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Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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On a system of non-linear differential equations

Consider the following system of coupled differential equations \begin{align} \dot{x}_{1}&= -b_1\sin(x_{1})+c(\sin(x_{2})-\sin(x_{3})) \\ \dot{x}_{2}&= a-2c\sin(x_{2})+b_1\sin(x_{1})-b_4\sin(...
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2answers
194 views

Behavior of a non-linear differential equation

Let us consider the following differential equation $$ \dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}. $$ My question. Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the ...
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1answer
360 views

Unusual problem of calculus-of-variations. Attempt 2

I already tried to ask the same question here Unusual problem of calculus-of-variations, but I did a huge mistake and have no time to understand it. In the past post, I tried to simplify the problem, ...
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2answers
105 views

ODE with Holder drift - Cauchy-Peano theorem

Consider the following ODE: $$ x′(t)=b(x(t)),\quad x(0)=x_0. $$ If $b$ is bounded and Holder continuous, then the Cauchy-Peano theorem ensures that there exists a solution to the above equation (but ...
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57 views

$N-$Green function in $\mathbb R^N$

Let $N \geq 3$. Does there exist solution of the following equation $$-\Delta_N G + G^{N-1} = \delta_0,$$ where $-\Delta_N = - \text{ div}(|\nabla \cdot |^{N-2} \nabla \cdot )$ denotes $N-$Laplace ...
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1answer
79 views

How to find all orthogonal coordinates in a space of dimension $n$

I have been thinking for a while how to determine all the orthogonal coordinate systems in linear spaces of an arbitrary dimension $n$. The motivation for such a task comes from physics: I am ...
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186 views

Unusual problem of calculus-of-variations

I have a domain $D:\quad$ $-1<x<1$ and $-f(x)<y<f(x)$.\ There is function $u(x,y)$ which is obey the equation $\Delta u(x,y)=0$, $\forall (x,y)\in D$ with the Dirichlet boundary condition ...
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Structural properties (higher order monotonicity) preserved by linear differential operators

Consider the following linear differential equation: $(\rho+x)\ f_i(x,y)+k(x-b)\ \partial_x\ f_i(x,y)=x\ f_{i-1}(x,y),\quad x\geq x_0>b>0$ where $i\geq1$, $f_0(x,y)=0$ for any $(x,y)\in\...
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2answers
166 views

Convexity of a solution of a first order linear ODE

I have a very simple linear first order ODE. $$v(x) = c x + A - B x(1-x) v'(x)$$ $c, A ,B \in(0,1)$. The domain is $(\underline{x}, 1)$. where $\underline x > 0$. I am guessing that for any ...
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1answer
430 views

Differential algebraic geometry vs Diffiety theory

Algebraic geometry is said to be useful to study not only specific solutions of polynomial equations but to understand the intrinsic properties of the totality of solutions of a system of equations. ...
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80 views

Proving that system is Hamiltonian

This question is moved from math stackexchange, seems like it is a more advanced question. Here the link from the original question: https://math.stackexchange.com/questions/2666194/proving-that-...
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1answer
145 views

Dirichlet fractional Laplacian and zero boundary conditions

Does there exists a non-zero function $$f\in C_0([0,1]):=\{f:[0,1]\to \mathbb R:\ f\text{ is continuous and } f(0)=f(1)=0\},$$ such that $(-\Delta)^{\frac\alpha 2}f\in C_0([0,1]) $, where $(-\Delta)^{\...
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0answers
47 views

The canonical form of the first Painlevé equation

The first Painlevé equation is traditionally written as $$y''=6y^2+x. $$ Using scaling in both the dependent and independent variables, one can transform this equation into $$Y''=aY^2+bx $$ for ...
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1answer
111 views

On local attractivity of a coupled non-linear differential equation

Consider a dynamical system described by the following coupled non-linear differential equation \begin{align} \dot{x}_1(t) &= v + a_{12}\sin(x_2(t)-x_1(t)) + a_{13}\sin(x_3(t)-x_1(t))\\ \dot{x}_2(...
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2answers
170 views

Reference request: a singular differential equation

I need the following result (which I believe to be true though I was too lazy to write down a complete proof). Let $f$ be a function of two complex variables analytic at the origin and $a\not\in\...
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0answers
76 views

A singular foliation analogy of the Riemann Hilbert problem

Note: In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$. ...
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34 views

Is there a way to set a system of logistic differential equations to sum to 0

so I am working on this personal project in which I have a set of differential equations that are all of logistic growth form $$ \dfrac{dS_k}{dt}=S_k\left(1-\dfrac{S_k}{\sigma_k(S_1,S_2,...,S_n)}\...
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41 views

Redundant boundary condition of a $1$st order ODE?

