## Publications

74 results found

Castro MM, Chemnitz D, Chu H,
et al., 2024, The conditioned Lyapunov spectrum for random dynamical systems, *L'Institut Henri Poincare, Annales B: Probabilites et Statistiques*, Pages: 1-36, ISSN: 0246-0203

We establish the existence of a full spectrum of Lyapunov exponents for memoryless random dynamical systems with absorption. To this end, we crucially embed the process conditioned to never being absorbed, the Q-process, into the framework of random dynamical systems, allowing us to study multiplicative ergodic properties. We show that the finite-time Lyapunov exponents converge in conditioned probability and apply our results to iterated function systems and stochastic differential equations.

Bassols Cornudella B, Lamb JSW, 2023, Noise-induced chaos: a conditioned random dynamics perspective, *Chaos: an interdisciplinary journal of nonlinear science*, Vol: 33, ISSN: 1054-1500

We consider transitions to chaos in random dynamical systems induced by an increase in noise amplitude. We show how the emergence of chaos (indicated by a positive Lyapunov exponent) in a logistic map with bounded additive noise can be analyzed in the framework of conditioned random dynamics through expected escape times and conditioned Lyapunov exponents for a compartmental model representing the competition between contracting and expanding behavior. In contrast to the existing literature, our approach does not rely on small noise assumptions, nor does it refer to deterministic paradigms. We find that the noise-induced transition to chaos is caused by a rapid decay of the expected escape time from the contracting compartment, while all other order parameters remain approximately constant.

CASTRO MM, GOVERSE VPH, LAMB JSW,
et al., 2023, On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape, *Ergodic Theory and Dynamical Systems*, Pages: 1-38, ISSN: 0143-3857

In this paper, we consider absorbing Markov chains Xn admitting a quasi-stationary measure μ on M where the transition kernel P admits an eigenfunction 0≤η∈L1(M,μ). We find conditions on the transition densities of P with respect to μ which ensure that η(x)μ(dx) is a quasi-ergodic measure for Xn and that the Yaglom limit converges to the quasi-stationary measure μ-almost surely. We apply this result to the random logistic map Xn+1=ωnXn(1−Xn) absorbed at R∖[0,1], where ωn is an independent and identically distributed sequence of random variables uniformly distributed in [a,b], for 1≤a<4 and b>4.

Gabel A, Klein V, Valperga R, et al., 2023, Learning lie group symmetry transformations with neural networks, 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) at the 40th International Conference on Machine Learning, Publisher: ML Research Press, Pages: 50-59

The problem of detecting and quantifying the presence of symmetries in datasets is useful for model selection, generative modeling, and data analysis, amongst others. While existing methods for hard-coding transformations in neural networks require prior knowledge of the symmetries of the task at hand, this work focuses on discovering and characterizing unknown symmetries present in the dataset, namely, Lie group symmetry transformations beyond the traditional ones usually considered in the field (rotation, scaling, and translation). Specifically, we consider a scenario in which a dataset has been transformed by a one-parameter subgroup of transformations with different parameter values for each data point. Our goal is to characterize the transformation group and the distribution of the parameter values. The results showcase the effectiveness of the approach in both these settings.

Bakrani S, Lamb JSW, Turaev D, 2022, Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in R4 with Z2-symmetry and integral of motion, *Journal of Differential Equations*, Vol: 327, Pages: 1-63, ISSN: 0022-0396

We consider a -equivariant flow in with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

Valperga R, Webster K, Klein V, et al., 2022, Learning reversible symplectic dynamics, 4th Annual Learning for Dynamics and Control Conference, Publisher: PLMR

Time-reversal symmetry arises naturally as a structural property in manydynamical systems of interest. While the importance of hard-wiring symmetry isincreasingly recognized in machine learning, to date this has eludedtime-reversibility. In this paper we propose a new neural network architecturefor learning time-reversible dynamical systems from data. We focus inparticular on an adaptation to symplectic systems, because of their importancein physics-informed learning.

