arXiv subjects

Thomas Breunung

Publications and source records attributed to Thomas Breunung.

4 recordsLinked to original sources

arXiv2024

Learning Global Linear Representations of Nonlinear Dynamics

While linear systems are well-understood, no explicit solution for general nonlinear systems exists. A classical approach to make the understanding of linear system available in the nonlinear setting is to represent a nonlinear system by a linear model. While progress has been made in extending linearization techniques to larger domains and more complex attractor geometries, recent work has highlighted the limitations of these techniques when applied to nonlinear dynamics, such as those with coexisting attractors. In this work, we show nonlinear dynamics with a continuous Koopman spectrum, a limit cycle, and coexisting solutions that can be globally linearized. To this end, we explicitly construct linear systems mimicking these nonlinear behaviors. Subsequently, we approximate transformations between linear and nonlinear systems with deep neural networks. This approach yields finite dimensional linearizations exceeding the phase space dimension of the underlying linear system by one at most.

math.DS↗

arXiv2019

When does a Periodic Response Exist in a Periodically Forced Multi-Degree-of-Freedom Mechanical System?

While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of numerical methods for the detection of periodic responses. We illustrate on examples that each condition of the existence criterion we discuss is essential.

math.DS↗

arXiv2018

Fast Computation of Steady-State Response for Nonlinear Vibrations of High-Degree-of-Freedom Systems

We discuss an integral equation approach that enables fast computation of the response of nonlinear multi-degree-of-freedom mechanical systems under periodic and quasi-periodic external excitation. The kernel of this integral equation is a Green's function that we compute explicitly for general mechanical systems. We derive conditions under which the integral equation can be solved by a simple and fast Picard iteration even for non-smooth mechanical systems. The convergence of this iteration cannot be guaranteed for near-resonant forcing, for which we employ a Newton--Raphson iteration instead, obtaining robust convergence. We further show that this integral-equation approach can be appended with standard continuation schemes to achieve an additional, significant performance increase over common approaches to computing steady-state response.

math.DS↗

arXiv2018

Explicit Backbone Curves from Spectral Submanifolds of Forced-Damped Nonlinear Mechanical Systems

Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here we extend these results to periodically or quasiperiodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two-dimensional) time dependent SSMs. A judicious choice of the parameterization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal form methods.

math.DS↗