Non-linear systems phase portrait and conserved quantity.

[gino492]

So for this question we have to find fixed points, classify them, find a potential and conserved quantity; then sketch a phase portrait.

I have tried to do this in latex and failed. When I use xdot I'm referring to x with a dot on top.

the system is xdotdot = x^5 -5x^3+4x.

Fatorising gives (x-1)(x-2)x(x+1)(x+2)

So the fixed points are 2,1,0,-1,-2.

To find the potential we use the formula f(x)=-dv/dx

so the negative integral of the original system is

-x^6/6 +5/4x^4-2x^2 << apologies for ugly looking.

We have sketched both the original system and classified the points, -2,0,2 are unstable, -1,1 are stable.

From here, how to we find conserved quantity? Any ideas how to sketch the phase portrait?

Any help much appreciated.

 

[HallsofIvy]

You are taking a course in differential equations, are you not? Do you not know what a "potential function" or "phase portrait" is? "Potential function" (which is really a physics term, from potential energy- the math term is "first integral") is usually applied to a system of first order equations. We can write this second order equation as two first order equations by defining y= x' so that \(\displaystyle x''= y'= x^5- 5x^3+ 4x\) and, of course, x'= y. We could, although it is not necessary, write that as a single vector equation: Let \(\displaystyle X= \begin{pmatrix}x \\ y\end{pmatrix}\) so that \(\displaystyle X'= \begin{pmatrix}x' \\ y'\end{pmatrix}= \begin{pmatrix} y \\ x^5- 5x^3+ 4x\end{pmatrix}\).

A "potential function" or "first integral" for that is a function F(x,y) such that \(\displaystyle \nabla F= <y, x^5- 5x^3+ 4x>\) So that we must have \(\displaystyle \frac{\partial F}{\partial x}= y\) and \(\displaystyle \frac{\partial F}{\partial y}= x^5- 5x^3+ 4x\). The first question should be "does this system have a potential function?" There will be a potential function if and only if the system is "conservative" (another physics term- the math term is "an exact differential"). In order that \(\displaystyle \frac{\partial F}{\partial x}= f(x,y)\) and \(\displaystyle \frac{\partial F}{\partial y}= g(x,y)\) be "conservative" or "exact", we must have that the second mixed derivatives are equal: \(\displaystyle \frac{\partial^2 F}{\partial x\partial y}= \frac{\partial f}{\partial y}= \frac{\partial g}{\partial x}= \frac{\partial^2 F}{\partial y\partial x}\). In this case, \(\displaystyle \frac{\partial f}{\partial y}= 1\) while \(\displaystyle \frac{\partial g}{\partial x}= 5x^4- 15x^2+ 4\). Those are NOT equal so this system does NOT have a "potential function".

I just noticed that you said

I have tried to do this in latex and failed. When I use xdot I'm referring to x with a dot on top.

but then

the system is xdotdot = x^5 -5x^3+4x.

Is your equation "xdot" or "xdotdot". Questions about "phase portraits" and "conserved quantities" usually involve first order equations. If yours is really a second order equation I recommend treating it as a "vector" problem. That is, define y= xdot and write it as
\(\displaystyle \begin{pmatrix}x' \\ y' \end{pmatrix}= \begin{pmatrix} y \\ x^5- 5x^3+ 4x\end{pmatrix}\)