In recent years, experimental and theoretical studies have shown that size effect is non-negligible in mechanical micro- and nano-structures. As a result, classical continuum theory cannot model these structures. To accurately predict the behavior of micro- and nano-structures, non-classical continuum theories should be used. In this paper, the effect of size on the chaotic region of resonators is studied by comparing the results of classical and strain gradient theories. A bifurcation diagram and Lyapunov exponent are used for detecting chaos in nano-resonators.

It is shown that the effect of size on the chaotic region in bifurcation diagram varies significantly depending on the bias voltage. Lyapunov exponent is utilized to validate the chaotic region and it is in best agreement with bifurcation diagram. For some DC voltage values, size effect can eliminate chaotic vibrations while in higher DC voltages, it can result in chaos at much lower amplitudes than those predicted by classical theories. Dynamic behaviors of nano-resonators are categorized based on their bias voltages and their chaotic behaviors are analyzed based on classical and non-classical theories in detail. The obtained numerical results are compared with Melnikov and Maximum Velocity methods and it is shown that all methods predict the lower critical actuation amplitude for strain gradient theory.

近年来，实验和理论研究表明，尺寸效应在机械微结构和纳米结构中不可忽略。结果，经典连续论无法对这些结构进行建模。为了准确地预测微观和纳米结构的行为，应使用非经典的连续体理论。通过比较经典和应变梯度理论的结果，研究了尺寸对谐振器混沌区域的影响。分叉图和李雅普诺夫指数用于检测纳米谐振器中的混沌。

结果表明，在分叉图中，大小对混沌区域的影响随偏压的变化而显着变化。利用李雅普诺夫指数来验证混沌区域，它与分叉图最吻合。对于某些直流电压值，尺寸效应可以消除混沌振动，而在较高的直流电压下，其幅度可能比经典理论所预测的低得多。纳米谐振器的动态行为根据其偏置电压进行分类，并基于经典和非经典理论对它们的混沌行为进行详细分析。将所获得的数值结果与梅尔尼科夫方法和最大速度方法进行了比较，结果表明，所有方法都可以预测应变梯度理论的较低临界致动幅度。