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

We introduce the α-Gauss-Logistic map, a new nonlinear dynamics constructed by composing the logistic and α-Gauss maps. Explicitly, our model is given by xₓ+₁=f₋ (xₓ) xₓ^-α-⌊f₋ (xₓ) xₓ^-α⌋, where f₋ (xₓ) =rxₓ (1-xₓ) is the logistic map and ⌊. . . ⌋ is the integer part function. Our investigation reveals a rich phenomenology depending solely on two parameters, r and α. For α<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≤α<2, the onset of chaos is abrupt, occurring without any prior bifurcations, and the resulting chaotic attractors emerge without stability windows. For α≥2, the regular behavior is absent. The special case of α=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 α=1 can exhibit gaps or be gapless. Surprisingly, the golden ratio Φ 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.