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This work proposes a computational approach that has its roots in the early ideas of local Lyapunov exponents, yet, it offers new perspectives toward analyzing these problems. The method of interest, namely abstract dynamics, is an indirect quantitative measure of the variations of the governing vector fields based on the principles of linear systems. The examples in this work, ranging from simple limit cycles to chaotic attractors, are indicative of the new interpretation that this new perspective can offer. The presented results can be exploited in the structure of algorithms (most prominently machine learning algorithms) that are designed to estimate the complex behavior of nonlinear systems, even chaotic attractors, within their horizon of predictability. 

Abstract A wide range of mechanical systems have gaps, cracks, intermittent contact or other geometrical discontinuities while simultaneously experiencing Coulomb friction. A piecewise linear model with discontinuous force elements is discussed in this paper that has the capability to accurately emulate the behavior of such mechanical assemblies. The mathematical formulation of the model is standardized via a universal differential inclusion and its behavior, in different scenarios, is studied. In addition to the compatibility of the proposed model with numerous industrial systems, the model also bears significant scientific value since it can demonstrate a wide spectrum of motions, ranging from periodic to chaotic. Furthermore, it is demonstrated that this class of models can generate a rare type of motion, called weakly chaotic motion. After their detailed introduction and analysis, an efficient hybrid symbolic-numeric computational method is introduced that can accurately obtain the arbitrary response of this class of nonlinear models. The proposed method is capable of treating high dimensional systems and its proposition omits the need for utilizing model reduction techniques for a wide range of problems. In contrast to the existing literature focused on improving the computational performance when analyzing these systems when there is a periodic response, this method is able to capture transient and nonstationary dynamics and is not restricted to only steady-state periodic responses. 

A general formulation of piecewise linear systems with discontinuous force elements is provided in this paper. It has been demonstrated that this class of nonlinear systems is of great importance due to their ability to accurately model numerous scientific and engineering phenomena. Additionally, it is shown that this class of nonlinear systems can demonstrate a wide spectrum of nonlinear motions and in fact, the phenomenon of weak chaos is observed in a mechanical assembly for the first time. Despite such importance, efficient methods for fast and accurate evaluation of piecewise linear systems’ responses are lacking and the methods of the literature are either incompatible, very slow, very inaccurate, or bear a combination of the aforementioned deficiencies. To overcome this shortcoming, a novel symbolic-numeric method is presented in this paper that is able to obtain the analytical response of piecewise linear systems with discontinuous elements in an efficient manner. Contrary to other efficient methods that are based on stationary steady state dynamics, this method will not experience failure upon the occurrence of complex motion and is able to capture the entirety of the dynamics.