. A method of approximation that was introduced for functional differential equations with a single state-dependent delay is now extended to handle multiple state-dependent delays. A discussion of the method of steps (MoS) is used to develop a version for the case of multiple state-dependent delays that aligns favorably with the method presented here. Some numerical results are presented to confirm the robustness of the implementation. Benefits of this method include producing a continuous function as the approximation, freedom from root finding techniques, and derivatives of the vector field not being required, although the delay needs to be continuously differentiable except at specific isolated points.
We derive variational integrators for stochastic Hamiltonian systems on Lie groups using a discrete version of the stochastic Hamiltonian phase space principle. The structure-preserving properties of the resulting scheme, such as symplecticity, preservation of the Lie-Poisson structure, preservation of the coadjoint orbits, and preservation of Casimir functions, are discussed, along with a discrete No ether theorem for subgroup symmetries. We also consider in detail the case of stochastic Hamiltonian systems with advected quantities, studying the associated structure-preserving properties in relation to semidirect product Lie groups. A full convergence proof for the scheme is provided for the case of the Lie group of rotations. Several numerical examples are presented, including simulations of the free rigid body and the heavy top.
. We investigate the benefits of higher-order time-integration for the quantized formulation of the Euler equations for incompressible fluid flow on the sphere. We show that higher-order isospectral symplectic Runge-Kutta (IsoSyRK) methods are a viable and efficient alternative to the widely used isospectral implicit midpoint method. When implemented using a block matrix approach, higher-order methods allow for effective parallelization and exhibit superior conservation of the Hamiltonian while reducing overall simulation time due to the larger time-step enabled by higher-order methods. We establish this with a series of numerical experiments that compare energy and enstrophy conservation, as well as execution time across integrators of different orders. A qualitative experiment illustrates further benefits of these integrators, showing that their enhanced conservation properties translate into more stable longtime behavior for a simulation starting from a sensitive initial condition.
. We compare three approaches for structure preserving numerical integration of isospectral flows on quadratic Lie algebras. Such flows originate from Hamiltonian dynamics on the cotangent bundle of the Lie group. It is known, via discrete reduction theory, that symplectic Runge-Kutta methods applied to the cotangent bundle formulation induce isospectral symplectic Runge-Kutta (ISOSYRK) schemes on the Lie algebra. Here, we show that the same symplectic Runge-Kutta method, but applied to the transport formulation of the flow on the Lie group, is equivalent to the corresponding ISOSYRK scheme. We also give numerical results suggesting that the formulation on the Lie group is more efficient for schemes with two or more intermediate stages.
Long-term predictions of large-scale flow features on an Earth-like planet are crucial for developing global atmospheric model systems. Such predictions require computational models, i.e., governing PDE systems combined with appropriate numerical methods, to preserve essential structures of the underlying physical model over extended simulation periods. We present a Lie-Poisson formulation of the multi-layer quasi-geostrophhic (QG) equations on the full globe, mimicking the dynamics in the troposphere extended over the first 10 km of the atmosphere. The chosen computational modeling ensures consistency with the underlying structure and enables long-term simulations without the need for additional regularization, forcing, or numerical dissipation. Recent advancements in Lie-Poisson discretization that preserve energy, enstrophy, and higher-order moments of potential vorticity are extended to stratified QG multilayer systems on the sphere. We adopt Zeitlin discretization, which yields a finite-dimensional dynamical system conserving all numerically resolved Casimirs with machine-precision. Particular attention is given to the convergence of critical latitude (phi cl) predictions upon increasing the spatial resolution per layer (N) and the number of layers in the model (M). A systematic parameter study quantifies (i) that the dependency of phi cl on the resolution per layer N scales quadratically, showing near grid-independency for N >= 96, and (ii) the critical latitude decreases with the number of layers M for modest radial resolutions of M <= 32.
