Thread: 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.

Proceed as you would with a differential equation: For example in your first equation isolating we get . Applying this to the other two, adding them up, condensing the constants and using that the logarithm opens products we arrive at or equivalently and since the exponential is always positive, none of the can be zero or change sign because of continuity.

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

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 we have:



This means that all points of the and axes are equilibrium points and the axis decomposes into three orbits. You can conclude.

Fernando Revilla