This paper extends the concept of dissipativity to a new class of dynamical systems known as negative imaginary systems. The paper also introduce a new definition for nonlinear negative imaginary systems to extend the existent definition. Two different quadratic dynamic supply rates are introduced to allow for extending the dissipativity concept to cover the negative imaginary systems. One of which is a differential operator and the other is an integral operator. Both supply rates are used in order to formulate the class of negative imaginary systems as dissipative systems. This extension allows for wider class of dynamical systems to be considered in the negative imaginary framework. We also show how the new definition extends the negative imaginary system to analyze a class of higher order evolutionary dynamics. In particular, we show that the second order replicator dynamics satisfy the negative imaginary property and hence we can conclude convergence with certain class of games. Also, a nonlinear negative imaginary lemma based on the above definitions is derived.

本文将耗散性的概念扩展到一类称为负虚构系统的动力学系统。本文还介绍了非线性负虚部系统的新定义，以扩展现有的定义。引入了两种不同的二次动态供给速率，以允许将耗散性概念扩展到涵盖负虚数系统。一个是微分算子，另一个是积分算子。两种供给率都用于将负虚构系统的类别表述为耗散系统。此扩展允许在否定的虚构框架中考虑更广泛的动力学系统类别。我们还展示了新定义如何扩展负虚构系统以分析一类高阶进化动力学。尤其是，我们表明二阶复制器动力学满足负虚构性质，因此我们可以得出与某些类游戏的收敛性。此外，基于上述定义，得出了非线性负虚假引理。