Consider the following $1$st order eigen ODE system of 2 components $(\alpha,\beta)(x)$ defined for $x\in[0,L]$ $$ F_{l+1}\beta+m\alpha=\lambda\alpha\\ F_{-l}\alpha-m\beta=\lambda\beta $$ where $\...
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Two variable delta-finite function

Let ore algebra $\mathbb{O}:=\mathbb{C}(x,y)[D_x , 1,D_x][D_y,1,D_y]$ Let $F(x,y):=\sum_{m,n}a_{m,n}x^m y^n$ is a $\partial$ finite function in two variable over the field of rational function $k:=\...
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46 views

Region of attraction of simple ODE with perturbation

Consider the following simplest example: $$\dot{x} = x(x-1)(x+1)$$ $[-1,1]$ is the ROA. Now consider the two dimensional case: \begin{equation} \begin{aligned} &\dot{x} = x(x-1)(x+1)\\ &...
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0answers
103 views

Stability when linearization fails

The dynamics of the $j$th system: \begin{equation} \begin{split} \dot{\overline r}_j &= h (\overline r_j) \,\, - \varepsilon \omega_{\mathrm{sw}}R_{\mathrm{Th}}\sqrt{\xi^2+\chi^2}\sum^N_{k=1} \...
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0answers
107 views

Does a smooth dynamical system always come with a metric

Warning: My education in formal mathematics is very weak so I apologize for any confusions/errors in the following, please don't hesitate to correct me. Question: Consider a smooth dynamical system $...
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1answer
151 views

Does asymptotic behavior guarantee uniqueness?

Suppose $w$ is a solution of $$\frac{d^2}{dx^2}w+\{u(x)+k^2\}w=0$$ with asymptotic condition $$\lim_{x\rightarrow \infty}w(x)e^{ikx}=1$$ and $u\in L^1_1(\mathbb{R})=\{f:\int_\mathbb{R}(1+|x|)|f|dx<...
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2answers
146 views

Rotation number of composition

Let $f,g:S^1 \to S^1$ be orientation-preserving homeomorphisms. Consider the lift $F,G:\mathbb R \to \mathbb R$. Let $\rho(G)$ and $\rho(F)$ be a rotation numbers. What we can say about rotation ...
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184 views

Canard limit cycle for certain singularly perturbed system(Is there a contradictory situation?)

From the figures of page 478 and 479 of this paper one find that the author probably means that we have a (canard) limit cycle for the system $$\begin{cases} x'=y-x^2\\ y'=\epsilon(a-x) ...
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0answers
34 views

Harnack type Estimates for a p-Poisson equation with constant source term

Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE $$ -\Delta_p u=1\,\,\mbox{in $B$} $$ Let another function $f$ be such that $$ \begin{cases} -\Delta_p f =1 \;\;\mbox{in $B$}\\ f=0 \...
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Examples of Differential Equations with Strongly Monotone Operator

Let $E \subset H \subset E^{*}$, where $E$ is a real Banach space, $E^{*}$ its dual and $H$ is a real Hilbert space. Embendings are continuous and dense. Let $\langle v_1,v_2\rangle$ be a dual pair ...
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1answer
50 views

Relation between symmetries and asymptotics for Painlevé equations

The first Painlevé equation $$P_I:y''=6y^2-x $$ has the symmetries $$x \mapsto \omega x \\y \mapsto \omega^3 y$$ for any fifth root of unity $\omega$. At the same time, the near-infinity asymptotics ...
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0answers
28 views

Implementing the Fair-Luke algorithm

The Fair-Luke algorithm, as appears in Rational Approximations to the Solution of the Second Order Riccati Equation, constructs rational functions which approximate solutions of the so-called "...
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2answers
144 views

Controlling solutions of a second order linear differential inequality

A slightly less general version of this question was asked, in a subsequent comment, by the OP of the question at Controlling subsolutions of a second order linear ODE Let $f:[0,\infty) \to \mathbb{...
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equivalence of covariance matrix construction using convolution method and diffusion operator method?

