Exploring Parametric Changes in ODEs: Analytic Way

[glebovg]

The defining equations are:

dx/dt = -(y + z)
dy/dt = x + ay
dz/dt = b + z(x - c)

where a = b = 0.2 and 2.6 ≤ c ≤ 4.2.

Is there an analytic way of showing that by changing the parameter c, we can get period-1 orbit, period-2 orbit, period-4 oribt, period-8 orbit, etc. and for c > 4.2 we get a chaotic attractor? I know you can construct a bifurcation diagram or use a projection onto the xy-plane and change c, but is there an analytic approach to this problem?

 

[lippyka]

Unfortunately, there is no analytical approach to this problem. Bifurcation analysis and numerical simulations of the system are the best way to show that changing the parameter c can yield different types of periodic orbits and chaotic attractors.