Questions tagged [nonlinear-analysis]

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

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Showing that a maximum exist for a semi lower sequentially continuous mapping from Hilbert space to R

Let H be a Hilbert space over $R$ , $r > 0$ and $F ∈ C^1(H, R)$ such that: 1)−F is weakly sequentially lower semicontinuous 2) $DF(u) = 0$ implies $u = 0$ (this is the Frechet derivative) 3) $F(0) =...
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How to solve for $\boldsymbol{\theta}$ in $[A]^{\theta}x = b$ if $\boldsymbol{[A]}$, $\boldsymbol{x}$, and $\boldsymbol{b}$ are known?

I am trying to find $\boldsymbol{\theta}$ when the output $\boldsymbol{b}$, input $\boldsymbol{x}$, and the matrix $\boldsymbol{[A]}$ are given. Here, $\boldsymbol{[A]}$ is a diagonal square matrix ...
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Prove $P|_{ker(C)}$ is a Fredholm operator

The following problem is from the course Nonlinear analyis, which used the reference written by Kung-Ching Chang. Precisely stating, Let $X, Y, Z$ be Banach spaces and $C: X \times Y \rightarrow Z$, $...
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Confusion about the arrival space of a multivalued operator

I have some confusion about the multivalued operator. Let $X$ Hilbert space. In some books, we find "$A:X\rightarrow 2^{X}$ is an operator." where $2^{X}$ is all subsets of set X. I ...
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Multivalued and single valued operator

let X Hilbert space et $A:D(A)\subset X \rightarrow 2^{X}$ multivalued nonlinear operator. The Cauchy problem is given by: \begin{eqnarray} U_t(t)+AU(t)&\in&f(t) \;t \in [0,T]\\ U(0)&=&...
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Global Existence

let X Hilbert space et $A:D(A)\subset X \rightarrow X$ non linear operator. The Cauchy problem is given by: \begin{eqnarray} U_t(t)+AU(t)&=&f(t) \;t \in [0,T]\\ U(0)&=&U_0 \end{...
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How to show sin(f(t) where $f(t)) \in L^2(0,1)$ is lipschitz continuous?

How to show sin(f(t)) where $f(t) \in L^2(0,1)$ is Lipchitz continuous? sin(f(t)) is a mapping from $L^2(0,1) \to L^2(0,1)$. I was surprised that this composition is even continuous in the first place....
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Existence of global solutions to a Bernoulli differential equation/inequality

I'm looking at some Bernoulli type differential equations at the moment and came across another question: First order nonlinear differential inequality. This got me thinking that the Bernoulli ...
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A condition on a fifth order differential equation

This may be very obvious but I am stuck trying to solve a boundary value problem. I am trying to solve the following differential equation : $$F^{3}F^{(5)}+F=1 \space (Eq.1)$$ I have been advised to ...
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Implicit finite difference scheme for a non-linear PDE

I am trying to write a finite difference scheme to solve numerically a 2-nd order non-linear equation : $$\boxed{\frac{\partial h}{\partial t} = A\frac{\partial}{\partial x}\left(h^{3}\frac{\partial h}...
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Is it possible to express a power series with squared coefficients as a function of the series without squared coefficients?

Suppose I have two sums, $P(x)$ and $Q(x)$: $$P(x)\equiv \sum_{n=0}^N a_n x^n$$ $$Q(x)\equiv \sum_{n=0}^N a_n^2 x^n$$ Is there a way to express $Q(x)$ as a function of $P(x)$? Context: I have a ...
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Solution to a second order semilinear PDE

Consider the following PDE:$$0=u_t+u_{yy}+u_{xx}+(x-y)u_y+y^{-\frac{3}{2}}u^2+1,$$ with $t \in [0,T], $ and a terminal condition $u_T=-1.$ I am looking for a way to tackle this nonlinear (in fact ...
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Solution to a second order semilinear PDE, linear in all derivatives.

I am trying to prove global existence and uniqueness, or even better, find an analytical solution, or at least some form to get rid of one of the dimensions to the following semilinear PDE $$ 0=u_{t}-...
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1 answer
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Can $f_1(x, y)-f_2(u, v)$ be written as $g(x-u, y-v)$?

when I do some calculation on the basic theory of diffractive neural networks,the question behind blocks my way.It's a pure math that i want to know, $f_1 (x, y)$ and $f_2(u, v)$ are both nonlinear ...
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Show u^2|u|^(p-3) is Holder continuous (derivative of power nonlinearity)

I am trying to show that the function $f:\mathbb{C}\rightarrow \mathbb{C}$ given by $f(u)=u^2|u|^{p-3}$ is (uniformly) Holder continuous for $p\in(1,2)$ (so $p-3\in(-2,-1)$, this ensures that $f$ is ...
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How can I find closed loop dynamics of this system?