We consider synchrony patterns in coupled phase oscillator networks that correspond to invariant tori. For specific nongeneric coupling, these tori are equilibria relative to a continuous symmetry action. We analyze how the invariant tori deform under forced symmetry breaking as more general network interaction terms are introduced. We first show in general that perturbed tori that are relative equilibria can be computed using a parametrization method; this yields an asymptotic expansion of an embedding of the perturbed torus, as well as the local dynamics on the torus. We then apply this result to a coupled oscillator network, and we numerically study the dynamics on the persisting tori in the network by looking for bifurcations of their periodic orbits in a boundary-value-problem setup. This way we find new bifurcating stable synchrony patterns that can be the building blocks of larger global structures such as hetero clinic cycles.
The Ghil-Zaliapin-Thompson (GZT) model, a scalar delay differential equation with seasonal periodic forcing and time-delayed feedback, captures key features of the El Nino-Southern Oscillation (ENSO) phenomenon. Numerical studies of the GZT model have revealed stable period-one orbits under strong forcing and locked, quasiperiodic, or even chaotic regimes under weaker forcing, but its analytical treatment remains challenging. To bridge this gap, we propose a piecewise smooth version of the GZT model with piecewise constant delayed feedback and continuous periodic forcing. For this piecewise smooth GZT model we explicitly construct solutions of initial value problems, and study the existence and properties of periodic orbits of period one. Period-one orbits, have period exactly matching that of the seasonal forcing, and correspond to seasonally dominated ENSO regimes in which the inter-annual variability is locked to the annual cycle. By studying the symmetries and possible phases of periodic solutions we are able to construct period-one solutions and the regions of parameter space in which they exist. We show that the stability of these orbits is governed by a linear mapping from which we find the Floquet multipliers for the periodic orbit and also the bifurcation curve along which these orbits lose stability. We show that for most values of the delay this occurs at a torus bifurcation, but that for small delays a fold bifurcation of period one orbits occurs. We then compare these analytical results with numerical continuation of the GZT model, showing that they align very closely.
This paper gives a basic demonstration of matrix hydrodynamics; the field pioneered by V. Zeitlin, where 2-D incompressible fluids are spatially discretized via quantization theory.
A discrete theory for implicit nonholonomic Lagrangian systems undergoing elastic collisions is developed. It is based on the discrete Lagrange-d'Alembert-Pontryagin variational principle and the dynamical equations thus obtained are the discrete nonholonomic implicit Euler-Lagrange equations together with the discrete conditions for the elastic impact. To illustrate the theory, variational integrators with collisions are built for several examples, including a bouncing ellipse and a nonholonomic spherical pendulum evolving inside a cylinder.
The reachable sets of nonlinear control systems can in general only be numerically approximated, and are often very expensive to calculate. In this paper, we propose an algorithm that tracks only the boundaries of the reachable sets and that chooses the temporal and spatial discretizations in a non-uniform way to reduce the computational complexity.
We consider synchrony patterns in coupled phase oscillator networks that correspond to invariant tori. For specific nongeneric coupling, these tori are equilibria relative to a continuous symmetry action. We analyze how the invariant tori deform under forced symmetry breaking as more general network interaction terms are introduced. We first show in general that perturbed tori that are relative equilibria can be computed using a parametrization method; this yields an asymptotic expansion of an embedding of the perturbed torus, as well as the local dynamics on the torus. We then apply this result to a coupled oscillator network, and we numerically study the dynamics on the persisting tori in the network by looking for bifurcations of their periodic orbits in a boundary-value-problem setup. This way we find new bifurcating stable synchrony patterns that can be the building blocks of larger global structures such as heteroclinic cycles.
Global information about dynamical systems can be extracted by analysing associated infinite-dimensional transfer operators, such as Perron-Frobenius and Koopman operators as well as their infinitesimal generators. In practice, these operators typically need to be approximated from data. Popular approximation methods are extended dynamic mode decomposition (EDMD) and generator extended mode decomposition (gEDMD). We propose a unified framework that leverages Monte Carlo sampling to approximate the operator of interest on a finite-dimensional space spanned by a set of basis functions. Our framework contains EDMD and gEDMD as special cases, but can also be used to approximate more general operators. Our key contributions are proofs of the convergence of the approximating operator under relaxed conditions. We also prove that in some cases eigenpairs of the approximating operators weakly converge to eigenpairs of the exact operator, in others they do not. Moreover, we derive explicit convergence rates and account for the presence of noise in the observations. Whilst all these results are broadly applicable, they also refine previous analyses of EDMD and gEDMD. We verify the analytical results with the aid of several numerical experiments.
