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Robert Szalai

Publications and source records attributed to Robert Szalai.

16 recordsLinked to original sources
arXiv2025

Data-driven modelling of autonomous and forced dynamical systems

The paper demonstrates that invariant foliations are accurate, data-efficient and practical tools for data-driven modelling of physical systems. Invariant foliations can be fitted to data that either fill the phase space or cluster about an invariant manifold. Invariant foliations can be fitted to a single trajectory or multiple trajectories. Over and underfitting are eliminated by appropriately choosing a function representation and its hyperparameters, such as polynomial orders. The paper extends invariant foliations to forced and parameter dependent systems. It is assumed that forcing is provided by a volume preserving map, and therefore the forcing can be periodic, quasi-periodic or even chaotic. The method utilises full trajectories, hence it is able to predict long-term dynamics accurately. We take into account if a forced system is reducible to an autonomous system about a steady state, similar to how Floquet theory guarantees reducibility for periodically forced systems. In order to find an invariant manifold, multiple invariant foliations are calculated in the neighbourhood of the invariant manifold. Some of the invariant foliations can be linear, while others nonlinear but only defined in a small neighbourhood of an invariant manifold, which reduces the number of parameters to be identified. An invariant manifold is recovered as the zero level set of one or more of the foliations. To interpret the results, the identified mathematical models are transformed to a canonical form and instantaneous frequency and damping information are calculated.

math.DS↗
arXiv2024

Machine-learning invariant foliations in forced systems for reduced order modelling

We identify reduced order models (ROM) of forced systems from data using invariant foliations. The forcing can be external, parametric, periodic or quasi-periodic. The process has four steps: 1. identify an approximate invariant torus and the linear dynamics about the torus; 2. identify a globally defined invariant foliation about the torus; 3. identify a local foliation about an invariant manifold that complements the global foliation 4. extract the invariant manifold as the leaf going through the torus and interpret the result. We combine steps 2 and 3, so that we can track the location of the invariant torus and scale the invariance equations appropriately. We highlight some fundamental limitations of invariant manifolds and foliations when fitting them to data, that require further mathematics to resolve.

math.DS↗
arXiv2024

Non-resonant invariant foliations of quasi-periodically forced systems

We show the existence and uniqueness of invariant foliations about invariant tori in analytic discrete-time dynamical systems. The parametrisation method is used prove the result. Our theory is a foundational block of data-driven model order reduction, that can only be carried out using invariant foliations. The theory is illustrated by two mechanical examples, where instantaneous frequencies and damping ratios are calculated about the invariant tori.

math.DS↗
arXiv2023

Uncountably many cases of Filippov's sewed focus

The sewed focus is one of the singularities of planar piecewise smooth dynamical systems. Defined by Filippov in his book 'Differential Equations with Discontinuous Righthand Sides' (Kluwer, 1988), it consists of two invisible tangencies either side of the switching manifold. In the case of analytic focus-like behaviour, Filippov showed that the approach to the singularity is in infinite time. For the case of non-analytic focus-like behaviour, we show that the approach to the singularity can be in finite time. For the non-analytic sewed centre-focus, we show that there are uncountably many different topological types of local dynamics, including cases with infinitely many stable periodic orbits, and show how to create systems with periodic orbits intersecting any bounded symmetric closed set.

math.DS↗
arXiv2023

Data-driven reduced order models using invariant foliations, manifolds and autoencoders

