We consider dynamics of an oscillatory mechanical system with 1.5 degrees of freedom. The system consists of two linear subsystems, a leader and a follower, which are coupled via a kinetic friction interface, and additionally driven by a proportional control term. The kinetic friction model accounts for the Coulomb friction and the Stribeck effect (as negative damping) at the passage from zero to nonzero relative velocity. The equations of motion are equivalent to a Lur’e system with a single sign nonlinearity. We show that at a critical value of the parameter the zero steady state blows up into an attracting continuum (segment) of equilibrium states. Moreover, two antipodal small- amplitude periodic orbits bifurcate from the end points of the segment of equilibrium states, creating stick-slip vibrations. We consider other possible attractors consisting of stick-slip solutions of the Lur’e system such as a Z₂-symmetric stick-slip periodic orbit and an attractor composed of a continuum of heteroclinic loops representing stick-slip motions towards a steady state.

We study the simultaneous homogenization and dimension reduction of an energy functional with linear growth defined on the space of manifold valued Sobolev functions. The study is carried out by Γ-convergence, providing an integral representation result in the space of manifold constrained functions with bounded variation.

We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE is then solved using standard integration in time schemes, and the high-dimensional solution is reconstructed from the low- dimensional one. We investigate the architecture of neural networks for constructing effective autoencoders that hold necessary properties to reconstruct the input manifold with exact representation capabilities. We study the convergence of the reduced-order model to the high-fidelity one. Numerical experiments show the robustness and accuracy of our approach in different scenarios, highlighting its effectiveness in highly complex and nonlinear settings without sacrificing accuracy. Moreover, we examine how the reduction influences the stability properties of the reconstructed high-dimensional solution.