# Is a cascaded chaotic system is 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?

Thank you.

-