This paper explores how to identify a reduced order model (ROM) from a physical system. A ROM captures an invariant subset of the observed dynamics. We find that there are four ways a physical system can be related to a mathematical model: invariant foliations, invariant manifolds, autoencoders and equation-free models. Identification of invariant manifolds and equation-free models require closed-loop manipulation of the system. Invariant foliations and autoencoders can also use off-line data. Only invariant foliations and invariant manifolds can identify ROMs, the rest identify complete models. Therefore, the common case of identifying a ROM from existing data can only be achieved using invariant foliations. Finding an invariant foliation requires approximating high-dimensional functions. For function approximation, we use polynomials with compressed tensor coefficients, whose complexity increases linearly with increasing dimensions. An invariant manifold can also be found as the fixed leaf of a foliation. This only requires us to resolve the foliation in a small neighbourhood of the invariant manifold, which greatly simplifies the process. Combining an invariant foliation with the corresponding invariant manifold provides an accurate ROM. We analyse the ROM in case of a focus type equilibrium, typical in mechanical systems. The nonlinear coordinate system defined by the invariant foliation or the invariant manifold distorts instantaneous frequencies and damping ratios, which we correct. Through examples we illustrate the calculation of invariant foliations and manifolds, and at the same time show that Koopman eigenfunctions and autoencoders fail to capture accurate ROMs under the same conditions.

math.DS↗
arXiv2023

Measurements and linearized models for golf ball bounce

A detailed set of experiments are described that capture over a 1000 different instances of the bounce of a golf ball. Video analysis is used to capture velocity and spin immediately prior to and subsequent to each bounce for a wide variety of landing conditions. Data are presented from two different turfs; one artificial and one from a typical tee. Measurement errors and repeatability are analysed. The data are compared to predictions from models of rigid bounce with friction, including Penner's modification to account for elasto-plasticity. Coefficients of restitution and friction, and Penner's effective contact angle are fit from the data. A better fit to the data is found using a non-physical piecewise-affine landing to lift-off relationship, which distinguishes between cases that bounce in pure slip from those that undergo rolling. Nevertheless, even balls that undergo rolling are typically found to lift off slipping, having undergone spin reversal. The findings suggest that further effort needs to be spent on finding simple physics-based models of golf-ball bounce.

physics.class-ph↗
arXiv2022

Analysis of point-contact models of the bounce of a hard spinning ball on a compliant frictional surface

Inspired by the turf-ball interaction in golf, this paper seeks to understand the bounce of a ball that can be modelled as a rigid sphere and the surface as supplying an elasto-plastic contact force in addition to Coulomb friction. A general formulation is proposed that models the finite time interval of bounce from touch-down to lift-off. Key to the analysis is understanding transitions between slip and roll during the bounce. Starting from the rigid-body limit with a an energetic or Poisson coefficient of restitution, it is shown that slip reversal during the contact phase cannot be captured in this case, which result generalises to the case of pure normal compliance. Yet, the introduction of linear tangential stiffness and damping, does enable slip reversal. This result is extended to general weakly nonlinear normal and tangential compliance. An analysis using Filippov theory of piecewise-smooth systems leads to an argument in a natural limit that lift-off while rolling is non-generic and that almost all trajectories that lift off, do so under slip conditions. Moreover, there is a codimension-one surface in the space of incoming velocity and spin which divides balls that lift off with backspin from those that lift off with topspin. The results are compared with recent experimental measurements on golf ball bounce and the theory is shown to capture the main features of the data.

math.DS↗
arXiv2020

Invariant spectral foliations with applications to model order reduction and synthesis

The paper introduces a technique that decomposes the dynamics of a nonlinear system about an equilibrium into low order components, which then can be used to reconstruct the full dynamics. This is a nonlinear analogue of linear modal analysis. The dynamics is decomposed using Invariant Spectral Foliations (ISF), which is defined as the smoothest invariant foliation about an equilibrium and hence unique under general conditions. The conjugate dynamics of an ISF can be used as a reduced order model. An ISF can be fitted to vibration data without carrying out a model identification first. The theory is illustrated on a analytic example and on free-vibration data of a clamped-clamped beam

