I should state all linear subspaces for which the solutions for t->inf or t->-inf converge against x=0.
I think I have an idea what you mean. Let's take $\displaystyle t\to\infty.$ Now, we don't know the signs of $\displaystyle a$ and $\displaystyle b,$ so we'll have to do different cases depending on which one we have. We may be able to do something clever with the signum function. I'll do the first component for $\displaystyle t\to\infty,$ to show you what's going on. We have
$\displaystyle x(t)=e^{at}(c_{1}\sin(bt)+c_{2}\cos(bt)).$
The trig functions are not going to either blow up or go to zero. Suppose $\displaystyle a>0.$ Then the exponential blows up. Since we can't have that, the only option is if $\displaystyle c_{1}=c_{2}=0.$ Otherwise, the $\displaystyle x$ component blows up. If, on the other hand, $\displaystyle a<0,$ then it doesn't matter what $\displaystyle c_{1}$ and $\displaystyle c_{2}$ are, the $\displaystyle x$ component will go to zero.
Conversely, let's say we are looking at $\displaystyle t\to-\infty.$ The situation will be precisely the opposite of the previous case. $\displaystyle a<0$ means a blow-up unless $\displaystyle c_{1}=c_{2}=0,$ and $\displaystyle a>0$ means the component goes to zero regardless of what $\displaystyle c_{1}$ and $\displaystyle c_{2}$ are.
Does that help?
Hmm. Your solution is a space curve in 3 dimensions, with parameter t, right? Any of those CAS's should be able to plot that, although I'm not sure how to do that in Maple. I could probably figure it out in Mathematica. Captain Black is your best bet for MATLAB. I don't know who is especially good at Maple on this forum. I would definitely consult the help manual for information on parametric plotting.