Eldering J, Lamb JSW, Pereira T,
et al., 2021, Chimera states through invariant manifold theory, *Nonlinearity*, Vol: 34, Pages: 5344-5374, ISSN: 0951-7715

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network of two symmetrically linked star subnetworks of identical oscillators with shear and Kuramoto–Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-star coupling strength is of order ɛ, the chimera states persist on time scales at least of order 1/ɛ in general, and on time-scales at least of order 1/ɛ2 if the intra-star coupling is of Kuramoto–Sakaguchi type. If the intra-star coupling configuration is sparse, the chimeras are asymptotically stable. The analysis relies on a combination of dimensional reduction using a Möbius symmetry group and techniques from averaging theory and normal hyperbolicity.

Lamb J, Lima M, Martins R,
et al., 2020, On the Hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in R^4, *Journal of Differential Equations*, Vol: 269, Pages: 11366-11395, ISSN: 0022-0396

This paper addresses the question whether normal forms of smooth reversible vector fields in R4 at an elliptic equilibrium possess a formal Hamiltonian structure. In the non-resonant case we establish a formal conjugacy between re-versible and Hamiltonian normal forms. In the case of non-semi-simple 1 : 1 resonance and p:q resonance with p+q >2 we establish a weaker form of equivalence, namely that of a formal orbital equivalence to a Hamiltonian normal formthat involves an additional time-reparametrization of orbits. Moreover, in case p+q >3 we show that no formal conjugacy to a Hamiltonian normal form exists.

Faranda D, Castillo IP, Hulme O,
et al., 2020, Asymptotic estimates of SARS-CoV-2 infection counts and their sensitivity to stochastic perturbation, *Chaos: An Interdisciplinary Journal of Nonlinear Science*, Vol: 30, Pages: 051107-1-051107-10, ISSN: 1054-1500

Despite the importance of having robust estimates of the time-asymptotic total number of infections, early estimates of COVID-19 show enormous fluctuations. Using COVID-19 data from different countries, we show that predictions are extremely sensitive to the reporting protocol and crucially depend on the last available data point before the maximum number of daily infections is reached. We propose a physical explanation for this sensitivity, using a susceptible–exposed–infected–recovered model, where the parameters are stochastically perturbed to simulate the difficulty in detecting patients, different confinement measures taken by different countries, as well as changes in the virus characteristics. Our results suggest that there are physical and statistical reasons to assign low confidence to statistical and dynamical fits, despite their apparently good statistical scores. These considerations are general and can be applied to other epidemics.COVID-19 is currently affecting over 180 countries worldwide and poses serious threats to public health as well as economic and social stability of many countries. Modeling and extrapolating in near real-time the evolution of COVID-19 epidemics is a scientific challenge, which requires a deep understanding of the non-linearities undermining the dynamics of the epidemics. Here, we show that real-time predictions of COVID-19 infections are extremely sensitive to errors in data collection and crucially depend on the last available data point. We test these ideas in both statistical (logistic) and dynamical (susceptible–exposed–infected–recovered) models that are currently used to forecast the evolution of the COVID-19 epidemic. Our goal is to show how uncertainties arising from both poor data quality and inadequate estimations of model parameters (incubation, infection, and recovery rates) propagate to long-term extrapolations of infection counts. We provide guidelines for reporting those unce

Engel M, Lamb J, Rasmussen M, 2019, Conditioned Lyapunov exponents for random dynamical systems, *Transactions of the American Mathematical Society*, Vol: 372, Pages: 6343-6370, ISSN: 0002-9947

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to tra-jectories that stay within a bounded domain for asymptotically long times. This is motivated by thedesire to characterize local dynamical properties in the presence of unbounded noise (when almost alltrajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context.The theory of conditioned Lyapunov exponents of stochastic differential equations builds on thestochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic dis-tributions. We show that conditioned Lyapunov exponents describe the asymptotic stability behaviourof trajectories that remain within a bounded domain and – in particular – that negative conditionedLyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum isintroduced and its main characteristics are established.

Chekroun MD, Lamb JSW, Pangerl CJ, et al., 2019, A Girsanov approach to slow parameterizing manifolds in the presence of noise