One can define a correlation by making convolutions of two radially symmetric functions. \begin{equation*} C_0(r)=B_1*B_2(r) [B_1*B_2(0)]^{-1} \end{equation*} On the other hand, one can also ...
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1answer
151 views

the asymptotic behaviour of function as $\lambda \to -\infty$

Let's consider the following differential equation on $\mathbb{R}$: $$-u''(x)+u(x)-V(x)u(x)=\lambda u(x),$$ where $\lambda<1$ and $V$ is a bounded. We consider only that solution $u(x) \in C^1$ ...
3
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1answer
82 views

Reference request - existence of movable essential singularities

On the Wikipedia page regarding the Painlevé transcendents it says: Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and found a special case of ...
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2answers
199 views

Can one obtain this ODE as an Euler-Lagrange equation?

Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...
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1answer
207 views

Controlling subsolutions of a second order linear ODE

Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality $$f'' - 2\alpha f' + 2\alpha f \leq 0$$ where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that $...
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2answers
293 views

Non-linear first order ODE $ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$

I am trying to solve an ODE which has the following form: $$ \dfrac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$$ with an initial condition $y(x_0) = y_0 \\ $....
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1answer
145 views

flow of holomorphic vector field [closed]

Let $(M,J)$ be a complex manifold, where $J$ is the integrable complex structure. Let $X$ be a holomorphic vector field on $M$ and let $\varphi_{t} : M\rightarrow M $ be its flow. Question: Is $\...
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0answers
92 views

Stochastic Approximation in Reproducing Kernel Hilbert Space

Consider an iterative algorithm with incremental updates \begin{align} x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}], \end{align} where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
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171 views

Location of the endpoints of two parametric curves

I have two curves, $C_1$ and $C_2$ parametrized by $\theta$, the angle of the outward normal with the X-axis. $C_1$ is given by the following equations (say $r = 0.2$): \begin{align*} \frac{dx}{d\...
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52 views

Existence of a couple of functions solution of a differential equation (with additional constraint)

I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$): $$...
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0answers
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How do I obtain the first fundamental form in terms of the arclength and unit normal of an arbitrary ray in the real plane?

My question is based on the 1981 paper "Computation of wave fields in inhomogenous media" by Cerveny et.al. (I will add any appropriately needed information to this posted question, so there is no ...
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0answers
54 views

When are Green's functions causal convolution kernels

Let $L$ be a linear differential operarator acting on distributions over $\mathbb{R}$ and $G(t, s)$ be a Green's function, i.e., a solution to $LG(t, s) =\delta(t-s)$. $G$ is said to be causal if $G(...
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0answers
83 views

Origins of the generalized shift operator exp(t*g(z)d/dz)

Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
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1answer
98 views

Basis for solutions hypergeometric differential equation

In the book "Theorie der gewöhnlichen Differentialgleichungen" by Bieberbach, page 240, there is a solution to the hypergeometric differential equation $z(z-1)w^{\prime \prime}+(2z-1)w^{\prime}+\frac{...
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2answers
198 views

Is this Riccati equation (“Josephson junction”) always phase-locked at integer rotation numbers?

Given parameters $(a,k,A) \in \mathbb{R}^3$, we consider on $\mathbb{S}^1$ the $2\pi$-periodic ODE $$ \dot{\theta} \ = \ - a\sin(\theta) + k + A\cos(t) \hspace{4mm} \mathrm{mod} \ 2\pi. $$ Identifying ...
2
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1answer
262 views

Gronwall's inequality for higher order derivatives

Gronwall's inequality says that solutions to the initial value problem $u'(t) \leq \beta(t)u(t)$ with $u(0)=u_0$ are bounded by solutions to the problem with inequality replaced with equality for $t\...
3
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0answers
78 views

Some quasi differential equations

This question is inspired by the concept of "Differential Inclusion". The ring of entire holomorphic functions is denoted by $Hol(\mathbb{C})$. Is there a complete classification of all $f\in Hol(\...
2
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2answers
119 views

Planar polynomial vector field for a harmonic pair of polynomials

Has the system of ODEs $$\frac{dx}{dt}=P(x,y)\\ \frac{dy}{dt}=Q(x,y) $$ been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of ...
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0answers
80 views

Relating to the work of Václav Hlavatý

Upon looking for a topic for my phd thesis, I came upon the work of Václav Hlavatý, whose work concerning unified field theory seemed very interesting to me. My question is this; what are some similar ...
6
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1answer
196 views

Pursuit solutions to the Rock-paper-scissors flow and delay differential equations

The Rock-paper-scissors flow is the following reaction-diffusion system $$r_t = \Delta r + rs-rp,$$ $$p_t = \Delta p + pr-ps,$$ $$s_t = \Delta s + sp-sr.$$ We can assume $r,p,s\geq 0$, $r+p+s$ is ...