Example of robust chaos in a smooth map

Robust chaos is defined by the absence of periodic windows and coexisting a ttractors in some neighborhood of the parameter space. The occurrence of ro bust chaos has been discussed in Phys. Rev. Lett., 78 ( 1997) 4561 nd Phys. Rev. Lett., 80 (1998) 3049. It has been shown that robust chaos can occur in piecewise smooth systems. Also, it has been conjectured that robust chao s cannot occur in smooth systems. However, here we give counterexample to t his conjecture. We present one-dimensional smooth map x(t+1) = f(x(t), alph a) that generates robust chaos in large domain of the parameter space (alph a).