Keeping track of limit cycles via certain second order differential operator

Inspired by the two posts which are linked bellow we ask the following question:

Question: For a vector field $ X$ on the plane we define the differential operator $ D$ on $ C^{\infty}(\mathbb{R}^2)$ with $ D(f)=(\Delta\circ L_X-L_X\circ \Delta)(f)$ where $ \Delta$ is the standard Laplacian.

Is there a vector field $ X$ on the plane with $ 2$ nested closed orbits $ \gamma_1 \subset \gamma_2$ such that there exist a smooth function $ f$ for which $ D(f)$ does not vanish on the closur of the amnular region surrounded by $ \gamma_1$ and $ \gamma_2$ ?

On equation $ \Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

Elliptic operators corresponds to non vanishing vector fields

Remark: The first words of the titles of this post is inspired by a paper by C.C. Pugh and J.P Francois ” Keeping track of Limit cycles”

Continuity of a differential of a Banach-valued holomorphic map

Originally posted on MSE.

Let $ U$ be an open set in $ \mathbb{C}^{n}$ let $ F$ be a Banach space (in my case even a dual Banach space), and let $ \varphi:U\to F$ be a holomorphic map. I seem to be able to prove that the differential map $ D\varphi:U\times\mathbb{C}^{n}\to F$ defined by $ $ D\varphi (z,v)= \lim\limits_{t\to 0}\frac{\varphi(z+tv)-\varphi(z)}{t}$ $ is holomorphic.

Is there a reference for this assertion? (Or at least for continuity)

I tried to look into some sources on infinite-dimensional holomorphicity and could not find such a statement, but some of those sources are rather complicated, and so it is likely I missed it.

Inclusion of the spectrum of two differential operators defined on $L^2[-a,a]$ and $L^2[0, \infty)$

Let $ T$ be the formal operator defined by $ $ Tu:= \sum_{j=0}^{2n} a_j\frac{d^ju}{dx^j}$ $ where $ a_j \in \mathbb{C}$ . Consider the differential operators $ T_a: D(T_a)\subseteq L^2[-a,a] \to L^2[-a,a]$ and $ T_\infty: D(T_\infty)\subseteq L^2[0,\infty) \to L^2[0,\infty)$ defined by $ $ T_af:=Tf, \ T_bg:=Tg, \ f \in D(T_a), g \in D(T_\infty),$ $ where $ $ D(T_a):=\{ f \in L^2[-a,a] : Tf \in L^2[-a,a], f^{(j)}(-a)=f^{(j)}(a)=0 \mbox{ for } 0 \leq j \leq n-1\}$ $ and $ $ D(T_\infty):=\{ f \in L^2[0,\infty) : Tf \in L^2[0,\infty) , f^{(j)}(0)=0 \mbox{ for } 0 \leq j \leq n-1\}.$ $

Can we say that $ \sigma(T_a) \subseteq \sigma(T_\infty)$ ?. I know that the inclusion is true if we take $ Tu:=u”$ or $ Tu:=-u”-2u’$ , for example.

Thanks in advance for any help you are able to provide.

Restriction of a differential operator on $L^2(\mathbb{R})$

I’m reading a proof of a paper and I don’t understand an argument of the proof. That argument is the following:

Let $ A: D(A) \subseteq L^2(\mathbb{R}) \to L^2(\mathbb{R}) $ be a constant coefficients differential operator of order $ 2n$ . The restriction of $ A$ to $ L^2(I)$ ($ I$ is an interval in $ \mathbb{R}$ ) in the sense of quadratic forms is precisely the restriction which satisfies the Dirichlet boundary conditions (that boundary conditions are $ u(c)=u'(c)= \cdots = u^{(n-1)}(c)=0$ at any finite boundary point $ c$ ).

Can you help me to show the last statement, please?

I don’t know what “restriction in the sense of quadratic forms” means. Can you give me a reference on that subject, please?

