Bifurcation from relative periodic solutions

Published versio

Learning reversible symplectic dynamics

Time-reversal symmetry arises naturally as a structural property in many dynamical systems of interest. While the importance of hard-wiring symmetry is increasingly recognized in machine learning, to date this has eluded time-reversibility. In this paper we propose a new neural network architecture for learning time-reversible dynamical systems from data. We focus in particular on an adaptation to symplectic systems, because of their importance in physics-informed learning

The Evolution of Compact Binary Star Systems

We review the formation and evolution of compact binary stars consisting of white dwarfs (WDs), neutron stars (NSs), and black holes (BHs). Binary NSs and BHs are thought to be the primary astrophysical sources of gravitational waves (GWs) within the frequency band of ground-based detectors, while compact binaries of WDs are important sources of GWs at lower frequencies to be covered by space interferometers (LISA). Major uncertainties in the current understanding of properties of NSs and BHs most relevant to the GW studies are discussed, including the treatment of the natal kicks which compact stellar remnants acquire during the core collapse of massive stars and the common envelope phase of binary evolution. We discuss the coalescence rates of binary NSs and BHs and prospects for their detections, the formation and evolution of binary WDs and their observational manifestations. Special attention is given to AM CVn-stars -- compact binaries in which the Roche lobe is filled by another WD or a low-mass partially degenerate helium-star, as these stars are thought to be the best LISA verification binary GW sources.Comment: 105 pages, 18 figure

Topological bifurcations of minimal invariant sets for set-valued dynamical systems

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

Synchronisation of chaos and its applications

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

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

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