In this work, we demonstrate numerically that two-frequency excitation is an effective method to produce chaotification over very large regions of the parameter space for the Duffing oscillator with single- and double-well potentials. It is also shown that chaos is robust in the last case. Robust chaos is characterized by the existence of a single chaotic attractor which is not altered by changes in the system parameters. It is generally required for practical applications of chaos to prevent the effects of fabrication tolerances, external influences, and aging that can destroy chaos. After showing that very large and continuous regions in the parameter space develop a chaotic dynamics under two-frequency excitation for the double-well Duffing oscillator, we demonstrate that chaos is robust over these regions. The proof is based upon the observation of the monotonic changes in the statistical properties of the chaotic attractor when the system parameters are varied and by its uniqueness, demonstrated by changing the initial conditions. The effects of a second frequency in the single-well Duffing oscillator is also investigated. While a quite significant chaotification is observed, chaos is generally not robust in this case.

在这项工作中，我们在数值上证明了双频激励是在具有单阱和双阱势能的Duffing振荡器的参数空间的很大区域上产生混沌的有效方法。还表明，在最后一种情况下，混沌是可靠的。鲁棒性混沌的特征是存在单个混沌吸引子，该吸引子不会因系统参数的变化而改变。通常需要在混沌的实际应用中防止制造公差，外部影响和可能破坏混沌的老化的影响。在表明参数空间中非常大且连续的区域在双频Duffing振荡器的两频激励下发展出混沌动力学之后，我们证明了在这些区域上的混沌是鲁棒的。该证明是基于对系统参数变化时混沌吸引子统计特性的单调变化的观察，并通过改变初始条件来证明其唯一性。还研究了第二频率在单阱Duffing振荡器中的影响。尽管观察到非常明显的混乱，但在这种情况下，混乱通常并不稳健。