Composing $\alpha$-Gauss and logistic maps: Gradual and sudden transitions to chaos

Abstract

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} - \lfloor f_L(x_t)x_t^{-\alpha} \rfloor $ where $f_L(x_t) = r x_t (1-x_t)$ is the logistic map and $ \lfloor \ldots \rfloor $ 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 $1 \leq \alpha < 2$, the onset of chaos is abrupt, occurring without any prior bifurcations, and the resulting chaotic attractors emerge without stability windows. For $\alpha \geq 2$, 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.