Here is a system of three nonlinear ODEs that has interesting behavior: with $0 < b < 1 < c$, consider
$$ u' = u(1u^2bv^2cw^2), $$ $$ v' = v(1v^2bw^2cu^2), $$ $$ w' = w(1w^2bu^2cv^2). $$
Notice that the three variables are all the same, treated cyclically. The system has a fixed point whenever one variable is equal to $1$ and the others are equal to $0$. Suppose, say, $u\approx 1$ and $v, w \approx 0$. Then $v$ will decay exponentially since $c>1$, whereas $w$ will increase exponentially since $b < 1$. For certain choices of $b$ and $c$ we end up with a kind of alternation, with $u, w, v, u, w, \dots$ taking values close to $1$ in turn.
Here is an illustration with $b=0.55$, $c=1.5$, and initial conditions $u=0.5$, $v=w=0.49$. The plot shows $v(t)$ for $t\in [0,800]$.
b = 0.55; c = 1.5; N = chebop(0,800); N.op = @(t,u,v,w) [ ... diff(u)  u.*(1  u.^2  b*v.^2  c*w.^2) diff(v)  v.*(1  v.^2  b*w.^2  c*u.^2) diff(w)  w.*(1  w.^2  b*u.^2  c*v.^2)]; N.lbc = @(u,v,w) [u0.5; v0.49; w0.49]; [u,v,w] = N\0; plot(v) ylim([0.5 1.5])
On a 3D plot, we can see how the orbit swings from one corner in the $u,v,w$ octant to the next to the next. Most of the time is spent near the corners, where the velocity is low. The orbit is approaching a heteroclinic limit cycle between the three fixed points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$.
clf, plot3(u,v,w,'k'), view(10,10), axis equal, grid on xlabel u, ylabel v, zlabel w
These equations come from [3], where a plot is given on p. 201. (The caption there reports initial conditions $v=w=0.499$, but this is a misprint; we thank Phil Holmes for confirming this by email 23 Feb. 2015.) The discussion in [3] is adapted from the paper [2] by Guckenheimer and Holmes, which in turn gives credit to a related discussion by Busse and Heikes [1] in the context of RayleighBenard convection.
Let's compute a longer orbit, to $t=2000$:
clf, N.domain = [0 2000]; [u,v,w] = N\0; plot(v), ylim([0.5 1.5])
The intervals are getting exponentially longer as the orbit winds in towards the limit cycle. To quantify this, we can compute three vectors of crossing times at which $u$, $v$, and $w$ pass through the value $0.5$ with positive derivative:
tu = roots(u0.5); up = diff(u); tu = tu(up(tu)>0); nu = length(tu); tv = roots(v0.5); vp = diff(v); tv = tv(vp(tv)>0); nv = length(tv); tw = roots(w0.5); wp = diff(w); tw = tw(wp(tw)>0); nw = length(tw);
A semilog plot of the differences between these numbers reveals the exponential growth, with red, green, and blue corresponding to $u$, $v$, and $w$.
clf semilogy(2/3+(2:nu),diff(tu),'.','color',[.9 0 0 ]), hold on semilogy(1/3+(2:nv),diff(tv),'.','color',[0 .7 0 ]) semilogy(0/3+(2:nw),diff(tw),'.','color',[0 0 1]) xlabel('crossing number'), ylabel('time') title ('Crossing times'), grid on, axis([5 28 6 600]) legend('u','v','w','location','southeast')
References

F. H. Busse and K. E. Heikes, Convection in a rotating layer: a simple case of turbulence, Science 208 (1980), 173175.

J. Guckenheimer and P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. Soc. 103 (1988), 189192.

P. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd ed., Cambridge, 2012.