Thanks in advance for any help.

Differential equation slope field and cauchy solution

Given the differential equation

$ 2y’=\frac{yx}{x^2+4}+\frac{x}{y}$

  • I have to draw the slope field in a rectangle called $ P$ , containing the point (2, 1).
  • Then in appropriate interval I have to find the solution of the Cauchy problem for the given differential equation with initial equation $ y(x_{0})=y_{0}$ where $ (x_{0}, y_{0})$ is inputed by clicking in the rectangle $ P$
  • and in the same rectangle $ P$ the graph of the found approximation of the Cauchy problem with given initial equation given above .

Here is my solution:

 function Plotslope x=-5:0.6:5; y=-6:0.6:6; delta=0.2;  hold on axis([-5,5,-6,6]) daspect([1,1,1])  for k=1:length(x)         for m=1:length(y)           eps=delta/(sqrt(1+ff(x(k),y(m))^2));           plot([x(k)-eps, x(k)+eps],...          [y(m)-eps*ff(x(k),y(m)),...            y(m)+eps*ff(x(k),y(m))],'k');         plot(x(k),y(m),'k.','LineWidth',0.2)          end end  [x0,y0]=ginput(1); plot(x0,y0,'bo') [T,Y]=ode45(@ff,[x0,5],y0); [T1,Y1]=ode45(@ff,[x0,-6],y0); plot(T,Y,'r',T1,Y1,'r')  function z=ff(x,y)      z=(y*x)/(2*(x^2+4))+x/2; end end   

I am sure I am finding the first 2 bullets, but how can I solve the 3rd one?

Differential equation tangent and slope field method

Given the equation $ y’=\frac{y}{3x}+\frac{x^2}{3y}$ I have to find :

1.The tangent to the integral curve of this equation which passes through the point $ (x_{0},y_{0}) ∈ R^2$

2.Describe a method for building slope field of the given equation.

For 1) my solution is the following :

so $ y'(x)=tg(\alpha)$ and let $ f(x_{0},y_{0})= \frac{y_{0}}{3x_{0}}+\frac{x_{0}^2}{3y_{0}}$ ,however $ tg(\alpha)=\frac{y-y_{0}}{x-x_{0}}$ so $ \frac{y-y_{0}}{x-x_{0}} = \frac{y_{0}}{3x_{0}}+\frac{x_{0}^2}{3y_{0}}$ . Then $ y= \frac{\frac{y_{0}}{3x_{0}}+\frac{x_{0}^2}{3y_{0}}}{\frac{y-y_{0}}{x-x_{0}}} + y_{0}$ which gives us the desired answer

For 2) my solution is the following : for the right side of the resulting equation we assosciate vector field in the plane, for which for every point $ (x,y) \in R$ of the domain of $ f(x,y)$ corresponds to the vector $ (1,f(x,y))$ ,however it is better to use lines – they are parts of the tangent throught every point $ (x,y) \in R^2$ from the domain of $ f(x,y)$ .The resulting lines make the slope field, so the method will be :

1.choose $ \delta$ >0

2.for each point $ (x_{k},y_{k})$ we compute $ \epsilon $ and we draw a line connecting the points $ (x_{k}-\epsilon,\bar{y_{m}})$ and $ (x_{k}+\epsilon,\underline{y_{m}})$

so let $ 2\delta$ be the length of the line ,then: $ tg(\alpha)=y'(x_{k})=f(x_{k},y_{m})$

$ tg(\alpha)=\frac{\sqrt{\delta^2-\epsilon^2}}{\epsilon}=f(x_{k},y_{m})$ from which we directly get that $ \epsilon=\frac{\delta}{\sqrt{1+f^2(x_{k},y_{m})}}$

Now $ y_{m}-\bar{y_{m}}=\sqrt{\delta^2-\epsilon^2}=\delta f(x_{k},y_{m})=>\underline{y_{m}}=y_{m}-\delta f(x_{k},y_{m})$

Do you think I have any mistakes in my solution and if yes please correct me.

