Sorry if I haven’t said my question well. Maybe the following example helps. Consider a chaotic system (flow, not map) which is described by
x1' = f1(x1, x2, ... , xn)
x2' = f2(x1, x2, ... , xn)
…..
xn' = fn(x1, x2, ... , xn)
Is there any chaotic system in which one of the equations be:
xj' =xj* g(x1, x2, ..., xn)
?
If no, why?
There are certainly some trivial examples such as
xj' = xj*g(x1/xj, x2/xj, ..., xn/xj)
and
x1', x2', x3' = (any chaotic system you like but not a function of x4)
x4' = x4*g(x,y,z)
but I assume you mean to exclude such cases.
An example of what you want is a 3-D Lotka-Volterra system such as the following (Eq. 3.29 in my Elegant Chaos book):
x' = x(1 - x - 9y)
y' = -y(1 - 6x - y + 9z)
z' = z(1 - 3x - z)
with initial conditions (0.6, 0.2, 0.01) and Lyapunov exponents (0.0122, 0, -0.3322). This may be the simplest chaotic Lotka-Volterra system. Such systems have been known since the work of Arneodo and collaborators in the 1980s.
