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Trying to learn math :)


Apr
11
answered Help Finding Saddle Connections of a System
Apr
11
revised 2nd order linear differential equation
added 117 characters in body
Apr
11
answered 2nd order linear differential equation
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
10
revised Find values of the parameters in Predator prey model
added 11 characters in body
Apr
10
answered Find values of the parameters in Predator prey model
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
19
awarded  Scholar
Mar
19
accepted Center manifold of sets of equilibria
Mar
19
asked Help understanding a theorem on Lagrangians and extremals
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
18
answered Topological conjugation between two flows
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
11
answered Consider the system $x'= \frac{-x}{2}; y' = 2y + x^2 $ to solve the system and find topologically conjugacy and show topologically conjugate
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.