arXiv subjects

Georg A. Gottwald

Publications and source records attributed to Georg A. Gottwald.

At least 37 records · Page 2Linked to original sources

arXiv2023

Canards in modified equations for Euler discretizations

Canards are a well-studied phenomenon in fast-slow ordinary differential equations implying the delayed loss of stability after the slow passage through a singularity. Recent studies have shown that the corresponding maps stemming from explicit Runge-Kutta discretizations, in particular the forward Euler scheme, exhibit significant distinctions to the continuous-time behavior: for folds, the delay in loss of stability is typically shortened whereas, for transcritical singularities, it is arbitrarily prolonged. We employ the method of modified equations, which correspond with the fixed discretization schemes up to higher order, to understand and quantify these effects directly from a fast-slow ODE, yielding consistent results with the discrete-time behavior and opening a new perspective on the wide range of (de-)stabilization phenomena along canards.

math.DS↗

arXiv2022

SINDy for delay-differential equations: application to model bacterial zinc response

We extend the data-driven method of Sparse Identification of Nonlinear Dynamics (SINDy) developed by Brunton et al, Proc. Natl. Acad. Sci USA 113 (2016) to the case of delay differential equations (DDEs). This is achieved in a bilevel optimization procedure by first applying SINDy for fixed delay and then subsequently optimizing the error of the reconstructed SINDy model over delay times. We test the SINDy-delay method on a noisy short data set from a toy delay differential equation and show excellent agreement. We then apply the method to experimental data of gene expressions in the bacterium {\it Pseudomonas aeruginosa} subject to the influence of zinc. The derived SINDy model suggests that the increase of zinc concentration mainly affects the time delay and not the strengths of the interactions between the different agents controlling the zinc export mechanism.

math.DS↗

arXiv2022

Heterogeneous nucleation in finite size adaptive dynamical networks

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, co-evolving with the nodes' dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite size adaptive network of heterogeneous phase oscillators. Depending on the nature of defects in the internal frequency distribution, we observe either an abrupt single-step transition to full synchronization or a more gradual multi-step transition. This observation has a striking resemblance to heterogeneous nucleation. We develop a mean-field approach to study the interplay between adaptivity and nodal heterogeneity and describe the dynamics of multicluster states and their role in determining the character of the phase transition. Our work provides a theoretical framework for studying the interplay between adaptivity and nodal heterogeneity.

nlin.AO↗

arXiv2022

On financial market correlation structures and diversification benefits across and within equity sectors

We study how to assess the potential benefit of diversifying an equity portfolio by investing within and across equity sectors. We analyse 20 years of US stock price data, which includes the global financial crisis (GFC) and the COVID-19 market crash, as well as periods of financial stability, to determine the `all weather' nature of equity portfolios. We establish that one may use the leading eigenvalue of the cross-correlation matrix of log returns as well as graph-theoretic diagnostics such as modularity to quantify the collective behaviour of the market or a subset of it. We confirm that financial crises are characterised by a high degree of collective behaviour of equities, whereas periods of financial stability exhibit less collective behaviour. We argue that during times of increased collective behaviour, risk reduction via sector-based portfolio diversification is ineffective. Using the degree of collectivity as a proxy for the benefit of diversification, we perform an extensive sampling of equity portfolios to confirm the old financial adage that 30-40 stocks provide sufficient diversification. Using hierarchical clustering, we discover a `best value' equity portfolio for diversification consisting of 36 equities sampled uniformly from 9 sectors. We further show that it is typically more beneficial to diversify across sectors rather than within. Our findings have implications for cost-conscious retail investors seeking broad diversification across equity markets.

q-fin.PM↗

arXiv2021

On spurious detection of linear response and misuse of the fluctuation-dissipation theorem in finite time series

