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

-4
votes
1answer
91 views

Hamiltonian functions [on hold]

I have a Hamiltonian function given by $$H(x,y)=(b/2) y^2+(ax-x^3)y+(a^2/ 2 b) x^2=C$$ ($C \in \mathbb R$). I would like to draw the curve of $H(x,y)-C$ by using critical curves of this curve. But ...
1
vote
0answers
69 views

Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...
2
votes
3answers
159 views

Complex differential equations

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs. Mostly, I'm just ...
1
vote
1answer
69 views

Solve nonlinear equation

Suppose that $f:E\to F$(between Banach spaces), is of the form $$f(x)=f(0)+D(x)+N(x).$$ Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\...
0
votes
0answers
32 views

Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element

In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks. Consider the system consisting of the controlled object and regulator. The ...
-3
votes
0answers
59 views

Transform a function with a homothecy [closed]

What expression describes how a function $y(x)$ is transformed under a general homothetic transformation? I'd like to prove that if I apply a homothetic transformation to a particular solution I ...
1
vote
1answer
23 views

Does stability of equilibrium point preserved by permutation matrix (symmetry)?

Given the following differential equations: \begin{equation} \begin{aligned} \dot{x}_1 &= f_1(x_1,\ldots,x_n) \\ \vdots \\ \dot{x}_n &= f_n(x_1,\ldots,x_n) \end{aligned} \end{equation} In ...
0
votes
1answer
29 views

Galerkin Finite element for solving third order time dependent partial differential equation inti weak form

How to solve the third order time dependent partial differential equation (i.e. u_t + 6u_x + u_xxx = 0) into weak form using galerkin finite different method?
1
vote
2answers
81 views

One inequality connected with the linear second order ODE

Is the following statement true? Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation $ \ddot{x}+a \dot{x}+bx=h$ with initial conditions $x(0)=u<0 , \dot{x}(0)...
5
votes
0answers
86 views
+50

Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related ...
0
votes
0answers
65 views

What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
0
votes
0answers
42 views

Euler-Lagrange Equation for system with holonomic constraints

I have a very basic question, but i am not sure, that i understand the formula correctly. I have the Lagrangian for a many particle system. The coordinate of the particles build a N-dimensional ...
6
votes
1answer
111 views

Ordinary differential operators satisfying braid relation?

Let $W$ be the algebra of linear ordinary differential operators with analytic coefficients $C^{\omega}(\mathbb{R})[\partial_x]$ (with multiplication given by composition). Do there exist two elements ...
1
vote
0answers
25 views

Mean of a periodic velocity field and trajectory displacement bound

Suppose $u(t,x)$ is a smooth velocity field on $[0,\infty)\times \mathbb{R}$ and periodic in space, i.e., $u(t,0)=u(t,1)$ $\forall t$. Assume that $\int_0^1 u(t,x) \,dx = c$, independent of time. Let $...
16
votes
1answer
2k views

Differential equation changing sign almost everywhere

Conjecture: Let $f:\mathbb{R}→\mathbb{R}$ be an everywhere differentiable function and assume that $f(x)+f′(x)∈ \{-1,1\}$ almost everywhere and $f'(0)=0$. Then is $f$ necessarily a constant function? ...
3
votes
1answer
106 views

Spectral decomposition of a specific operator

To understand a crucial example in representation theory, I need the explicit spectral decomposition of the differential operator $$ Df(x)=(1+x^2)f''(x)+2xf'(x) $$ on $L^2({\mathbb R})$. I'm not an ...
10
votes
0answers
244 views

Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?