Leafwise de Rham cohomology(A true definition of Differential forms along leaves)

For a foliated space $ (M, \mathcal{F})$ , one associate a leafwise de Rham cohomology. This cohomology and trace class operators on this cohomology and trace interpretations for closed orbits of certain flow on $ M$ is the main object of this paper”Number theory and dynamical system of foliated manifolds.

But in the later paper, I did not find a very precise definition of “Differential forms along leaf”.

So I try to find other papers or talk to find a precise definition for this concept. Then I found a definition at page 8 of this talk “Lefschetz trace formula for flow on foliated manifolds” which give a local representation for such forms. But my problem is the following:

I think that such representation, which is quoted below, is NOT invariant under foliation charts:

$ $ \omega\sum_{\alpha_1<\alpha_2<\ldots<\alpha_k} a_{\alpha}(x,y) dx_{\alpha_1}\wedge dx_{\alpha_2}\wedge \ldots\wedge dx_{\alpha_k}$ $

Am I mistaken?

What is a precise definition and precise local representations of “Differential forms along leaves”?

How to convert delay differential equations from continuous time to discrete time

I am building an agent-based model in discrete time. To do this, I need to convert continuous time delay and ordinary differential equations to discrete time equations. Here are the equation system:

Equation system

How can I rewrite these continuous time equations to dicrete time equations? What discretization method should be used. In the ABM, each time step represents 1 day. I am beginner in ODEs and DDEs. Any help would be greatly appreciated.

System of Differential Equations (Linear Algebra – Eigenvalues/vectors)

Q: In tracking the propagation of a disease, a population can be divided into 3 groups: the portion that is susceptible, S(t), the portion that is infected, F(t), and the portion that is recovering, R(t). Each of these will change according to a differential equation:

S′  =  − S/4 F′  =  S/4  −  F/6 R′  =  F/6

so that the portion of the population that is infected is increasing in proportion to the number of susceptible people that contract the disease, and decreasing as a proportion of the infected people who recover. If we introduce the vector y  =  [S F R]T, this can be written in matrix form as y′  =  Ay.

If one of the solutions is

y  =  x1  +  400 e^(−t/a)x2  +  600 e^(−t/c) x3 ,

where x1  =  [0 0 50,000]T, x2  =  [0 −1 1]T, and x3  =  [b 24 −16]T, what are the values of a, b, and c?

Enter the values of a, b, and c into the answer box below, separated with commas.

So I don’t really know where to start, I was looking through my textbook to see if anything would help and thought that this would see image below

I don’t really know how it’ll really play out, could someone be kind enough, to guide me on how to solve this or give hints?

I really appreciate it!

System of Differential Equations (Linear Algebra – Eigenvalues/vectors)

Q: In tracking the propagation of a disease, a population can be divided into 3 groups: the portion that is susceptible, S(t), the portion that is infected, F(t), and the portion that is recovering, R(t). Each of these will change according to a differential equation:

S′  =  − S/4 F′  =  S/4  −  F/6 R′  =  F/6

so that the portion of the population that is infected is increasing in proportion to the number of susceptible people that contract the disease, and decreasing as a proportion of the infected people who recover. If we introduce the vector y  =  [S F R]T, this can be written in matrix form as y′  =  Ay.

If one of the solutions is

y  =  x1  +  400 e^(−t/a)x2  +  600 e^(−t/c) x3 ,

where x1  =  [0 0 50,000]T, x2  =  [0 −1 1]T, and x3  =  [b 24 −16]T, what are the values of a, b, and c?

Enter the values of a, b, and c into the answer box below, separated with commas.

So I don’t really know where to start, I was looking through my textbook to see if anything would help and thought that this would see image below

I don’t really know how it’ll really play out, could someone be kind enough, to guide me on how to solve this or give hints?

I really appreciate it!