Tobias Schneider

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The topology of a chaotic attractor in the Kuramoto-Sivashinsky equation

The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within a chaotic attractor and the topology of that attractor, for three-dimensional systems. In certain cases, the fractal dimension of a ch ...

2025

Accurate Implementation of Rotating Magneto-Hydrodynamics in a Channel Geometry Using an Influence Matrix Method

Tobias Schneider

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 ...

MDPI AG2025

Search for Unstable Relative Periodic Orbits in Channel Flow Using Symmetry-reduced Dynamic Mode Decomposition

Tobias Schneider, Omid Ashtari

In the dynamical systems approach to turbulence, unstable periodic orbits (UPOs) provide valuable insights into system dynamics. Such UPOs are usually found by shooting-based Newton searches, where constructing sufficiently accurate initial guesses is diff ...

CAMBRIDGE UNIV PRESS2025

Ghost States Underlying Spatial and Temporal Patterns: How Nonexistent Invariant Solutions Control Nonlinear Dynamics

Tobias Schneider, Zheng Zheng, Pierre Beck, Omid Ashtari, Jeremy Peter Parker

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-d ...

AMER PHYSICAL SOC2025

Oscillating Turing Rolls as Periodic Orbits of the Lugiato-Lefever Equation

Tobias Kippenberg, Tobias Schneider

The Lugiato-Lefever equation (LLE) is a fundamental nonlinear model of pattern formation in damped-driven optical Kerr cavities with applications ranging from fiber to chip-integrated optical resonators. In standard dimensionless form the LLE describes dyn ...

IEEE2025

Following marginal stability manifolds in quasilinear dynamical reductions of multiscale flows in two space dimensions

Tobias Schneider, Alessia Ferraro

We derive a two-dimensional (2D) extension of a recently developed formalism for slow-fast quasilinear (QL) systems subject to fast instabilities. The emergent dynamics of these systems is characterized by a slow evolution of (suitably defined) mean fields ...

2025

Data-driven Guessing and Gluing of Unstable Periodic Orbits

Tobias Schneider, Pierre Beck

Unstable periodic orbits (UPOs) are believed to be the underlying dynamical structures of spatiotemporal chaos and turbulence. Finding these UPOs is, however, notoriously difficult. Matrix-free loop convergence algorithms deform entire space-time fields (l ...

AMER PHYSICAL SOC2025

Natural convection in a vertical channel. Part 1. Wavenumber interaction and Eckhaus instability in a narrow domain

Tobias Schneider, Zheng Zheng

In a vertical channel driven by an imposed horizontal temperature gradient, numerical simulations (Gao et al., Phys. Rev. E, vol. 88, 2013, 023010; Phys. Rev. E, vol. 91, 2015, 013006; Phys. Rev. E, vol. 97, 2018, 053107) have previously shown steady, time ...

Cambridge University Press2024

Natural convection in a vertical channel. Part 2. Oblique solutions and global bifurcations in a spanwise-extended domain

Tobias Schneider, Zheng Zheng

Vertical thermal convection is a non-equilibrium system in which both buoyancy and shear forces play a role in driving the convective flow. Beyond the onset of convection, the driven dissipative system exhibits chaotic dynamics and turbulence. In a three-d ...

Cambridge University Press2024

Identifying invariant solutions of wall-bounded three-dimensional shear flows using robust adjoint-based variational techniques

Tobias Schneider, Omid Ashtari

Invariant solutions of the Navier-Stokes equations play an important role in the spatiotemporally chaotic dynamics of turbulent shear flows. Despite the significance of these solutions, their identification remains a computational challenge, rendering many ...

Cambridge University Press2023

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Natural convection in a vertical channel. Part 2. Oblique solutions and global bifurcations in a spanwise-extended domain

Tobias Schneider, Zheng Zheng

Cambridge University Press2024

Identifying invariant solutions of wall-bounded three-dimensional shear flows using robust adjoint-based variational techniques

Tobias Schneider, Omid Ashtari

Cambridge University Press2023