Publications by Tobias Schneider

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

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

Predicting chaotic statistics with unstable invariant tori

Tobias Schneider, Omid Ashtari, Jeremy Peter Parker

It has recently been speculated that long-time average quantities of hyperchaotic dissipative systems may be approximated by weighted sums over unstable invariant tori embedded in the attractor, analogous to equivalent sums over periodic orbits, which are ...

American Institute of Physics2023

Jacobian-free variational method for computing connecting orbits in nonlinear dynamical systems

Tobias Schneider, Omid Ashtari

One approach for describing spatiotemporal chaos is to study the unstable invariant sets embedded in the chaotic attractor of the system. While equilibria, periodic orbits, and invariant tori can be computed using existing methods, the numerical identifica ...

American Institute of Physics2023

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

Tobias Schneider, Jeremy Peter Parker

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, bas ...

Cambridge University Press2022

Invariant tori in dissipative hyperchaos

Tobias Schneider, Jeremy Peter Parker

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 ...

American Institute of Physics2022

Constructing periodic orbits of high-dimensional chaotic systems by an adjoint-based variational method

Tobias Schneider, Sajjad Azimi, Omid Ashtari

Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatiotemporal systems including fluid turbulence is supported by nonchaotic, exactly recurring time-periodic solutions of the governing equations ...

2022

Superspreading events suggest aerosol transmission of SARS-CoV-2 by accumulation in enclosed spaces

Tobias Schneider, John Martin Kolinski

Viral transmission pathways have profound implications for public safety; it is thus imperative to establish a complete understanding of viable infectious avenues. Mounting evidence suggests SARS-CoV-2 can be transmitted via the air; however, this has not ...

AMER PHYSICAL SOC2021

Modified snaking in plane Couette flow with wall-normal suction

Tobias Schneider, Sajjad Azimi

A specific family of spanwise-localised invariant solutions of plane Couette flow exhibits homoclinic snaking, a process by which spatially localised invariant solutions of a nonlinear partial differential equation smoothly grow additional structure at the ...

Cambridge University Press2021

Periodic orbits exhibit oblique stripe patterns in plane Couette flow

Tobias Schneider, Florian Reetz

Spatiotemporally chaotic dynamics of transitional plane Couette flow may give rise to regular turbulent-laminar stripe patterns with a large-scale pattern wavelength and an oblique orientation relative to the laminar flow direction. A recent dynamical syst ...

AMER PHYSICAL SOC2021

Nondestructive Prediction of the Buckling Load of Imperfect Shells

Tobias Schneider, Emmanuel Georges Albert Virot, Emilio José Lozano Vazquez

From soda cans to space rockets, thin-walled cylindrical shells are abundant, offering exceptional load carrying capacity at relatively low weight. However, the actual load at which any shell buckles and collapses is very sensitive to imperceptible defects ...

2020

Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states

Tobias Schneider, Florian Reetz

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number . Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of ...

Cambridge University Press2020