math.DS↗
arXiv2018

Model reduction of non-densely defined piecewise-smooth systems in Banach spaces

In this paper a model reduction technique is introduced for piecewise-smooth (PWS) vector fields, whose trajectories fall into a Banach space, but the domain of definition of the vector fields is a non-dense subset of the Banach space. The vector fields depend on a parameter that can assume different discrete values in two parts of the phase space and a continuous family of values on the boundary that separates the two parts of the phase space. In essence the parameter parametrizes the possible vector fields on the boundary. The problem is to find one or more values of the parameter so that the solution of the PWS system on the boundary satisfies certain requirements. In this paper we require continuous solutions. Motivated by the properties of applications, we assume that when the parameter is forced to switch between the two discrete values, trajectories become discontinuous. Discontinuous trajectories exist in systems whose domain of definition is non-dense. It is shown that under our assumptions the trajectories of such PWS systems have unique forward-time continuation when the parameter of the system switches. A finite-dimensional reduced order model is constructed, which accounts for the discontinuous trajectories. It is shown that this model retains uniqueness of solutions and other properties of the original PWS system. The model reduction technique is illustrated on a nonlinear bowed string model.

math.DS↗
arXiv2016

Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations

In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude-frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.

math.DS↗
arXiv2015

Nonlinear dynamics of the mammalian inner ear

A simple nonlinear transmission-line model of the cochlea with longitudinal coupling is introduced that can reproduce Basilar membrane response and neural tuning in the chinchilla. It is found that the middle ear has little effect on cochlear resonances, and hence conclude that the theory of coherent reflections is not applicable to the model. The model also provides an explanation of the emergence of spontaneous otoacoustic emissions (SOAEs). It is argued that SOAEs arise from Hopf bifurcations of the transmission-line model and not from localized instabilities. The paper shows that emissions can become chaotic, intermittent and fragile to perturbations.

physics.bio-ph↗
arXiv2014

Non-deterministic dynamics of a mechanical system

A mechanical system is presented exhibiting a non-deterministic singularity, that is, a point in an otherwise deterministic system where forward time trajectories become non-unique. A Coulomb friction force applies linear and angular forces to a wheel mounted on a turntable. In certain configurations the friction force is not uniquely determined. When the dynamics evolves past the singularity and the mechanism slips, the future state becomes uncertain up to a set of possible values. For certain parameters the system repeatedly returns to the singularity, giving recurrent yet unpredictable behaviour that constitutes non-deterministic chaotic dynamics. The robustness of the phenomenon is such that we expect it to persist with more sophisticated friction models, manifesting as extreme sensitivity to initial conditions, and complex global dynamics attributable to a local loss of determinism in the limit of discontinuous friction.

math.DS↗
arXiv2014

Impact mechanics of elastic bodies with point contact

This paper introduces a modeling framework that is suitable to resolve singularities of impact phenomena encountered in applications. The method involves an exact transformation that turns the continuum, often partial differential equation description of the contact problem into a delay differential equation. The new form of the physical model highlights the source of singularities and suggests a simple criterion for regularity. To contrast singular and regular behavior the impacting Euler-Bernoulli and Timoshenko beam models are compared.

math.DS↗
arXiv2013

Modelling elastic structures with strong nonlinearities with application to stick-slip friction

An exact transformation method is introduced that reduces the governing equations of a continuum structure coupled to strong nonlinearities to a low dimensional equation with memory. The method is general and well suited to problems with point discontinuities such as friction and impact at point contact. It is assumed that the structure is composed of two parts: a continuum but linear structure and finitely many discrete but strong nonlinearites acting at various contact points of the elastic structure. The localised nonlinearities include discontinuities, e.g., the Coulomb friction law. Despite the discontinuities in the model, we demonstrate that contact forces are Lipschitz continuous in time at the onset of sticking for certain classes of structures. The general formalism is illustrated for a continuum elastic body coupled to a Coulomb-like friction model.

math.DS↗
arXiv2011

Characteristic matrices for linear periodic delay differential equations

Szalai et al. (SIAM J. on Sci. Comp. 28(4), 2006) gave a general construction for characteristic matrices for systems of linear delay-differential equations with periodic coefficients. First, we show that matrices constructed in this way can have a discrete set of poles in the complex plane, which may possibly obstruct their use when determining the stability of the linear system. Then we modify and generalize the original construction such that the poles get pushed into a small neighborhood of the origin of the complex plane.

math.DS↗