How to determine whether dynamical systems are dissipative, conservative or expanding?

Question

I have the dynamical systems:

1. $dx/dt=y$,
   $dy/dt=-w^2x$
2. $dx/dt=y$,
   $dy/dt=-by-w^2x$
3. $dx/dt=a(y-x)$,
   $dy/dt=x(b-z)-y$,
   $dz/dt=xy-cz$
   with $a,b,c,w\in\mathbb{R}$

How do I determine whether these are dissipative, conservative or expanding? What is the general way to do this?

Answer 1

Given the relation $$ (\dot{x},\dot{y})=F(x,y) $$ you can check the divergence of the vector field $F$: $$ \operatorname{div}F = \nabla\cdot{F} = ( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, ) \cdot (F_1,F_2) = \frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y}. $$

• If $\operatorname{div}F=0$ everywere then the flow is area-preserving,
• if $\operatorname{div}F<0$ everywere then the flow is dissipative,
• if $\operatorname{div}F>0$ everywere then the flow is expanding,
• if $\operatorname{div}F$ changes the sign then the flow has no one of these three properties.