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?
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.