76 results found
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.
Castro MM, Chemnitz D, Chu H, et al., 2022, The Lyapunov spectrum for conditioned random dynamical systems, Pages: 1-36
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.
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.
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  on periodically kicked limit cycles to the stochastic context.
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).
Lamb JSW, 2018, Foreword
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
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
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
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
Makarenkov O, Lamb JSW, 2012, Preface: Dynamics and Bifurcations of Nonsmooth Systems, PHYSICA D-NONLINEAR PHENOMENA, Vol: 241, Pages: 1825-1825, ISSN: 0167-2789
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
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
Buono P-L, Helmer M, Lamb JSW, 2009, On the zero set of G-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
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
Homburg AJ, Jukes AC, Knobloch J, et al., 2008, Saddle-nodes and period-doublings of Smale horseshoes: a case study near resonant homoclinic bellows, BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN, Vol: 15, Pages: 833-850, ISSN: 1370-1444
Guo S, Lamb JSW, 2008, Equivariant Hopf bifurcation for neutral functional differential equations, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 136, Pages: 2031-2041, ISSN: 0002-9939
JSW Lamb, I Melbourne, 2007, Normal form theory for relative equilibria and relative periodic solutions, Trans. Amer. Math. Soc., Vol: 359, Pages: 4537-4556
Homburg AJ, JSW Lamb, 2006, Symmetric homoclinic tangles in reversible systems., Ergodic Theory Dynam. Systems, Vol: 26, Pages: 1769-1789
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