Using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems. A goodness-of-fit statistics is developed for a linear statistical model of the observations, based on results on central limit theorems for deterministic dynamical systems, and used to detect linear response breakdown. We apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series, the magnitude of the perturbation and on the choice of the observable. We find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed. Even for simple systems such as the logistic map, one needs of the order of $10^6$ observations to reliably detect the breakdown, if less observations are available one may be falsely led to conclude that linear response theory is valid. The amount of data required is larger the smaller the applied perturbation. For judiciously chosen observables the necessary amount of data can be drastically reduced, but requires detailed {\em{a priori}} knowledge about the invariant measure which is typically not available for complex dynamical systems. Furthermore we explore the use of the fluctuation-dissipation theorem (FDT) in cases with limited data length or coarse-graining of observations. The FDT, if applied naively to a system without linear response, is shown to be very sensitive to the details of the sampling method, resulting in erroneous predictions of the response.

nlin.CD↗

arXiv2021

Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations

We present a supervised learning method to learn the propagator map of a dynamical system from partial and noisy observations. In our computationally cheap and easy-to-implement framework a neural network consisting of random feature maps is trained sequentially by incoming observations within a data assimilation procedure. By employing Takens' embedding theorem, the network is trained on delay coordinates. We show that the combination of random feature maps and data assimilation, called RAFDA, outperforms standard random feature maps for which the dynamics is learned using batch data.

cs.LG↗

arXiv2021

A collective coordinate framework to study solitary waves in stochastically perturbed Korteweg-de Vries equations

Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to describe the effect of random perturbations on coherent solitary waves. We present a collective coordinate approach to describe the effect on coherent solitary waves in stochastically perturbed KdV equations. The collective coordinate approach allows one to reduce the infinite-dimensional stochastic partial differential equation (SPDE) to a finite-dimensional stochastic differential equation for the amplitude, width and location of the solitary wave. The reduction provides a remarkably good quantitative description of the shape of the solitary waves and its location. Moreover, the collective coordinate framework can be used to estimate the time-scale of validity of stochastically perturbed KdV equations for which they can be used to describe coherent solitary waves. We describe loss of coherence by blow-up as well as by radiation into linear waves. We corroborate our analytical results with numerical simulations of the full SPDE.

nlin.PS↗

arXiv2021

Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation

Data-driven prediction and physics-agnostic machine-learning methods have attracted increased interest in recent years achieving forecast horizons going well beyond those to be expected for chaotic dynamical systems. In a separate strand of research data-assimilation has been successfully used to optimally combine forecast models and their inherent uncertainty with incoming noisy observations. The key idea in our work here is to achieve increased forecast capabilities by judiciously combining machine-learning algorithms and data assimilation. We combine the physics-agnostic data-driven approach of random feature maps as a forecast model within an ensemble Kalman filter data assimilation procedure. The machine-learning model is learned sequentially by incorporating incoming noisy observations. We show that the obtained forecast model has remarkably good forecast skill while being computationally cheap once trained. Going beyond the task of forecasting, we show that our method can be used to generate reliable ensembles for probabilistic forecasting as well as to learn effective model closure in multi-scale systems.

physics.data-an↗

arXiv2020

A model for Dansgaard-Oeschger events and millennial-scale abrupt climate change without external forcing

We propose a conceptual model which generates abrupt climate changes akin to Dansgaard-Oeschger events. In the model these abrupt climate changes are not triggered by external perturbations but rather emerge in a dynamic self-consistent model through complex interactions of the ocean, the atmosphere and an intermittent process. The abrupt climate changes are caused in our model by intermittencies in the sea-ice cover. The ocean is represented by a Stommel two-box model, the atmosphere by a Lorenz-84 model and the sea-ice cover by a deterministic approximation of correlated additive and multiplicative noise (CAM) process. The key dynamical ingredients of the model are given by stochastic limits of deterministic multi-scale systems and recent results in deterministic homogenisation theory. The deterministic model reproduces statistical features of actual ice-core data such as non-Gaussian $α$-stable behaviour. The proposed mechanism for abrupt millenial-scale climate change only relies on the existence of a quantity, which exhibits intermittent dynamics on an intermediate time scale. We consider as a particular mechanism intermittent sea-ice cover where the intermittency is generated by emergent atmospheric noise. However, other mechanisms such as freshwater influxes may also be formulated within the proposed framework.