There is controller, where $a>0$. Now, if $v>0$, then $u = \ddot\theta(t)$ and if $v<0$, then $-u=\ddot\theta(t)$. However, how can I detect whether $v$ is positive or negative?
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Computing positive definite matrices of a generalized Lipschitz condition

Let a function $f:X\rightarrow \mathbb{R}^n$, where $X \subseteq \mathbb{R}^n$, satisfy the Lipschitz continuity condition $$ \|f(x) - f(y) \| \leq L \|x-y\|, \quad \forall x,y\in X $$ where $L\geq 0$ ...
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Theory for solving nonlinear matrix equation

I'm trying to solve a system of equations for $\theta\in \mathbb{R}^m$ of the form $$ X V_\theta (X^\top \theta - b) = 0 $$ where $X \in \mathbb{R}^{m \times n}, V_\theta \in \mathbb{R}^{n \times n}, ...
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Holder regularity for solutions to the Poisson equation

I am dealing with a nonlinear PDE of the form $$-\Delta u=f(u),\quad in\,\,\,\mathbb{R}^n$$ (where $f(u)$ is a nonliear function) I would like to ask you which regularity results do exist in the case ...
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Is the rate of change of Finite-Duration Solutions always bounded?

Is the rate of change of Finite-Duration Solutions always bounded? (between times $[t_0;\,t_F]$) I have found recently a paper Finite Time Differential Equations (V. T. Haimo - 1985), where its proved ...
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Analysis and Stability Non Linear System

I have this system and i want study stability: $$ \left\{ \begin{array}{c} \dot x_1 = x_3 \\ \dot x_2=x_4\\ \dot x_3 =\frac{1}{I}[u-bx_3-k(x_1-x_2)]\\ \dot x_4 =\frac{1}{mL^2}[k(x_1-x_2)-mgL\sin(x_2)]...
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1 vote
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How Achieve This Specific Non-Linear Mapping

Let's say in a computer program, we have a Slider whereby a user selects a value. The slider widget itself produces a value, X, from 0.0 to 1.0. This value then maps to some other range. Most people ...
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1 vote
1 answer
64 views

Topology, limit cycles, Bendixson's criterion, and simply connected regions

I have some doubts about the criterion of Bendixson that regards the non-existence of limit cycles. According to Bendixson's theorem, let $$x'=P(x,y),\quad y'=Q(x,y).\label{*}\tag{1}$$ be a two-...
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Finding the center manifold for a 2D dynamical system

I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the ...
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Is it true that every PDE is a system of first order quasi-linear PDEs?

As shown in Folland (Introduction to PDE, page 48), a fully non-linear PDE of the form $$ \partial_{x_n}^ku=G(x,(\partial^\beta u)_{|\beta |\leq k, \beta _n<k}) $$ ($G$ smooth) is equivalent to a ...
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Potential Extension of variable metric Quasi-Fejér monotone: interesting convergence analysis tool with iteration-dependent norms?

In [1], there are many theorems and propositions on (quasi) Fejér monotonicity. Let us focus on finite-dimensional spaces, for instance Euclidean space. Theorem 3.3 and Proposition 4.1-4.3 in [1] (...
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1 vote
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Regularity theorem of minimizers

During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them. The function $f:[-1,1] \times \mathbb R ...
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2 votes
1 answer
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Linearization of The Ginzburg-Landau (GL) equation

I am studying the LG equation, $$\partial_t u = (1 + i \alpha) \partial_x^2 u + u - (1+ i \beta)|u|^2 u, \quad \alpha, \beta, x \in \mathbb R, \text{ and }\,\, u(x,t) \in \mathbb C. \tag{1} $$ To find ...
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Approximating expansion for non-smooth or nonlinear functions?

Are there some examples or research trends to find approximating expansions to nonlinear or non-smooth functions that have some nice properties from the Taylor expansion - e.g. possibility to use some ...
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analytic solution to second-order nonlinear ordinary differential equation

Is there any way to solve this equation analytically ? $m\ddot{x} = B_0 \left( \frac{1}{x^4} - \frac{1}{(L-x)^4} \right)$ its supposed to describe this system
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1 vote
1 answer
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Help proving that the derivative $Df(x)$ is continuous

In what follows, $X, Y$ are Banach and $U \subseteq X$ is open. Consider the map $f: U \to Y$. We say that $f$ is locally uniformly differentiable at $x \in U$ and $h \in X$ if given $\epsilon > 0$ ...
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1 vote
1 answer
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Non-trivial solution of $x = r\sin\pi x$

To compute the fixed points of a sine map, I need to solve $$x = r\sin\pi x$$. The question asks me to find the value of r for which the non-trivial fixed point (a second solution of the above ...
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1 vote
0 answers
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Comparing two nonlinear regression models with related parameters

We have a nonlinear regression model with $m$ parameters ($\alpha_1,\alpha_2, ..., \alpha_m$) and $n$ regressors $(X_1, .... X_n)$, predicting an outcome Y: $Y = f(\alpha_1,\alpha_2, ..., \alpha_m; ...
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5 votes
2 answers
132 views

