The definition to identify a non linear equation, as mentioned in the book 'M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos and Patterns (Springer - Verlag, Berlin, 2003)' is as follows:
"If each of the terms of a given differential equation, after rationalization, has a total degree either 1 or 0 in the dependent variables and their derivatives, then, it is a linear differential equation. Even if one of the terms has a degree different from 0 or 1 in the dependent variables (and their derivatives), then it is nonlinear. Note that the presence of the independent variable does not affect the linearity/nonlinearity nature."
So, how can we say that the following system equations are non linear? Image of the system of differential equations
According to the definition there are no powers other than 0 or 1 for the dependent variables. Is it because their products are involved they are non linear? How can one realise the meaning of the definition of non linearity in the context of presence of products of dependent variables in equation?