We consider a three-dimensional slow-fast system arising in fluid dynamicswith quadratic nonlinearity and additive noise. The associated deterministicsystem of this stochastic differential equation (SDE) exhibits a periodic orbitand a slow manifold. We show that in presence of noise, the deterministic slowmanifold can be viewed as an approximate parameterization of the fast variableof the SDE in terms of the slow variables, for certain parameter regimes. Weexploit this fact to obtain a two dimensional reduced model from the originalstochastic system, which results into a Hopf normal form with additive noise.Both, the original as well as the reduced system admit ergodic invariantmeasures describing their respective long-time behaviour. It is then shown thatfor a suitable Wasserstein metric on a subset of the space of probabilitymeasures on the phase space, the discrepancy between the marginals along theradial component of each invariant measure is controlled by a parameterizationdefect measuring the quality of the parameterization. An important technical tool to arrive at this result is Girsanov's theoremthat allows us to derive such error estimates in presence of an oscillatoryinstability. This approach is finally extended to parameter regimes for whichthe variable to parameterize is no longer evolving on a faster timescale thanthat of the resolved variables. There also, error estimates involving theWasserstein metric are derived but this time for reduced systems obtained fromstochastic parameterizing manifolds involving path-dependent coefficients tocope with such challenging regimes.

Engel M, Lamb J, Rasmussen M, 2019, Bifurcation analysis of a stochastically driven limit cycle, *Communications in Mathematical Physics*, Vol: 365, Pages: 935-942, ISSN: 0010-3616

We establish the existence of a bifurcation from an attractive random equilibrium to shear-inducedchaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent.This relates to an open problem posed by Kevin Lin and Lai-Sang Young in [11, 16], extending resultsby Qiudong Wang and Lai-Sang Young [14] on periodically kicked limit cycles to the stochastic context.

Doan TS, Engel M, Lamb J,
et al., 2018, Hopf bifurcation with additive noise, *Nonlinearity*, Vol: 31, ISSN: 0951-7715

We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent.We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (amplitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).

Callaway M, Doan TS, Lamb JSW,
et al., 2017, The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise, *Annales de l’Institut Henri Poincaré Probabilités et Statistiques*, Vol: 53, Pages: 1548-1574, ISSN: 0246-0203

We develop the dichotomy spectrum for random dynamical systems and demonstrate its use in the characterization of pitchfork bifurcations for random dynamical systems with additive noise.Crauel and Flandoli (J. Dynam. Differential Equations10 (1998) 259–274) had shown earlier that adding noise to a system with a deterministic pitchfork bifurcation yields a unique attracting random equilibrium with negative Lyapunov exponent throughout, thus “destroying” this bifurcation. Indeed, we show that in this example the dynamics before and after the underlying deterministic bifurcation point are topologically equivalent.However, in apparent paradox to (J. Dynam. Differential Equations10 (1998) 259–274), we show that there is after all a qualitative change in the random dynamics at the underlying deterministic bifurcation point, characterized by the transition from a hyperbolic to a non-hyperbolic dichotomy spectrum. This breakdown manifests itself also in the loss of uniform attractivity, a loss of experimental observability of the Lyapunov exponent, and a loss of equivalence under uniformly continuous topological conjugacies.Nous développons le spectre de dichotomie pour les systèmes dynamiques aléatoires et nous démontrons son utilité pour la caractérisation des bifurcations de fourches dans des systèmes dynamiques aléatoires avec du bruit additif.Crauel et Flandoli (J. Dynam. Differential Equations10 (1998) 259–274) ont précédemment montré que l’ajout de bruit additif à un système comprenant une bifurcation de fourche déterministe produit un unique équilibre aléatoire attractif avec un exposant de Lyapunov négatif partout, « détruisant » ainsi cette bifurcation. En effet, nous montrons dans cet exemple que la dynamique avant et après le point de bifurcation déterministe sous-jacent sont t

Eroglu D, Lamb JSW, Pereira T, 2017, Synchronisation of chaos and its applications, *Contemporary Physics*, Vol: 58, Pages: 207-243, ISSN: 0010-7514

Dynamical networks are important models for the behaviour of complex systems, modelling physical, biological and societal systems, including the brain, food webs, epidemic disease in populations, power grids and many other. Such dynamical networks can exhibit behaviour in which deterministic chaos, exhibiting unpredictability and disorder, coexists with synchronisation, a classical paradigm of order. We survey the main theory behind complete, generalised and phase synchronisation phenomena in simple as well as complex networks and discuss applications to secure communications, parameter estimation and the anticipation of chaos.