Convexity of infimum function

This question asks about the convexity of the function $$ g(x)=\inf_{y\in\Re^n} f(x,y) $$ where $f \colon \Re^n \times \Re^m \to \Re$ is convex in $(x,y)$. I would ask a more advanced case that if the ...
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3 votes
0 answers
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Hopf bifurcation computation fail

For few days I am working on Hopf bifurcation of a system like this: $$ \frac{dx}{dt}=\alpha\frac{x^2 y + a x y}{x^2+bx+1}-1\\ \frac{dy}{dt}=\frac{1-y-cx^2 y }{1+x^2} $$ above $\alpha$ is the ...
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1 vote
0 answers
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Numerical Method for solving a nonlinear PDE with initial boundary conditions

I'm currently trying to solve the following nonlinear partial differential equation for $f(r,\theta)$: $$ \frac{r^2f_{,rr}+rf_{,r}+f_{,\theta\theta}}{r\sqrt{1+f_{,r}^2+\frac{f_{,\theta}^2}{r^2}}}-\...
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7 votes
1 answer
128 views

Maximum of $ F(f)=\int_0^1 |f(x)|^2\; dx-\left(\int_0^1 f(x)\; dx\right)^2 $ over a subset of continuous functions on $[0,1]$

Let $X$ be a subset of $C([0,1])$ with $$ X=\big\{f\in C([0,1]): 0\le f(x)\le x,\text{$f$ is a polynomial}\big\} $$ where $C([0,1])$ denotes the space of continuous real-valued functions on $[0,1]$. ...
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3 votes
1 answer
166 views

2D bifurcation problem

I come across this problem which is about bifurcation. I am trying to take all the cases. I am expecting Hopf bifurcation to occur here but the last case I could not find the fixed point. Could you ...
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Does this paper make valid statements regarding Non linear dynamics?

This paper claims non linear dynamics has something to do with the brain, does he claims it makes in "Tension Domain " make sense? Is there really anything called "Phase Tension ...
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3 votes
2 answers
80 views

3rd orderCanonical form of nonlinear dynamical system

I am have been solving this problem since a month. I solved a more difficult ones but I do not know why I stuck at this one. There is a clue I can not understand. I solved the first point and stuck ...
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1 answer
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Linearization of system of Nonlinear reaction diffusion equation

I came across many problems in my course and I solved them but the forth one, it seems the hardest for me. I will show the problem I want to solve at first, after that I will show my solution for ...
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1 answer
140 views

A regularity result for semilinear PDE of the form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III.

Under the assumption that $$\partial_{u} f(x, u) = 0 \text{ for } |u| \geq K \quad(1.6)$$ Michael E. Taylor said that (proposition $(1.3)$) For $k=1,2,..$, if $g \in H^{k+1 / 2}(\partial M)$ then any ...
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1 vote
1 answer
37 views

Melnikov's method, homoclinic orbits, and bifurcation values

In nonlinear dynamics, Melnikov's approach provides an intriguing way to detect homoclinic bifurcations and bifurcation values, i.e., the values of the parameter at which a dynamical system exhibits ...
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2 votes
0 answers
30 views

Show $h$ and $g$ are commutative in the canonical form of ODE.

I come a cross this problem in my nonlinear analysis course. I know how to find the normal forms of any order. However, the commutative isometry! And in the third point the professor put two ...
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1 vote
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How to check if the given nonlinear curve or curved surface are included into higher dimension inequality area.

For example, I attached the figure. The first figure located at the left showed that the red nonlinear curve is always in the black inequality region. However, in the right figure, the red nonlinear ...
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nonlinear features mapping to the RKHS

Edited: We usually use linear features mapping in RKHS. But how to handle a nonlinear features mapping? For example: if we are given a non linear mapping $$\phi:R^d\to R^{d+d(d-1)/2}: x = \begin{...
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0 answers
69 views

Solution to Fisher-KPP equation

This is problem I solved but stuck with the third requirement. Could you please check my solution and help with the third point, namely $0 <\phi(\xi)<1$. Consider the following PDE \begin{...
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0 answers
42 views

Inverse Function Theorem for Functions That Aren't Continuously Differentiable

I am trying to show that given the vectors $\mathbf{a}$ and $\mathbf{b}$, the system of equations given by $$\mathbf{a}=\mathbf{x}_2 - \mathbf{x}_1$$ $$\mathbf{b}=\frac{\mathbf{x}_1}{\lVert \mathbf{x}...
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0 votes
1 answer
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Uniqueness of Solution to Nonlinear System of Equations

I am trying to show that given non-zero vectors $\mathbf{a}$ and $\mathbf{b}$, the system of equations given by $$\mathbf{a}=\mathbf{x}_2 - \mathbf{x}_1$$ $$\mathbf{b}=\frac{\mathbf{x}_1}{||\mathbf{x}...
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0 votes
1 answer
115 views

How to use LaSalle principle for this Lyapunov (Hamiltonian)?

I am analyzing this problem, and some questions has appeared to me. 1- In case 2 and 3, which Hamiltonian should I choose? 2- I did not understand case 4? Could you enlighten me please. Let $\alpha$ ...
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