## 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.

## 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!