A three-degrees-of-freedom aeroelastic system with concentrated structural nonlinearities is considered. The aerodynamic loads in the model are simulated by using unsteady aerodynamic forces. First, we evaluate the linear stability region of the aerodynamic system employing the Lienard–Chipart criterion so that the stability boundaries of the center of gravity position and the uncoupled pitch natural frequency are obtained. Next, we present the distribution of the first Lyapunov coefficient in the parameter space by using the Poincaré projection method to determine the type of Hopf bifurcation (subcritical or supercritical). We identify three types of hysteresis loops associated with Hopf bifurcations: the first type contains one stable equilibrium point and one stable limit cycle, the second type consists of one stable equilibrium point and two stable limit cycles, and the third type comprises two stable limit cycles. Nonlinear analysis shows that the variations in hysteresis loop types are mainly caused by the nonlinear flutter critical points, namely the saddle–node bifurcation points of limit cycles. Furthermore, we determine a degenerate Hopf bifurcation type known as generalized Hopf bifurcation (Bautin bifurcation) and its corresponding second Lyapunov coefficient, . Subsequently, we conduct nonlinear flutter analyses for positive and negative values of by using bifurcation diagrams and phase portraits. The results show that when the uncoupled pitch natural frequency is used as the control parameter, the occurring generalized Hopf bifurcations are all supercritical (). However, when the center of gravity position is used as the control parameter, the generalized Hopf bifurcations that occur have both supercritical and subcritical () modes.

中文翻译：

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具有集中非线性的三自由度气动弹性系统中的滞回行为和广义 Hopf 分岔


考虑具有集中结构非线性的三自由度气动弹性系统。模型中的气动载荷采用非定常气动力进行模拟。首先，我们采用 Lienard-Chipart 准则评估气动系统的线性稳定区域，从而获得重心位置和非耦合俯仰固有频率的稳定边界。接下来，我们使用庞加莱投影方法呈现参数空间中第一李雅普诺夫系数的分布，以确定 Hopf 分岔的类型（亚临界或超临界）。我们确定了与 Hopf 分岔相关的三种类型的磁滞回线：第一种类型包含一个稳定平衡点和一个稳定极限环，第二种类型包含一个稳定平衡点和两个稳定极限环，第三种类型包含两个稳定极限环。非线性分析表明，磁滞回线类型的变化主要是由非线性颤振临界点，即极限环的鞍节点分岔点引起的。此外，我们确定了一种简并的 Hopf 分岔类型，称为广义 Hopf 分岔（Bautin 分岔）及其相应的第二 Lyapunov 系数 。随后，我们利用分岔图和相图对 的正值和负值进行非线性颤振分析。结果表明，当以非耦合俯仰固有频率为控制参数时，发生的广义Hopf分岔均为超临界()。然而，当以重心位置作为控制参数时，出现的广义Hopf分岔同时具有超临界和亚临界()模式。