标题： 组合$α$-高斯映射和逻辑映射：向混沌的渐进和突然过渡 显示英文标题

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

摘要： 我们介绍了$\alpha$-高斯-逻辑映射，这是一种通过组合逻辑映射和$\alpha$-高斯映射构建的新非线性动力系统。 具体而言，我们的模型由$x_{t+1} = f_L(x_t)x_t^{-\alpha} - \lfloor f_L(x_t)x_t^{-\alpha} \rfloor $给出，其中$f_L(x_t) = r x_t (1-x_t)$是逻辑映射，$ \lfloor \ldots \rfloor $是取整函数。 我们的研究揭示了仅依赖于两个参数$r$和$\alpha$的丰富现象学。 对于$\alpha < 1$，系统表现出随着参数$r$的增加而逐渐倍周期分岔到混沌的过程，并且在混沌吸引子内穿插着稳定性窗口。 相比之下，对于$1 \leq \alpha < 2$，混沌的出现是突然的，没有经历任何先前的分岔，并且由此产生的混沌吸引子没有稳定性窗口。 对于$\alpha \geq 2$，不存在规则行为。 特殊情况$\alpha = 1$允许进行解析处理，通过使用 Perron-Frobenius 方程得到李雅普诺夫指数的闭式公式以及精确均匀不变密度的条件。 $\alpha = 1$的混沌状态可能有间隙或无间隙。 令人惊讶的是，黄金比例$\Phi$标志着最大间隙消失的阈值。 此外，在突变过渡区域的混沌边缘，不变密度接近于$q$-高斯分布，其中$q=2$，对应于柯西分布。 显示英文摘要

摘要： 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 a gradual period-doubling route 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.