Composing α-Gauss and logistic maps: Gradual and sudden transitions to chaos

We introduce the alpha-Gauss-Logistic map, a new nonlinear dynamics constructed by composing the logistic and alpha-Gauss maps. Explicitly, our model is given by x(t+1) = f(L)(x(t))x(t )(-alpha) - [f(L)(x(t))x(t)(-alpha)], where f(L)(x(t))=rx(t)(1-x(t)) is the logistic map and [...] is the integer part function. Our investigation reveals a rich phenomenology depending solely on two parameters, r and alpha. For alpha < 1, the system exhibits multiple period-doubling cascades to chaos as the parameter r is increased, interspersed with stability windows within the chaotic attractor. In contrast, for 1 <= alpha < 2, the onset of chaos is abrupt, occurring without any prior bifurcations, and the resulting chaotic attractors emerge without stability windows. For alpha >= 2, the regular behavior is absent. The special case of alpha = 1 allows an analytical treatment, yielding a closed-form formula for the Lyapunov exponent and conditions for an exact uniform invariant density, using the Perron-Frobenius equation. Chaotic regimes for alpha = 1 can exhibit gaps or be gapless. Surprisingly, the golden ratio Phi marks the threshold for the disappearance of the largest gap in the regime diagram. Additionally, at the edge of chaos in the abrupt transition regime, the invariant density approaches a q-Gaussian with q = 2, which corresponds to a Cauchy distribution.