A new procedure to construct symplectic methods for constrained mechanical systems is developed in this paper. The definition of a map coming from the notion of retraction maps allows to adapt the continuous problem to the discretization rule rather than viceversa. As a result, the constraint submanifold is exactly preserved by the discrete flow and the extension of the methods to the case of non-linear configuration spaces is straightforward.
This work develops a functional analytic framework for computer assisted existence proofs for transverse connecting orbits between hyperbolic periodic solutions of analytic ordinary differential equations. We exploit FourierTaylor approximations of the local stable/unstable manifolds of the periodic orbits, combined with a numerical method for solving two point boundary value problems via Chebyshev series approximations. A-posteriori analysis exploiting interval arithmetic leads to mathematically rigorous bounds on all discretization and truncation errors, providing both abstract existence results and quantitative information about the true heteroclinic solution. The arguments require some adjustment when the system conserves a first integral, and is readily adapted for non-polynomial nonlinearities. To illustrate the utility of the method we prove a number of theorems about the Lorenz system and the Hill restricted four body problem.
. Continuous piecewise affine (CPA) functions have been successfully used to construct Lyapunov functions for autonomous ordinary differential equations with an exponentially stable equilibrium. However, for time-periodic systems this method cannot be used. In this paper, we develop a method to compute a Lyapunov function for a time-periodic ordinary differential equation with a (known) exponentially stable periodic solution. While the usual approach with a triangulation works away from the periodic solution, a new approach is required in a neighborhood of the periodic solution. We derive sufficient conditions for a Lyapunov function and, moreover, we prove a converse theorem, showing that these conditions can always be satisfied if the triangulation is sufficiently fine. Finally, we formulate the construction problem as a linear programming problem and apply it to two examples. This paper deals with the case of one spatial dimension, but we believe that the method can be generalised to any dimension, which will be done in a subsequent paper.
We investigate theoretically and numerically transport noise-induced diffusion in flows on the sphere. Previous analysis on the torus has shown that suitably chosen transport noise in the Euler equations leads to diffusive behavior resembling the Navier-Stokes equations. Here, we analyze dynamics on the sphere with noise-induced differential operator dissipation and characterize their energy and enstrophy decay properties. Through structure-preserving numerical simulations with the Zeitlin discretization, we demonstrate that appropriately scaled transport noise induces energy dissipation while preserving enstrophy and coadjoint orbits. The presented analysis provides a foundation for further theoretical investigation of transport noise and supports the calibration of transport noise models as a parametrization for unresolved processes in geophysical fluid simulations.
. The Koopman operator has become an essential tool for datadriven analysis, prediction, and control of complex systems. The main reason is the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. This equally applies to ordinary, stochastic, and partial differential equations (PDEs). Until now, with a few exceptions only, the PDE case is mostly treated rather superficially, and the specific structure of the underlying dynamics is largely ignored. In this paper, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to significantly increase the model efficacy. Moreover, the situation where we only have access to partial observations-i.e., measurements, as is very common for experimental data-has not been treated to its full extent, either. Moreover, we address the highly-relevant case where we cannot measure the full state, where alternative approaches (e.g., delay coordinates) have to be considered. We derive rigorous statements on the required number of observables in this situation, based on embedding theory. We present numerical evidence using various numerical examples including the wave equation and the Kuramoto-Sivashinsky equation.
Any commutative connected graded Hopf algebra gives rise to a specific notion of rough path. This article provides a review of the different Hopf algebras, providing the various kinds of rough paths which have appeared in recent literature: geometric and quasi-geometric rough paths, branched and planarly branched rough paths, and multi-index rough paths. Relations between all these variants are also described.
