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Trying to learn math :)


Apr
10
comment Principal direction in Dynamical System
Indeed, you could translate principal direction as the direction of the eigenvector. The term "eigen" is german, however it is widely used I guess in many languages.
Apr
8
comment Uniqueness of solution to differential equation
Fixed point theorem together with the assumption that the right hand side of the ODE is Lipschitz should be useful.
Mar
26
comment Can one exchange fibre and base space in a fibre bundle?
@EarthliƋ : Did you find an answer to this? I am also very interested in such a thing.
Mar
18
comment Bifurcation in coupled differential equations
Bifurcations normally have to do with parameters. When such parameters make the spectrum of a system cross the imaginary axis, then it is said that a bifurcation occurs. I do not think the definition of a bifurcation is only that 1 or more eigenvalues are 0. Your plot might be correct with the system, but something is missing to be able to talk of bifurcations. More precisely there is a $1$-dimensional center manifold (a local invariant) that contains the non-hyperbolic equilibrium point $(x,y)=(0,11)$
Mar
18
comment Linear Differential Equation with Quasiperiodic Coefficient: A Question
I am not expert but just from the sentence I guess there is a supposition (must probably quasi periodicity) that implies $\omega_i$ is irrational. Otherwise there exists an integer $A$ such that $A\omega_i=\omega_j$, which would mean that $a(t)$ is periodic and not quasi periodic (as stated in the hypothesis).
Mar
18
comment Principal direction in Dynamical System
What are your thoughts? The fixed points are those $x$ where $F(x)=0$. At each fixed point $x^*$ you can compute the linear part $J=\partial F/\partial x (x^*)$. Such matrix $J$ gives a lot of information on the local behaviour of the solutions, provided $J$ is not too degenerate. The eigenvalues and eigenvectors of $J$ are very important to understand the solutions $x(t)$ "very" near the equilibrium points.
Mar
12
comment Reference request: Nonlinear dynamics graduate reference
I would add the book Geometric theory of dynamical systems by Palis and de Melo, and maybe Dynamical Systems and chaos by Broer and Takens. There are plenty of references of a graduate level, but dynamical systems (in particular nonlinear dynamics) is a very broad topic. There is the encyclopaedia of dynamical systems as well (by V.I. Arnold), which treats many nonlinear phenomena in a rigorous way. Another applied mathematics book would be Nonlinear Differential Equations by Jordan and Smith.
Mar
7
comment Find a nonlinear system conjugate the linear system $\overrightarrow x' = \left(\begin{array}{cc} 1 & 2\\0&-4 \end{array}\right) \overrightarrow x$.
I think you are looking for the Hartman-Grobman theorem. In such case any nonlinear vector field with the linear part you wrote answers your question.
Feb
13
comment Lagrange optimimzation with a Diff.eq constraint
I guess you should carry out the operations inside the summation sign. Then $L$ is just a scalar function and you proceed as usual with the partial derivatives.
Feb
8
comment Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $a > 0$ and $x(0) = 0.$
What you need to do is to check wether the function $f(x)=x^a$ is Lipschitz or not.
Feb
6
comment How to prove orbit periodicity in some conservative systems?
It occurs to me, although it's just and idea, that from the properties of $g(x)$ you could write the second equation as $\dot y=x+O(x^2)$. And then, at least locally, you have periodic orbits ... ¿? Another idea is, can you use a Poincarè section and prove the existence of a fixed point?
Dec
6
comment Center manifold of sets of equilibria
But, as I understand, the function you write is tangent to the centre space only at the origin. Shouldn't it be then $h:\mathbb{R}^2\to\mathbb{R}$ of the form $h(x,0)=Dh(x,0)=0$??? Maybe I am complicating myself too much :)
Nov
19
comment How to calculate the relevancy
Are you looking for a specific, very accurate statistical method? Or something rather empirical? In your first example I'd go as follows. You have a fixed chain of words, which if I counted correctly are 28. Assume each word has the same probability to appear. Then $P(Lorem)=1/28$. Since it appeared 3 times then its relevance is $R(Lorem)=3/28$. Of course this is not very formal, but maybe the idea helps. Then you can compare, for example $R(Lorem)=R(Ipsum)>R(text)$. Note: I used probability just to give a "feeling" not really meaning it was a mathematically correct argument.
Nov
7
comment Relationship between solutions of $y'''+y=0$
As suggested, convert to 1st order ODE. Also know as state space. Define $u=y, \, v=y', \, w=y''$. From there the result will follow.
Nov
6
comment Compute coefficients of a rational expansion
Ok. I think I've got it. in the series, first I write $(u+1)^{7/2}$ in its binomial expansion (which I did't know holds for arbitrary complex exponents). After that I am able to play a little bit with the exponents, re-arrange, shift, and finally collect common coefficients.
Nov
1
comment Integral curves in the plane
do you need the analytic solutions? If that is the case you have an uncoupled system which can be solved. one equation is $\dot x=x$ and the other is $\dot y=-y$. In case you only need the phase portrait note that you have a linear system, and the eigenvalues are ${-1,1}$ so you have a saddle.
Oct
30
comment How do you find the time-1 map of an autonomous differential equation?
Are you referring to Poincaré sections?
Oct
30
comment Differentiating a non-linear functional with respect to a vector
My bad, you are completely right. I'll correct.
Oct
18
comment Not able to solve the below mentioned inequality. Someone please explain me it's solution.
I can see the image just OK.
Sep
26
comment Pollution of lakes, differential equation
If I understand correctly you want to express everything in therms of $\frac{m_i(t)}{Q_i}$. Then why not just define the variable $c_i(t)=\frac{m_i(t)}{Q_i}$ (concentration) and then you have $\frac{d}{dt}c_i(t)=c_i(t)\frac{v}{Q_i}$.