 answers
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Can these two concave functions on unit interval intersect at most three points?

Consider two concave functions on the unit interval $[0,1]$ of the form: $f_{p_1,q_1}(x)=H_b(xp_1+(1-x)q_1)-xH_b(p_1)-(1-x)H_b(q_1)$ $f_{p_2,q_2}(x)=H_b(xp_2+(1-x)q_2)-xH_b(p_2)-(1-x)H_b(q_2)$ where $...
Sushant Vijayan's user avatar
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0 answers
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Nonlinear Coordinate Transform via Intersections?

I've defined a sort of 'warp' procedure for 2d shapes, and I'm curious whether it's familiar within the math canon or even has a name. Given its simplicity I'm sure that there's a formal definition ...
Ray's user avatar
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1 answer
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Fitting a line to the outermost plot on a graph

I am plotting the nonlinear regression on some fatigue data. My goal is to create a line that adheres to the slope line (in red). This line should sit beneath the lowest point on the left side of the ...
Willy's user avatar
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6 votes
1 answer
191 views

Has anyone looked at the ODE $x_{ssss} - x_{ss} x = c$ before?

In my research, I've come across the following inhomogenous nonlinear ODE ($c \geq 0$ is an undetermined constant): $$x_{ssss} - x_{ss} x = c$$ It has boundary conditions $$x_s(0) = x_{sss}(0) = x(1) =...
Ron Shvartsman's user avatar
2 votes
1 answer
85 views

Prove that you cannot solve clearly for y in $ (x^2+y^2)^2+3x^2y-y^3=0 $

I am supposed to decide whether or not you can solve the equation clearly $ f(x, y)=(x^2+y^2)^2+3x^2y-y^3=0 $ for $x$ or $y$, with $x, y, \in U$, $(0, 0) \in U$ and $U$ open. When looking at the ...
Cake's user avatar
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0 answers
81 views

Observability Gramian of an Unscented Kalman Filter not Matching Estimation Results

I am running an unscented Kalman filter on my system and am able to estimate the states within 4% of their true values. This is true with a Monte Carlo simulation consisting of 1000 runs. However, ...
DerekB's user avatar
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Growth conditions for elliptic quasilinear equations in divergence form

Let us consider the following elliptic quasilinear problem with mixed boundary conditions $$ \left.\begin{alignedat}{2}-\textrm{div}[a(x,u,\nabla u)]+c(x,u,\nabla u) & =g & & \textrm{in}\,...
RiemannGauss's user avatar
4 votes
1 answer
405 views

(Frechet) Differentiability of Implicit function in Banach spaces

I'm looking at the classical implicit function theorem in Banach spaces. So $X,Y,Z$ are Banach spaces and $F: U_{x_0}\times V_{y_0} \to Z$ continuous and continuously differentiable with respect to y. ...
Petar's user avatar
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0 answers
13 views

Is nonlinear transform of a vector space a connected set

Consider $b_i=(\theta_p-\theta_q)\sin((\theta_p-\theta_q)\alpha)$ where $p,q\in\{1,\cdots,w\}$ and $N=\frac{w(w-1)}{2}$. In order to guarantee the one-to-one correspondence between $(\theta,\alpha)$ ...
happyle's user avatar
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1 vote
1 answer
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Solve $a\frac{d^3y}{dx^3}+by\frac{d^2y}{dx^2}+c\frac{dy}{dx}=0$

Is there any specific method of solving nonlinear differential equations? I want to solve the following differential $$a\frac{d^3y}{dx^3}+by\frac{d^2y}{dx^2}+c\frac{dy}{dx}=0$$ by the method of ...
MANI's user avatar
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1 vote
0 answers
39 views

Existence of all stable nonlinear system.

Consider a nonlinear dynamical system that evolves with a continuously differentiable vector field that is there exists a compact set within which the solution trajectories exist and are unique. Is it ...
Priyan Bhattacharya's user avatar
2 votes
0 answers
88 views

Extending Atiyah-Singer for selfadjoint Fredholm operators

In the paper "Index theory for skew-adjoint Fredholm operators" by Atiyah and Singer, the corollary at page 3 states that the space $\mathcal{F}_{s}(H)$ has two contractible components, ...
Paul Thorwarth's user avatar
7 votes
0 answers
165 views

Can this non-linear boundary-value problem be solved analytically?

I am trying to solve $$ [y^2(x)]''+\frac{x}{2} y'(x) = 0 $$ on $x\in\mathbb{R}$, with the conditions $$ \lim_{x\rightarrow-\infty} y(x) = a,\qquad \lim_{x\rightarrow\infty} y(x) = b. $$ I am ...
ArrhythmicTiling's user avatar
0 votes
1 answer
60 views

Help with 2nd order ordinary differential equation and solutions [closed]

I came across an ordinary differential equation of the form: $y''+yy' + Ay + A = 0,$ Where $y=y(t)$, $A$ is some constant, positive or negative. So far the equation is solvable for $A=0$. Could ...
Georges Leukic's user avatar
0 votes
1 answer
90 views

Critical points of an autonomous dynamical systems depending on a parameter

Consider the autonomous dynamical system: $$ \left\{ \begin{array}{l} \dot{x}=x+y-x^3-\alpha xy^2\\ \dot{y}=y-x-x^2y-y^3 \end{array} \right. $$ Prove that $(0,0)$ is the unique critical point if $\...
Sergio Ferrer's user avatar
0 votes
1 answer
49 views

Let $a$ be continuous such that $|a(x)| \le C |x|^{p/q}$. Then $A:L^p(\Omega) \to L^q(\Omega), u \mapsto a \circ u$ is continuous

Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. Let $p, q \in [1, \infty)$. Let $a:\mathbb R \to \mathbb R$ be continuous such that $$ |a(x)| \le C |x|^{p/q} \quad \forall x \in \...
Akira's user avatar
  • 17.2k
3 votes
1 answer
415 views

How do I input a vector form of ODE's on Runge-Kutta-4 for generality to solve in matlab

I have come across a system of ODE's that are written on vector/Matrix format such that; $Ax'=b$ For simplicity, say the system of ODE's has a vector $x'$ containing first order derivatives of 2-...
Amatics's user avatar
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