The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
6
votes
1answer
479 views

(In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations $$ \dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t) = -\gamma x_2(t) - \cos(\...
2
votes
0answers
148 views

A question about whether an operator can be lipschitz or not

Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$. Now define the operator $ \mathcal{A} : C^{‎\sigma‎, \sigma‎/2‎}(‎X‎) \to C^{‎\sigma‎, \...
1
vote
0answers
73 views

2nd oder evolution equations and regularity results of their solution

I am interested in regularity results for solutions to 2nd order evolution equations in the shape of $$ u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\ u(0) = u_0 \text{ in } H, u'(0)...
1
vote
1answer
79 views

Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
1
vote
1answer
154 views

Classification of a system of two second order PDEs with two dependent and two independent variables

If we have a second order quasilinear PDE of the form $A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...
1
vote
1answer
75 views

Positive Solutions of second-order ODE

Consider second-order ordinary differential equations of the form $u''(t)=a(t)u(t)-2$ I'm interested in general criteria on the function $a(t)$, which guarantee respectively rule out the existence ...
1
vote
0answers
50 views

Well-posedness for a general linear grad-div PDE

Let $U \subset \mathbb{R}^n$ be an open bounded domain with smooth boundary ($\partial U$ is a closed manifold) and let $A(x)\in C^1(U, M_{n \times n}(\mathbb{R}))$, $b(x) \in C^0(U, \mathbb{R}^n)$ ...
0
votes
1answer
51 views

Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?

Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE: $$u_t = grad[V(u)]$$ For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-...
6
votes
0answers
96 views

Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$

I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...
3
votes
0answers
62 views

Parametrix of external product of elliptic operators

Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...
10
votes
0answers
383 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
1
vote
0answers
68 views

Reference for numerical solutions for differential equations like $f'(x)=f(x+1)+f(x-1)$

One can solve a delay differential equation (like for example $f'(x)=f(x-1)$) if we have a function as a bounded condition (in my example we need to know $f$ on $[0,1)$) and then use a simple forward ...
7
votes
2answers
364 views

Generating function of $SO(N)$ random matrix

I am interested in the generating function of $SO(N)$ random matrix, that is, I want to compute $$ Z_N[J]=\int dM e^{{\rm Tr} (J^T M)}, $$ where $dM$ is the $SO(N)$ Haar measure, and $J$ is an ...
3
votes
0answers
93 views

If the sum of everywhere linearly independent vector fields are periodic, are the component vector fields periodic?

I feel like the above must be true but embarrassingly cannot seem to prove it. Take linearly independent, commuting vector fields $X$ and $Y$ on a manifold and corresponding flows $\Phi^t_X$, $\Phi^...
3
votes
2answers
232 views

Are there vector fields which are gradients with respect to one metric but not another? [closed]

Is it possible for a vector field on a smooth manifold $M$ to be a gradient with respect to a Riemannian metric $g$, but not a gradient with respect to a different Riemannian metric $h$? For ...
-1
votes
1answer
68 views

explicit answer of and initial condition ODE with delta input

Assume this initial value problem ODE with constant coefficient: $\mathcal{D}[u] = \sum_{n=0}^N {a_n u^{(n)}}=0$ $u(0)=u_0\hspace{0.2cm} ;\hspace{0.2cm} u'(0)=u_1\hspace{0.2cm} ;\hspace{0.2cm} ... \...
4
votes
1answer
120 views

Which utility functions are linearly transformed by normal perturbations?

I'll ask this question as pure economics, as pure math, and showing the translation. Economics (micro): Which utilities have the property that whenever $EU(X)>EU(Y)$, the same is true after ...
7
votes
0answers
77 views

An upper bound for the solution of an integro-differential equation

For those who are interested in "motivations" this has something to do with modeling flames in turbulent jets. However the question itself is irritatingly elementary and requires no mathematical or ...
0
votes
0answers
56 views

Caratheodory Differential Equations ( Existence of Solution )

I'm working with filippov's book , "Differential Equations with Discontinuous Right Hand Sides ". Theorem 1) For $ t_{0} \leq t \leq t_{0} + a ~~~,|x-x_{0}| \leq b $ let the function $ f(t,x) $ ...
0
votes
0answers
45 views

Change of polynomial eigenvalues between polynomials

Given the polynomial eigenvalue problem $$ p_t(z) = det ( P(z) + Q(t) ) = 0, $$ where $P(z) = \sum_{i=0}^k P_i z^i$ with $P_i \in \mathbb{C}^{n \times n}$ and $Q(t) \in \mathbb{C}^{n \times n}$. The ...
0
votes
0answers
56 views