physics.ao-ph↗

arXiv2020

Stochastically perturbed bred vectors

The breeding method is a computationally cheap procedure to generate initial conditions for ensemble forecasting which project onto relevant synoptic growing modes. Ensembles of bred vectors, however, often lack diversity and align with the leading Lyapunov vector, which severely impacts their statistical reliability. In previous work we developed stochastically perturbed bred vectors (SPBVs) and random draw bred vectors (RDBVs) in the context of multi-scale systems. Here we explore when this method can be extended to systems without scale separation, and examine the performance of the stochastically modified bred vectors in the single scale Lorenz 96 model. In particular, we show that the performance of SPBVs crucially depends on the degree of localisation of the bred vectors. It is found that, contrary to the case of multi-scale systems, localisation is detrimental for applications of SPBVs in systems without scale-separation when initialised from assimilated data. In the case of weakly localised bred vectors, however, ensembles of SPBVs constitute a reliable ensemble with improved ensemble forecasting skills compared to classical bred vectors, while still preserving the low computational cost of the breeding method. RDBVs are shown to have superior forecast skill and form a reliable ensemble in weakly localised situations, but in situations when they are strongly localised they do not constitute a reliable ensemble and are over-dispersive.

physics.ao-ph↗

arXiv2020

Model Reduction for the Kuramoto-Sakaguchi Model: The Importance of Non-entrained Rogue Oscillators

The Kuramoto-Sakaguchi model for coupled phase oscillators with phase-frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the collective dynamics of finite size Kuramoto-Sakaguchi models. We find that the inclusion of the effects of rogue oscillators is essential to obtain an accurate description, in contrast to the original Kuramoto model where we show that their effects can be ignored. We further introduce a more accurate ansatz function to describe the shape of synchronized oscillators. Our results from this extended collective coordinate approach reduce in the thermodynamic limit to the well-known mean-field consistency relations. For finite networks we show that our model reduction describes the collective behavior accurately, reproducing the order parameter, the mean frequency of the synchronized cluster, and the size of the cluster at given coupling strength, as well as the critical coupling strength for partial and for global synchronization.

nlin.AO↗

arXiv2020

Simulation of non-Lipschitz stochastic differential equations driven by $α$-stable noise: a method based on deterministic homogenisation

We devise an explicit method to integrate $α$-stable stochastic differential equations (SDEs) with non-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the Lévy noise, we use a deterministic map which has the desired SDE as its homogenised limit. Moreover, our method naturally overcomes difficulties in expressing the Marcus integral explicitly. We present an example of an SDE with a natural boundary showing that our method respects the boundary whereas Euler-Maruyama discretisation fails to do so. As a by-product we devise an entirely deterministic method to construct $α$-stable laws.

math.DS↗

arXiv2019

Linear response for macroscopic observables in high-dimensional systems

The long-term average response of observables of chaotic systems to dynamical perturbations can often be predicted using linear response theory, but not all chaotic systems possess a linear response. Macroscopic observables of complex dissipative chaotic systems, however, are widely assumed to have a linear response even if the microscopic variables do not, but the mechanism for this is not well-understood. We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional coupled deterministic dynamical systems, where the coupling is via a mean field and the microscopic subsystems may or may not obey linear response theory. We derive stochastic reductions of the dynamics of these observables from statistics of the microscopic system, and provide conditions for linear response theory to hold in finite dimensional systems and in the thermodynamic limit. In particular, we show that for large systems of finite size, linear response is induced via self-generated noise. We present examples in the thermodynamic limit where the macroscopic observable satisfies LRT, although the microscopic subsystems individually violate LRT, as well a converse example where the macroscopic observable does not satisfy LRT despite all microscopic subsystems satisfying LRT when uncoupled. This latter, maybe surprising, example is associated with emergent non-trivial dynamics of the macroscopic observable. We provide numerical evidence for our results on linear response as well as some analytical intuition.