Cherubini AM, Lamb JSW, Rasmussen M,
et al., 2017, A random dynamical systems perspective on stochastic resonance, *Nonlinearity*, Vol: 30, Pages: 2835-2853, ISSN: 1361-6544

We study stochastic resonance in an over-damped approximation of the stochastic Duffing oscillator from a random dynamical systems point of view. We analyse this problem in the general framework of random dynamical systems with a nonautonomous forcing. We prove the existence of a unique global attracting random periodic orbit and a stationary periodic measure. We use the stationary periodic measure to define an indicator for the stochastic resonance.

knobloch J, Lamb JSW, webster KN, 2017, Shift dynamics near non-elementary T-points with real eigenvalues, *Journal of Difference Equations and Applications*, Vol: 24, Pages: 609-654, ISSN: 1563-5120

We consider non-elementary T-points in reversible systems in R2n+1. We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.

Giles W, Lamb J, Turaev D, 2016, On homoclinic orbits to center manifolds of elliptic-hyperbolic equilibria in Hamiltonian systems, *Nonlinearity*, Vol: 29, ISSN: 1361-6544

We consider a Hamiltonian system which has an elliptic-hyperbolic equilibriumwith a homoclinic loop. We identify the set of orbits which are homoclinic tothe center manifold of the equilibrium via a Lyapunov- Schmidt reductionprocedure. This leads to the study of a singularity which inherits certainstructure from the Hamiltonian nature of the system. Under non-degeneracyassumptions, we classify the possible Morse indices of this singularity,permitting a local description of the set of homoclinic orbits. We alsoconsider the case of time-reversible Hamiltonian systems.

Lamb JSW, Rasmussen M, Rodrigues CS, 2015, Topological bifurcations of minimal invariant sets for set-valued dynamical systems, *Proceedings of the American Mathematical Society*, Vol: 143, Pages: 3927-3937, ISSN: 1088-6826

We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are naturally satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous appearances. We also characterise these transitions by properties of Morse-like decompositions.

Hamzi B, Lamb JSW, Lewis D, 2015, A Characterization of Normal Forms for Control Systems, *JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS*, Vol: 21, Pages: 273-284, ISSN: 1079-2724

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- Citations: 8

Knobloch J, Lamb JSW, Webster KN, 2014, Using Lin's method to solve Bykov's problems, *JOURNAL OF DIFFERENTIAL EQUATIONS*, Vol: 257, Pages: 2984-3047, ISSN: 0022-0396

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- Citations: 16

Gonchenko SV, Lamb JSW, Rios I,
et al., 2014, Attractors and repellers near generic elliptic points of reversible maps, *DOKLADY MATHEMATICS*, Vol: 89, Pages: 65-67, ISSN: 1064-5624

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- Citations: 12

Dingle K, Lamb JSW, Lazaro-Cami J-A, 2013, Knudsen's law and random billiards in irrational triangles, *NONLINEARITY*, Vol: 26, Pages: 369-388, ISSN: 0951-7715

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- Citations: 5

Makarenkov O, Lamb JSW, 2012, Dynamics and bifurcations of nonsmooth systems: A survey, *PHYSICA D-NONLINEAR PHENOMENA*, Vol: 241, Pages: 1826-1844, ISSN: 0167-2789

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- Citations: 207

Makarenkov O, Lamb JSW, 2012, Preface: Dynamics and Bifurcations of Nonsmooth Systems, *PHYSICA D-NONLINEAR PHENOMENA*, Vol: 241, Pages: 1825-1825, ISSN: 0167-2789

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- Citations: 5

Homburg AJ, Jukes AC, Knobloch J,
et al., 2011, BIFURCATION FROM CODIMENSION ONE RELATIVE HOMOCLINIC CYCLES, *TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY*, Vol: 363, Pages: 5663-5701, ISSN: 0002-9947

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- Citations: 5

Buono P-L, Helmer M, Lamb JSW, 2009, On the zero set of <i>G</i>-equivariant maps, *MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY*, Vol: 147, Pages: 735-755, ISSN: 0305-0041

Guo S, Lamb JSW, Rink BW, 2009, Branching patterns of wave trains in the FPU lattice, *NONLINEARITY*, Vol: 22, Pages: 283-299, ISSN: 0951-7715

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- Citations: 7

Buono P-L, Lamb JSW, Roberts M, 2008, Bifurcation and branching of equilibria in reversible-equivariant vector fields, *NONLINEARITY*, Vol: 21, Pages: 625-660, ISSN: 0951-7715

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- Citations: 6

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