Reference for Existence and uniqueness of an Integro-Differential Equation

I have an Integro-Differential Equation (IDE) of the following form: $$ x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds, $$ I have found this classical reference, but the IDEs considered therein ...
0
votes
0answers
60 views

Perturbed trajectory of non-autonomous ode

Does the existence of strict Liapunov function guarantee that limit point of perturbed trajectory will be close to an isolated equilibrium for non-autonomous ode ? Can someone help me with some ...
0
votes
1answer
74 views

On the (qualitative) behavior of a coupled differential equation

Let $\mathbf{x}(t):=[x_1(t),\dots,x_n(t)]^\top$, $n>1$, $A\in\mathbb{R}^{n\times n}$ be a nonnegative matrix and $\mathbf{b}\in\mathbb{R}^n$ be a positive vector. Consider the following ...
0
votes
1answer
105 views

Continuity of the differential flow under a perturbation of the vector field

Suppose $v$ is a (possibly time-dependent) vector field on a compact manifold $M$. Its flow is a mapping $g: M \times \mathbb{R} \rightarrow M$, where $g$ satisfies the following conditions (written ...
1
vote
1answer
113 views

Asymptotic behavior of a solution of an ODE

I am not quite sure if this question is appropriate for this site as it might be not of a research level. I am interested in the following ordinary differential equation on the real line $$f’’(x)+(x-...
2
votes
1answer
113 views

A nodal theorem in 1D

Consider a 1D zero-energy Schrödinger equation on the half-line, $(-\partial_x^2 + V(x))\psi(x)=0, \quad x \in (0, \infty)$ with a zero boundary condition $\psi(0) = 0$. Is it true that if the zero-...
2
votes
0answers
157 views

Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems

I have a query regarding two equalities in the lemma in the book. But first I'll provide two definitions that one needs for this lemma. Definition 4.2.4: Consider the vector field $f(x,t)$ with $f:\...
3
votes
2answers
130 views

Perturbed behavior of a differential equation

Let $a$, $b$ be two real positive parameters with $a>b$, and consider the following nonlinear differential equation: \begin{align} \dot{x}_{\varepsilon}(t) = a - b\sin(x_{\varepsilon}(t))+\...
2
votes
0answers
50 views

Stability of ODEs with exponentials in the vector field

What is known about fine stability properties of ODEs of the following kind? $$ \dot{x} = Ax + b + \phi(x),\quad x\in \mathbb{R}^d ,$$ where $d\geq 1$; $A$ is a constant matrix with all e.v. having ...
4
votes
1answer
148 views

Is there a Feynman-Kac formula for vector-valued Schrödinger operators?

Given a vector function $$f=(f_1,\ldots,f_n)\in L^2(\mathbb R,\mathbb R^n)$$ (for some $n\in\mathbb N$), let us define $$\Delta f:=(\Delta f_1,\ldots,\Delta f_n),$$ where $\Delta$ is the Laplacian ...
3
votes
0answers
53 views

Generalized viscosity sub(super)solution and it's convolution

Suppose that $\Gamma \subsetneq \mathbb{R}^n$ is an open symmetric convex cone containing positive orthant. Note that $\Gamma \subset \left\{x=(x_1,...,x_n) \in \mathbb{R}^n | \sum_{i=1}^{n} x_i > ...
4
votes
1answer
145 views

Convolution of viscosity solutions and subharmonic functions

Suppose that $u : \mathbb{C}^n \rightarrow \mathbb{R}$ is continuous. We say that $u$ is a viscosity subsolution (resp. viscosity supersolution) for the Laplace's equation if for all $\varphi \in C^2$ ...
2
votes
0answers
66 views

Differential equation's flow [closed]

I'm struggling to solve the following problem: Find the image of $(1,0)$ vector in $(0,0)$ point under the action of system's flow: \begin{cases} x'= e^{3x}-e^{4y} \\ y'= e^{x}-e^{y} \end{cases} I ...