Tobias M. Schneider

Invariant solutions of the Navier–Stokes equations, such as fixed points, traveling waves, and periodic orbits, provide a dynamical systems framework for turbulence. Although dynamically unstable, these solutions organize chaotic dynamics and capture coherent structures observed in experiments and simulations. Over the past three decades, substantial progress has been made in computing invariant solutions, initially focusing on isolated fixed points and traveling waves and, more recently, shifting toward libraries of periodic orbits that offer a more complete description of the dynamics. Computing these solutions is challenging because of their instability, the high dimensionality arising from spatial discretization, and the increasing complexity in more turbulent regimes. This review provides an overview of the numerical methods developed to address these challenges. We discuss matrix-free Newton–Krylov approaches, more recent variational methods designed to improve robustness in the computation of unstable periodic orbits, and emerging strategies that combine equation-based algorithms with data-driven techniques to generate improved initial guesses. Together, these developments provide increasingly powerful tools for exploring the dynamical structure of turbulence.

Self-injection locking (SIL) of a diode laser to a high-quality-factor microresonator has recently become increasingly important in hybrid integrated photonics, providing access to compact sub-Hz linewidth lasers. It was also shown to facilitate access to dissipative Kerr solitons—the key to a low-noise coherent frequency comb on a photonic chip. However, the existence and stability ranges of SIL soliton states in experimentally controlled parameters are still not fully understood. Here we study the bifurcation structure of solutions in a model of soliton SIL in the weak-backscattering limit. We show that SIL produces soliton-number-dependent existence ranges of multi-soliton solutions in free-laser detuning and feedback phase parameters. We identify exclusive single-soliton existence regions and demonstrate dynamical access to single solitons in this region by direct numerical simulations using prescribed parameter sweeps.

Coupled-microresonator systems, enabled by recent breakthroughs in the nanofabrication of low-loss integrated photonic microresonators, deliver significant performance benefits for coherent frequency comb generation beyond single resonators. They also display emergent nonlinear phenomena including soliton hopping—a dynamic process in which solitons periodically transit between coupled resonators. We employ a nonlinear dynamical systems approach, leveraging numerical techniques developed in the context of hydrodynamics, to identify exact periodic orbit solutions of the coupled Lugiato-Lefever equations underlying soliton hopping in photonic dimers and trimers. A bifurcation-theoretic origin of hopping reveals a fundamental difference in dimers and trimers: In dimers, hopping emerges from a branch of stable soliton solutions, while in trimers it originates from an unstable branch. This distinction lowers pump power requirements for trimers, rendering them promising for experimental demonstrations of hopping. We further show that subcritical Hopf bifurcations of unstable equilibria explain hysteresis, coexistence of multiple attractors, and path-dependent access to different dynamical regimes as observed in simulated laser scans mimicking typical experimental investigations. Our findings provide insights into the nonlinear dynamics of coupled multimode microresonators, offering a foundation for the controlled implementation of soliton-based technologies in future integrated photonic systems.

Unstable periodic orbits (UPOs) are the non-chaotic, dynamical building blocks of spatio-temporal chaos, motivating a first-principles based theory for turbulence ever since the discovery of deterministic chaos. Despite their key role in the ergodic theory approach to fluid turbulence, identifying UPOs is challenging for two reasons: chaotic dynamics and the high-dimensionality of the spatial discretization. We address both issues at once by proposing a loop convergence algorithm for UPOs directly within a low-dimensional embedding of the chaotic attractor. The convergence algorithm circumvents time-integration, hence avoiding instabilities from exponential error amplification, and operates on a latent dynamics obtained by pulling back the physical equations using automatic differentiation through the learned embedding function. The interpretable latent dynamics is accurate in a statistical sense, and, crucially, the embedding preserves the internal structure of the attractor, which we demonstrate through an equivalence between the latent and physical UPOs of both a model PDE and the 2D Navier-Stokes equations. This allows us to exploit the collapse of high-dimensional dissipative systems onto a lower dimensional manifold, and identify UPOs in the low-dimensional embedding.

Vertical thermal convection exhibits weak turbulence and spatio-temporally chaotic behaviour. For this configuration, we report seven new equilibria and 26 new periodic orbits. These orbits, together with four previously studied in Zheng et al. ( J. Fluid Mech. , 2024 b , vol. 1000, p. A29) bring the number of periodic-orbit branches computed so far to 30, all solutions to the fully nonlinear three-dimensional Navier–Stokes equations. These new and unstable invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects. These interesting and important fluid mechanical processes in a small flow unit are shown to also appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle-node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. Furthermore, these orbits are shown to be able to reconstruct statistically the core part of the attractor, so that these results may contribute to a quantitative description of transitional fluid turbulence using periodic orbit theory.

We obtain and investigate theoretically a broad family of stable and unstable time-periodic orbits-oscillating Turing rolls (OTR)-in the Lugiato-Lefever model of optical cavities. Using the dynamical systems tools developed in fluid dynamics, we access the OTR solution branches in parameter space and elucidate their bifurcation structure. By tracking these exact invariant solutions deeply into the chaotic region of the modulation instability, we connect the main dynamical regimes of the Lugiato-Lefever model: continuous waves, Turing rolls, solitons, and breathers, which completes the classical phase diagram of the optical cavity. We then demonstrate that the OTR periodic orbits play a fundamental role as elementary building blocks in the regime of the intracavity field transition from stable Turing rolls to fully developed turbulent regimes. Depending on the cavity size, we observe that the chaotic intracavity field driven by modulation instability displays either spatiotemporal or purely temporal intermittancy between chaotic dynamics and different families of the OTR solutions, exhibiting locally the distinctive wave patterns and large amplitude peaks. This opens avenues for a theoretical description of optical turbulence within the dynamical systems framework.

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 ordinary differential equations (ODEs), the influence of such "ghosts" on the temporal behavior of the system, namely delayed transitions, has been studied previously. We consider spatiotemporal partial differential equations (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 spatiotemporally chaotic Kuramoto-Sivashinsky PDE, the buckling of an elastic arc, and 3D Rayleigh-Bénard convection.

We numerically study wall-bounded convectively driven magneto-hydrodynamic (MHD) flows subject to rotation in a Cartesian periodic channel. For the accurate treatment of the rotating MHD equations, we develop a pseudo-spectral simulation code with accurate treatment of boundary conditions for both velocity and magnetic fields. The solenoidal condition on the magnetic field is enforced by the addition of a fictitious magnetic pressure. This allows us to employ an influence matrix method with tau correction for the treatment of velocity and magnetic fields subject to Robin boundary conditions at the confining walls. We validate the developed method for the specific case of no slip velocity and perfectly conducting magnetic boundary conditions. The validation includes the accurate reproduction of linear stability thresholds and of turbulent statistics. The code shows favorable parallel scaling properties.