On the nonlocal use of the results of local analysis of dynamic systems

Numerical methods of integration of differential equations of motion of dynamic systems, despite their extremely widespread use in engineering and scientific activities, have certain disadvantages. So, having obtained a numerical solution of differential equations for a specific point in the space of parameters of a dynamic system (which, from the point of view of engineering practice, corresponds to a specific design of a technical object), in the general case, it cannot be guaranteed that the results will be valid for other points of this space (for other structures). Such a generalization, non-local use of the results of local analysis, is possible only if the dynamical system under study has certain properties. The article considers the question: under what conditions the results obtained in the construction of the laws of motion of the investigated dynamic system by numerical integration of the differential equations of motion of its mathematical model (that is, by “calculating” one point of the parameter space) can be used “nonlocally”, i.e. can be extended to the entire space of design parameters of the investigated dynamic system? To solve the problem of the possibility of “non-local use of the results of local analysis of dynamical systems”, it is sufficient to bring the equations of motion of the dynamical system under study to a normal form and, further, make sure that in the expanded space of design parameters of the dynamical system under study, the right-hand sides of the above-mentioned normal form satisfy the Lipschitz conditions. In this article, using the example of a dynamic system describing the movement of a vehicle with an adaptive suspension along a non-straight road profile, the issue of the possibility of generalizing the results of local analysis to a non-local area is considered.