In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.

中文翻译：

---

基于扩展Liu系统的分数阶隐式吸引子

在本文中，提出了一种新的相应的分数阶混沌振荡器。基于Liu系统，提出了具有弱反馈项的数学模型，称为次生流。随着系统参数的改变，隐藏的吸引子将没有平衡点或线平衡。更有趣的是，在无法获得平衡点的情况下，在某些初始条件的引导下，相轨迹可以收敛到最小场，类似于固定点。我们称其为虚拟平衡点。另一方面，当参数值可以产生无限数量的平衡点时，线平衡点是非双曲线的。除此之外，还有一些共存的吸引子，它们可以表现出超混沌，混沌，周期，和虚拟平衡点。分析了系统的动态特性，并研究了参数估计。然后，构建了该系统的电子电路实现，表明了该系统的可行性。最后，针对具有隐藏吸引子的分数系统，基于分数系统的有限时间稳定性理论进行了系统的有限时间同步控制。并通过数值仿真验证了该控制器的有效性。基于分数系统的有限时间稳定性理论进行了系统的有限时间同步控制。并通过数值仿真验证了该控制器的有效性。基于分数系统的有限时间稳定性理论进行了系统的有限时间同步控制。并通过数值仿真验证了该控制器的有效性。