Robert Szalai

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.

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.

We classify Filippov’s type 3 singular points of planar bimodal piecewise smooth systems. These singular points consist of fold or cusp tangencies of the vector fields to both sides of a switching surface. For isolated analytic type 3 singular points there are 25 topological classes, up to time reversal. For isolated general type 3 singular points there are 40 topological classes, up to time reversal.

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. MSC codes 34A34 34C45 37M21 37C86

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.

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.

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, and 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.

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 a viscoelastic 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 an energetic or Poisson coefficient of restitution, it is shown that slip reversal during the contact phase cannot be captured in this case, which generalizes 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 the 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.

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.

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 Foliation (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.