We study the Duffing equation and its generalizations with polynomial nonlinearities. Recently, we have demonstrated that metamorphoses of the amplitude response curves, computed by asymptotic methods in implicit form as $F\left( Ω,\ A\right) =0$, permit prediction of qualitative changes of dynamics occurring at singular points of the implicit curve $F\left(Ω,\ A\right) =0$. In the present work we determine a global structure of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point. We connect our work with independent research on tangential points on amplitude profiles, associated with jump phenomena, characteristic for the Duffing equation. We also show that our techniques can be applied to solutions of form $Ω_{\pm }=f_{\pm }\left( A\right) $, obtained within other asymptotic approaches.
Bifurcation and chaos in the double well Duffing-van der Pol oscillator: Numerical and analytical studies
研究了驱动双阱Duffing-van der Pol（DVP）振荡器在特定参数选择（$\mid α\mid =β$）下的行为.在系统参数（$f-ω$）域中存在不同的吸引子进行检查和固定阻尼的各种稳定状态的详细说明。报道了通过混沌振荡从准周期运动到周期运动的转变。介入混沌制度进一步显示，具有岛屿的锁相状态和周期窗口（包括倍周期区域），边界危机，所有三个类的混沌，和瞬态混沌。我们还观察到在从拟周期解到周期解的过渡过程中存在间歇突变型的局部-全局分支和蓝天突变型的全局分支。利用微扰周期解，对各种形式的不稳定性进行了研究，可以预测$(f-ω)$平面上的Neimark不稳定性，并最终得到混沌区域的近似预测准则.
Bifurcation Structure of Double Hysteresis and Isolated Branches in a Parametric Duffing Nanoresonator
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；TUM Center for Quantum Engineering (ZQE)（慕尼黑工业大学量子工程中心）；Ain Shams University（艾因夏姆斯大学）
On Coupled Delayed Van der Pol-Duffing oscillators
研究一类时滞微分耦合Duffing-Van der Pol振子方程的动力学性质.用Lindstedt方法导出了同相模解，然后用双变量摄动法得到了控制同相模稳定性的慢流方程。利用级数展开法求解慢流方程，得到了系统发生Hopf分岔的条件，并研究了同相模的稳定性。最后，我们研究了原点的稳定性和分岔。我们对这个系统感兴趣是因为它与耦合激光振荡器有关。
Experimental observation of chaotic and multistable dynamics in a Duffing--Holmes--type analog circuit: antiperiodicity and attractor-coexistence signatures
