One of the most important issues in controller design and analysis of a nonlinear system is its degree of nonlinearity. Helbig et al. proposed a method in which the system, along with a linear reference model (which is the sum of a couple of first order transfer functions), is stimulated with a set of input patterns. The outputs of these systems are then compared and the nonlinearity measure of the system is obtained as the normalized difference between the outputs of these two systems. In this paper, the linear reference model is replaced with two simpler linear ones. The proposed method has been used for assessment of the nonlinearity measure of various nonlinear systems that are used as standard benchmarks by the nonlinear process control research community. The results show that, despite the use of this simpler model with less parameters, the calculated nonlinearity measure is almost similar to the one obtained by Helbig's method. Furthermore, the nonlinearity measure obtained by the proposed method can be calculated very much faster.

The purpose of this paper is to analyze in detail a special nonlinear partial differential equation (nPDE) of the second order which is important in physical, chemical and technical applications. The present nPDE describes nonlinear diffusion and is of interest in several parts of physics, chemistry and engineering problems alike. Since nature is not linear intrinsically the nonlinear case is therefore the general. We determine the classical Lie point symmetries including algebraic properties whereas similarity solutions are given as well as nonlinear transformations could derived. In addition, we discuss the nonclassical case which seems to be not solvable. Moreover we show how one can deduce approximate symmetries modeling the nonlinear part and we deduce new generalized symmetries of lower symmetry. The analysis allows one to deduce wider classes of solutions either of practical and theoretical usage in different domains of science and engineering.

Because of difficulty in inspection and retrofit of foundation in comparison with other elements, the common design philosophy is to avoid any nonlinear deformation in the foundation. This paper shows that by employing controlled foundation nonlinearity, in predetermined sections with arrangements for inspection and retrofit, it is possible to reduce seismic demand on superstructure. Localizing nonlinear deformation to pre-specified zones in the foundation, it is possible to avoid wide spread nonlinear deformation across various members in the superstructure in the case of strong ground motions. To evaluate the efficiency of the proposed model, the response of steel braced frames is examined on rigid foundation, rocking elastic foundation and finally on rocking foundation with controlled nonlinear deformation. Results show that while rocking could be used to protect the superstructure elements from possible overloading during large earthquakes for low-rise structures; it has no remarkable effect on the response of high-rise structures. However, the proposed model with nonlinearity in the foundation could be used in both cases (low-rise and high-rise structures) to effectively control the response of structure.