November 9, 2004 (modified February 26, 2005)

Chaos has been observed in the spatiotemporal Lotka-Volterra model described by

dx_{i}/dt
= x_{i}(1 - ax_{i}_{-2} - x_{i} - bx_{i}_{+1})

for a = b = 1 and i = 1 to N with N > 65 and periodic boundary
conditions. The system is described in detail in another
publication. The system exhibits a quasiperiodic route to chaos
with a first Hopf bifurcation that occurs at a value of approximately a = b = 0.889. Below that value, the
orbit approaches the equilibrium at x

If we set *s* = *a* = *b*
then s is a
parameter that multiplies the off-diagonal elements of the interaction
matrix. This
animation shows how the eigenvalues grow
as *s* increases with N
= 100. Using this parameter we can examine the
routes to chaos by changing *s* or by changing N. For each value of N the Hopf bifurcation occurs at a
different
value of s. The exact
position oscillates but is damped as N
increases:

The eigenvalues for a given s value fall on a curve that does not change with N. The number of eigenvalues, however, does change, and it is a rotation of these eigenvalues to accommodate the new ones that produces the change in the position of the Hopf bifurcation. As N increases the curve becomes filled, so that the oscillations shrink. In the limit of large N the s value at the Hopf bifurcation is asymptotic to ~0.8888916. This animation shows the rotation of the eigenvalues as new ones are added (emanating from the fixed point).