Synchronization of Rossler system - the Rossler Attractor

Question

I am studying synchronization of Rossler system given by the following set of two linear ODEs and one nonlinear ODE:

$\dot{x_1} = -x_2 - x_3$

$\dot{x_2} = x_1 + ax_2$

$\dot{x_3} = c + x_3(x_1 - b)$

where $a, b, c > 0$. It is required to mathematically prove that for any given initial condition of $x_3(0)>0$, the state $x_3(t)>0$, $\forall t$.

I think that integrating the third equation would give me the required condition. However the equation being a nonlinear ODE, I do not have enough mathematical background to solve this nonlinear ODE which I am sure is tough in general. How do I approach this? Any suggestions would be helpful.

Answer 1

For the first $T>0$ where $x_3(T)=0$ we must have $x_3'(T)\leq 0$ since we start out being positive, $x_3(0)>0$.

But, inserting $x_3(T)=0$ in your third equation gives $x_3'(T)=c>0$, so that is not possible. Hence $x_3(t)>0$ for all $t>0$.