math.DS↗

arXiv2019

Chaos in networks of coupled oscillators with multimodal natural frequency distributions

We explore chaos in the Kuramoto model with multimodal distributions of the natural frequencies of oscillators and provide a comprehensive description under what conditions chaos occurs. For a natural frequency distribution with $M$ peaks it is typical that there is a range of coupling strengths such that oscillators belonging to each peak form a synchronized cluster, but the clusters do not globally synchronize. We use collective coordinates to describe the inter- and intra-cluster dynamics, which reduces the Kuramoto model to $2M-1$ degrees of freedom. We show that under some assumptions, there is a time-scale splitting between the slow intracluster dynamics and fast intercluster dynamics, which reduces the collective coordinate model to an $M-1$ degree of freedom rescaled Kuramoto model. Therefore, four or more clusters are required to yield the three degrees of freedom necessary for chaos. However, the time-scale splitting breaks down if a cluster intermittently desynchronizes. We show that this intermittent desynchronization provides a mechanism for chaos for trimodal natural frequency distributions. In addition, we use collective coordinates to show analytically that chaos cannot occur for bimodal frequency distributions, even if they are asymmetric and if intermittent desynchronization occurs.

nlin.AO↗

arXiv2019

Detecting regime transitions in time series using dynamic mode decomposition

We employ the framework of the Koopman operator and dynamic mode decomposition to devise a computationally cheap and easily implementable method to detect transient dynamics and regime changes in time series. We argue that typically transient dynamics experiences the full state space dimension with subsequent fast relaxation towards the attractor. In equilibrium, on the other hand, the dynamics evolves on a slower time scale on a lower dimensional attractor. The reconstruction error of a dynamic mode decomposition is used to monitor the inability of the time series to resolve the fast relaxation towards the attractor as well as the effective dimension of the dynamics. We illustrate our method by detecting transient dynamics in the Kuramoto-Sivashinsky equation. We further apply our method to atmospheric reanalysis data; our diagnostics detects the transition from a predominantly negative North Atlantic Oscillation (NAO) to a predominantly positive NAO around 1970, as well as the recently found regime change in the Southern Hemisphere atmospheric circulation around 1970.

physics.ao-ph↗

arXiv2019

A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations

We propose a formal framework based on collective coordinates to reduce infinite-dimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finite-dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group. We study SPDEs arising in population dynamics with multiplicative noise and additive symmetry breaking noise. The collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour, which would otherwise only be available through costly computational experiments. We corroborate our analytical results with numerical simulations of the full SPDE.

nlin.PS↗

arXiv2019

Edgeworth expansions for slow-fast systems with finite time scale separation

We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter $\varepsilon$. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in $\varepsilon$ and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation.

nlin.CD↗

arXiv2018

Stochastically perturbed bred vectors in multi-scale systems

The breeding method is a computationally cheap way to generate flow-adapted ensembles to be used in probabilistic forecasts. Its main disadvantage is that the ensemble may lack diversity and collapse to a low-dimensional subspace. To still benefit from the breeding method's simplicity and its low computational cost, approaches are needed to increase the diversity of these bred vector (BV) ensembles. We present here such a method tailored for multi-scale systems. We describe how to judiciously introduce stochastic perturbations to the standard bred vectors leading to stochastically perturbed bred vectors. The increased diversity leads to a better forecast skill as measured by the RMS error, as well as to more reliable ensembles quantified by the error-spread relationship, the continuous ranked probability score and reliability diagrams. Our approach is dynamically informed and in effect generates random draws from the fast equilibrium measure conditioned on the slow variables. We illustrate the advantage of stochastically perturbed bred vectors over standard BVs in numerical simulations of a multi-scale Lorenz 96 model.

physics.ao-ph↗