Thread: Problems with Continuous Dynamical Systems

1. Problems with Continuous Dynamical Systems

Hi there, I'm new to this but I was wondering if you could help me with my understanding of Dynamical Systems.

I have a continuous dynamical system given by the equations:

dx1/dt = x1(x2-x3)
dx2/dt = x2(x3-x1)
dx3/dt = x3(x1-x3)

Which I need to analyse. There are a few problems I have but the main ones are centred around orbits.

My question is why can an orbit starting in the region x1 >= 0, x2 >= 0 , x3 >= 0 never cross one of the lines x_i = 0?

Any help on the matter would be greatly appreciated I'm drawing a complete blank with this and could really do with some help.

2. Proceed as you would with a differential equation: For example in your first equation isolating $\displaystyle x_1$ we get $\displaystyle \ln(x_1)=\int (x_2-x_3)dt+C_1$. Applying this to the other two, adding them up, condensing the constants and using that the logarithm opens products we arrive at $\displaystyle \ln (x_1x_2x_3)=\int (x_2-x_3)dt+C$ or equivalently $\displaystyle x_1x_2x_3=e^{C\int (x_2-x_3)dt}$ and since the exponential is always positive, none of the $\displaystyle x_i$ can be zero or change sign because of continuity.

PS. I'm not too familiar with dynamical systems so check everything carefully.

3. Originally Posted by wattsup
dx1/dt = x1(x2-x3)
dx2/dt = x2(x3-x1)
dx3/dt = x3(x1-x3)
My question is why can an orbit starting in the region x1 >= 0, x2 >= 0 , x3 >= 0 never cross one of the lines x_i = 0?
An alternative. For the given system $\displaystyle x'=v(x)$ we have:

$\displaystyle \begin{Bmatrix}v(\alpha,0,0)=(0,0,0)\\ v(0,\beta,0)=(0,0,0) \\ v(0,0,\gamma)=(0,0,-\gamma^2)\end{matrix}$

This means that all points of the $\displaystyle x_1$ and $\displaystyle x_2$ axes are equilibrium points and the $\displaystyle x_3$ axis decomposes into three orbits. You can conclude.

Fernando Revilla