Is a cascaded chaotic system still chaotic? I am curious whether a new system which cascades two individual chaotic systems is always chaotic. 
My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that 
$${\rm range}( S_1 ) = {\rm domain}(S_2)$$ 
and 
$${\rm range}( S_2 ) = {\rm domain}(S_1)$$ 
then their composite system $S' = S_1\circ S_2$ is also chaotic. I guess should use proof by contradiction. However, I donot know whether this claim is indeed true.
For example, a logistic map $L(\cdot)$ is defined as 
$$x_{i+1}=L(x)=rx_i(1-x_i)$$
and a tent map $T(\cdot)$ is defined as
$$x_{i+1}=T(x)=\left\{\begin{array}{lr}{ux_i\over c}&, x_i<c\\{u(1-x_i)\over (1-c)}&, x_i\geq c\end{array}\right.$$
Assume parameters of both maps are carefully chosen to ensure the map chaotic behaviors (both range and domain are $(0,1)$). Is the following system also chaotic?
$$G(c)=L\circ T(x) = L(\,T(x)\,)$$
Can anyone provide any idea here?
 A: Cascading "same" chaotic systems is a well known method as you can see in paper "A chaotic direct-sequence spreadspectrum communication system". And you can see examples of cascading "different" chaotic systems here and there like in "A Cryptosystem Based on Multiple Chaotic Maps" (You can google it). 
Neither of them rely on a mathematical proof. They are just engineering methods assuming that cascading chaotic maps will still be chaotic. Obviously, this case can be proven for same chaotic maps but cascading different chaotic maps is not that easy. Even though, their defined intervals are same, chaotic value of one map may correspond to a theoretical fixed point of the other. Even though this situation may not be so important in an engineering application. I think, it complicates a "general" rigorous mathematical proof.     
A: At least empirically, your conjecture does not hold:
The logistic maps for $r_1=3.7352$ and $r_2=3.8921$ are both chaotic ($λ=0.37$ and $λ=0.49$, respectively), while the composite of both maps is not ($λ=-0.32$).
I found these values with the following Python script appended below.
It takes random values from $[3.6,4.0]$ for $r_1$ and $r_2$ and uses Lyapunov exponents to check whether the respective logistic maps are chaotic and the composite isn’t (with rather conservative thresholds).
The rate at which complying values are found is not very low, which suggests that this is not a coincidence.
On the other hand, most of the maps composed from two chaotic maps in this case are again chaotic.
Appendix: Python Script
import numpy as np
from sympy import Lambda, lambdify
from sympy.abc import x

def logistic(r):
    return Lambda(x,r*x*(1-x))

def lyapunov(f,nsteps=10000,pre=1000):
    Map = lambdify(x,f(x))
    Der = lambdify(x,f(x).diff(x))

    y = np.random.uniform(0.01,0.99)
    for _ in range(pre):
        y = f(y)

    diffs = np.empty(nsteps)
    for i in range(nsteps):
        y = f(y)
        diffs[i] = Der(y)

    return np.mean(np.log(np.abs(diffs)))

while True:
    r_1,r_2 = np.round( np.random.uniform(3.6,4.0,2), 4 )
    map_1 = logistic(r_1)
    map_2 = logistic(r_2)
    composite = Lambda(x,map_1(map_2(x)))

    for n in [100,1000,10000]:
        if (
                   lyapunov(composite,n,n) > -0.2
                or lyapunov(map_1,n,n) < 0.2
                or lyapunov(map_2,n,n) < 0.2
            ):
            break
    else:
        print(r_1,r_2,lyapunov(map_1),lyapunov(map_2),lyapunov(composite))
```

