Critical point analysis for ${x}^{\prime}=y,{y}^{\prime}={x}^{2}-y-\u03f5$

bucstar11n0h
2022-11-21
Answered

Critical point analysis for ${x}^{\prime}=y,{y}^{\prime}={x}^{2}-y-\u03f5$

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Tasinazzokbc

Answered 2022-11-22
Author has **17** answers

There is a theorem which says so long as the critical (or equilibrium) point is not a center then the nature of this critical point in the system There is a theorem which says so long as the critical (or equilibrium) point is not a center then the nature of this critical point in the system x′,y′, is the same as that point in the linearization i.e,, is the same as that point in the linearization i.e,

${\stackrel{~}{x}}^{\prime}={p}_{x}({x}_{0},{y}_{0})\cdot x+{p}_{y}({x}_{0},{y}_{0})\cdot y$

${\stackrel{~}{y}}^{\prime}={q}_{x}({x}_{0},{y}_{0})\cdot x+{q}_{y}({x}_{0},{y}_{0})\cdot y$

where ${x}^{\prime}=p(x,y)$ and ${y}^{\prime}=q(x,y)$. Now you have a linear system. Find the associated matrix to this system, compute it's eigenvalues and use the Painleve Analysis.

${\stackrel{~}{x}}^{\prime}={p}_{x}({x}_{0},{y}_{0})\cdot x+{p}_{y}({x}_{0},{y}_{0})\cdot y$

${\stackrel{~}{y}}^{\prime}={q}_{x}({x}_{0},{y}_{0})\cdot x+{q}_{y}({x}_{0},{y}_{0})\cdot y$

where ${x}^{\prime}=p(x,y)$ and ${y}^{\prime}=q(x,y)$. Now you have a linear system. Find the associated matrix to this system, compute it's eigenvalues and use the Painleve Analysis.

asked 2022-11-08

Let $f:U\to R$, $U\subseteq {R}^{n}$ is open. if $\overrightarrow{a}\in U$ is local Maxima or local minima, then $\overrightarrow{a}$ is a critical point.

a proof assumes $(\mathrm{\nabla}f)(\overrightarrow{a})\ne {\overrightarrow{0}}^{\text{}t}$, and therefore for small enough $h\in R$, $h>0$ we get:

$f(\overrightarrow{a}+h((\mathrm{\nabla}f)(\overrightarrow{a}){)}^{t})>f(\overrightarrow{a})$

which is in contradiction to maxima.

why is have to be larger the $f(\overrightarrow{a})$

a proof assumes $(\mathrm{\nabla}f)(\overrightarrow{a})\ne {\overrightarrow{0}}^{\text{}t}$, and therefore for small enough $h\in R$, $h>0$ we get:

$f(\overrightarrow{a}+h((\mathrm{\nabla}f)(\overrightarrow{a}){)}^{t})>f(\overrightarrow{a})$

which is in contradiction to maxima.

why is have to be larger the $f(\overrightarrow{a})$

asked 2022-10-30

What type of critical point is $(0,0)$ in the following function?

$f(x,y)\in {C}^{2},{T}_{2}(x,y)=3+2x+0.5{x}^{2}-xy-{y}^{2}$

$f(x,y)\in {C}^{2},{T}_{2}(x,y)=3+2x+0.5{x}^{2}-xy-{y}^{2}$

asked 2022-11-11

Why is ${f}^{\prime}(c)=\text{does not exist}$ does not exist a critical point?

asked 2022-07-15

Find the critical point of

$f(x)=\frac{5x}{x-3}$

$f(x)=\frac{5x}{x-3}$

asked 2022-08-19

$\{\begin{array}{l}\dot{x}=-y-{y}^{3}\\ \dot{y}=x\end{array}$

where $x,y\in \mathbb{R}$. Show that the critical point for the linear system is a center. Prove that the type of the critical point is the same for the nonlinear system.

where $x,y\in \mathbb{R}$. Show that the critical point for the linear system is a center. Prove that the type of the critical point is the same for the nonlinear system.

asked 2022-11-11

What is the 'stability' of a critical point of a ($1$ dimensional) dynamical system?

asked 2022-08-31

Find all the critical points of f when $f(x)={x}^{4/5}(x-{5}^{2})$