How Can I Determine the Bounding Dimensions of a Nonlinear Dynamical System?

[coolnessitself]

I'm working on a visualizer of sorts for a system:
$$x_{n+1} = sin(a y_n) - cos(b x_n)$$
$$y_{n+1} = sin(c x_n) - cos(d y_n)$$
with $$a,b,c,d \in [-2.5, 2.5]$$
So for whatever initial $$(x_0,y_0)$$ I give the system, I know the next iteration will have both x and y between -2 and 2, and that will be true for all n>0.
However, for certain values of a,b,c,d, you could say that all $$x_{n>0}$$ and $$y_{n>0}$$ will be within some other, possibly smaller, area. How can I find these dimensions given a,b,c,d?

(I'll use this to scale the area on which the plot is drawn, so for those values of a,b,c,d which result in a small area, the plot will fill the entire space)

 

[mathador]

Hi,

Try to express where the four lines x=0, x=2, y=0, y=2 (l1,..,l4) and their four intersections (p1,..,p4) get mapped under your map (M). The four new points (M(p1),...,M(p4)) determine a trapezoid, say T1. You can calculate the slopes of the lines connecting M(p1),...,M(p4).
The images of the lines l1,..,l4 will be curves, but you could try to find points on these curves with slopes equal to those of the sides of the trapezoid. Simply move the sides of T1 out to these new points to create a larger trapezoid. This object might serve as a "bounding box". Since this map is nonlinear, this might not work, though...

You might also want to read the awesome book:
Chaos in discrete dynamical systems
By Ralph Abraham, Laura Gardini, C. Mira

Mathador