The effect of the asymmetric parameter in the dynamics, both of the forced Duffing van der Pol (DVdP) system and of the forced generalized Bonhoeffer-van der Pol (BVdP) system is investigated, namely the possibility for these systems to execute dynamics chaos even for large values of this asymmetric parameter. We have shown that the dynamics of the associated non dissipative and unforced system can be interpreted by means of an effective energy potential which may exhibit a two-hump or a two-well configuration according to the magnitude of the parameters of the system. In the two-well configuration, a pair of pulses of different amplitudes describing homoclinic orbits useful in the prediction of irregular dynamics for the associated forced system are analytically derived as well as the expression of the single pulse describing the single homoclinic orbit in the case of two-hump configuration and which leads to a kink-antikink pair (heteroclinic orbits) in the absence of the asymmetric parameter. By means of the Melnikov theory, the conditions for the existence of the transverse intersection of stable and unstable orbits or dynamics chaos are also derived. In the particular case for which the system can exhibit a pair of homoclinic orbits, the presence of the asymmetric parameter induces two types of domains among which a more favorable domain for the existence of chaotic behavior. In both cases, the asymmetric parameter reduces the domain range in which the system can execute regular dynamics and consequently increases its domain of a possible dynamics chaos. The accuracy of these analytical results is checked through the bifurcation diagrams and the corresponding Lyapunov exponents resulting from numerical simulations. It appears that the background properties of the unforced system, namely its capacity to exhibit autonomous oscillations and the stability of its equilibrium points which are closely connected to the magnitude of the asymmetric parameter, influence the accuracy of the analytical predictions.

研究了非对称参数在动力学中的影响，包括强制 Duffing van der Pol (DVdP) 系统和强制广义 Bonhoeffer-van der Pol (BVdP) 系统，即这些系统执行动力学混沌的可能性，甚至对于这个不对称参数的大值。我们已经表明，相关的非耗散和非受力系统的动力学可以通过有效能势来解释，根据系统参数的大小，该能势可能表现出双峰或双井配置。在两井配置中，一对不同振幅的脉冲描述同宿轨道可用于预测相关受迫系统的不规则动力学，以及在双峰配置情况下描述单同宿轨道的单个脉冲的表达式，这导致在没有不对称参数的情况下，变成扭结-反扭结对（异斜轨道）。借助梅尔尼科夫理论，还推导出稳定轨道与不稳定轨道横向交叉或动力学混沌的存在条件。在系统可以表现出一对同宿轨道的特殊情况下，不对称参数的存在导致了两种类型的域，其中一个域更适合混沌行为的存在。在这两种情况下，非对称参数减小了系统可以执行常规动力学的域范围，从而增加了其可能的动力学混沌域。这些分析结果的准确性通过分岔图和数值模拟产生的相应李雅普诺夫指数进行检查。似乎非受力系统的背景特性，即其表现出自主振荡的能力及其与非对称参数大小密切相关的平衡点的稳定性，影响了分析预测的准确性。这些分析结果的准确性通过分岔图和数值模拟产生的相应李雅普诺夫指数进行检查。似乎非受力系统的背景特性，即其表现出自主振荡的能力及其与非对称参数大小密切相关的平衡点的稳定性，影响了分析预测的准确性。这些分析结果的准确性通过分岔图和数值模拟产生的相应李雅普诺夫指数进行检查。似乎非受力系统的背景特性，即其表现出自主振荡的能力及其与非对称参数大小密切相关的平衡点的稳定性，影响了分析预测的准确性。