arXiv subjects

Jeremy P Parker

Publications and source records attributed to Jeremy P Parker.

8 recordsLinked to original sources

arXiv2026

From synthetic turbulence to true solutions: A deep diffusion model for discovering periodic orbits in the Navier-Stokes equations

Generative artificial intelligence has shown remarkable success in synthesizing data that mimic complex real-world systems, but its potential role in the discovery of mathematically meaningful structures in physical models remains underexplored. In this work, we demonstrate how a generative diffusion model can be used to uncover previously unknown solutions of a nonlinear partial differential equation: the two-dimensional Navier-Stokes equations in a turbulent regime. Trained on data from a direct numerical simulation of turbulence, the model learns to generate time series that resemble physically plausible trajectories. By carefully modifying the temporal structure of the model and enforcing the symmetries of the governing equations, we produce synthetic trajectories that are periodic in time, despite the fact that the training data did not contain periodic trajectories. These synthetic trajectories are then refined into true solutions using an iterative solver, yielding 111 new periodic orbits (POs) with very short periods. Our results reveal a previously unobserved richness in the PO structure of this system and suggest a broader role for generative AI: not as replacements for simulation and existing solvers, but as a complementary tool for navigating the complex solution spaces of nonlinear dynamical systems.

physics.flu-dyn↗

arXiv2026

Computation of attractor dimension and maximal sums of Lyapunov exponents using polynomial optimization

Two approaches are presented for computing upper bounds on Lyapunov exponents and their sums, and on the Lyapunov dimension, among all trajectories of a dynamical system governed by ordinary differential equations. The first approach expresses a sum of Lyapunov exponents as a time average in an augmented dynamical system and then applies methods for bounding time averages. This generalizes the method of Oeri \& Goluskin (Nonlinearity 36:5378--5400, 2023) for bounding the single largest Lyapunov exponent. The second approach considers a different augmented dynamical system, where bounds on sums of Lyapunov exponents are implied by stability of certain sets, and such stability is verified using Lyapunov function methods. Both of our approaches also can be adapted to directly compute bounds on Lyapunov dimension, which in turn imply bounds on the fractal dimension of a global attractor. For systems of ordinary differential equations with polynomial right-hand sides, all of our bounding formulations lead to polynomial optimization problems with sum-of-squares constraints. These sum-of-squares problems can be solved computationally for a chosen system to yield numerical bounds, provided the number of variables and degree of polynomials are not prohibitive. Most of our bounding formulations are proved to be sharp under mild assumptions. In the case of the polynomial optimization problems, sharpness means that upper bounds converge to the quantities being bounded as polynomial degrees are raised. Computational examples demonstrate upper bounds that are sharp to several digits, including for a six-dimensional dynamical system where sums of Lyapunov exponents are maximized on periodic orbits.

math.DS↗

arXiv2024

Ghost states underlying spatial and temporal patterns: how non-existing invariant solutions control nonlinear dynamics

Close to a saddle-node bifurcation, when two invariant solutions collide and disappear, the behavior of a dynamical system can closely resemble that of a solution which is no longer present at the chosen parameter value. For bifurcating equilibria in low-dimensional ODEs, the influence of such 'ghosts' on the temporal behavior of the system, namely delayed transitions, has been studied previously. We consider spatio-temporal PDEs and characterize the phenomenon of ghosts by defining representative state-space structures, which we term 'ghost states,' as minima of appropriately chosen cost functions. Using recently developed variational methods, we can compute and parametrically continue ghost states of equilibria, periodic orbits, and other invariant solutions. We demonstrate the relevance of ghost states to the observed dynamics in various nonlinear systems including chaotic maps, the Lorenz ODE system, the spatio-temporally chaotic Kuramoto-Sivashinsky PDE, the buckling of an elastic arc, and 3D Rayleigh-Bénard convection.

math.DS↗

arXiv2024

The Lorenz system as a gradient-like system

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc. do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for $0\leqρ\leq12$ when $σ=10$ and $β=8/3$, significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.

nlin.CD↗

arXiv2022

Koopman analysis of the periodic Korteweg-de Vries equation

The eigenspectrum of the Koopman operator enables the decomposition of nonlinear dynamics into a sum of nonlinear functions of the state space with purely exponential and sinusoidal time dependence. For a limited number of dynamical systems, it is possible to find these Koopman eigenfunctions exactly and analytically. Here, this is done for the Korteweg-de Vries equation on a periodic interval, using the periodic inverse scattering transform and some concepts of algebraic geometry. To the authors' knowledge, this is the first complete Koopman analysis of a partial differential equation which does not have a trivial global attractor. The results are shown to match the frequencies computed by the data-driven method of dynamic mode decomposition (DMD). We demonstrate that in general DMD gives a large number of eigenvalues near the imaginary axis, and show how these should be interpretted in this setting.

nlin.SI↗

arXiv2022

Invariant tori in dissipative hyperchaos

One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems including fluid turbulence, while higher-dimensional invariant tori representing quasi-periodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; that tori can be numerically identified via bifurcations of unstable periodic orbits and that their parametric continuation and characterization of stability properties is feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos.

nlin.CD↗

arXiv2022

Variational methods for finding periodic orbits in the incompressible Navier-Stokes equations

Unstable periodic orbits are believed to underpin the dynamics of turbulence, but by their nature are hard to find computationally. We present a family of methods to converge such unstable periodic orbits for the incompressible Navier-Stokes equations, based on variations of an integral objective functional, and using traditional gradient-based optimisation strategies. Different approaches for handling the incompressibility condition are considered. The variational methods are applied to the specific case of periodic, two-dimensional Kolmogorov flow and compared against existing Newton iteration-based shooting methods. While computationally slow, our methods converge from very inaccurate initial guesses.

physics.flu-dyn↗

arXiv2021

A study of the double pendulum using polynomial optimization

In dynamical systems governed by differential equations, a guarantee that trajectories emanating from a given set of initial conditions do not enter another given set can be obtained by constructing a barrier function that satisfies certain inequalities on phase space. Often these inequalities amount to nonnegativity of polynomials and can be enforced using sum-of-squares conditions, in which case barrier functions can be constructed computationally using convex optimization over polynomials. To study how well such computations can characterize sets of initial conditions in a chaotic system, we use the undamped double pendulum as an example and ask which stationary initial positions do not lead to flipping of the pendulum within a chosen time window. Computations give semialgebraic sets that are close inner approximations to the fractal set of all such initial positions.

nlin.CD↗