arXiv subjects

Tobias M. Schneider

Publications and source records attributed to Tobias M. Schneider.

At least 19 recordsLinked to original sources

arXiv2026

The Origin of Imperfection Sensitivity in the Buckling of Cylindrical Shells

Buckling of thin cylindrical shells under axial compression, a classical example of a subcritical instability, is highly sensitive to small imperfections, with minute geometric variations causing large changes in buckling threshold. To uncover the origin of this sensitivity, we use numerical continuation and bifurcation analysis while systematically varying the depth and size of a single localized Gaussian defect. We show that the instabilities of the imperfect shell originate from localized equilibria already present in the perfect shell. By breaking translation symmetry, the defect pins these equilibria and changes how they connect to the imperfect base state. Small changes in defect geometry can thereby switch the bifurcation that triggers buckling, producing non-monotonic and discontinuous changes in buckling threshold and abrupt changes in buckling mode. Imperfection sensitivity is therefore not simply sensitivity to imperfection magnitude, but sensitivity of the underlying bifurcation structure to imperfection geometry.

nlin.PS↗

arXiv2026

Identifying recurrent flows in high-dimensional dissipative chaos from low-dimensional embeddings

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.

nlin.CD↗

arXiv2025

Local and global bifurcations to large-scale oblique patterns in inclined layer convection

In the inclined layer convection system, thermal convection in a Rayleigh--Bénard cell tilted against gravity, the flow is subject to competing buoyancy and shear forces. For varying inclination angle ($γ$) and Rayleigh number ($Ra$), a variety of spatio-temporal patterns is observed. We investigate the switching diamond panes (SDP) pattern, observed at $(γ, Ra)\simeq(100\degree,10000)$, which exhibits large-scale oblique features and is one of the five complex tertiary patterns at Prandtl number $Pr=1.07$. First, we study the linear instability of the secondary-state transverse convection rolls and the five branches including two travelling waves and three periodic orbits, bifurcating simultaneously from it. These non-generic bifurcations arise from the breaking of the spatial $D_4$ and $O(2)$ symmetries of transverse rolls, and the resulting bifurcated solutions show large-scale diamond-shaped amplitude modulations. Second, we explore a periodic orbit that captures both the large-scale structure and small-scale defects of modulated rolls. Parametric continuation in $Ra$ reveals the global homoclinic bifurcation via which this periodic orbit emerges. Third, the boundary of the basins of attraction of two dynamically relevant periodic orbits has been characterised. Specifically, additional steady and time-periodic solutions are identified on the basin boundary and their bifurcation structures are analysed. Together, using non-linear invariant solutions and their bifurcations, we take a further step toward understanding the emergence and dynamics of SDP far from the onset of convection, where linear methods have not been applied successfully.

physics.flu-dyn↗

arXiv2025

Natural convection in a vertical channel. Part 3. Bifurcations of many (additional) unstable periodic orbits and their dynamical relevance

Vertical thermal convection system exhibits weak turbulence and spatio-temporally chaotic behaviour. In this system, we report seven equilibria and 26 periodic orbits, all new and linearly unstable. These orbits, together with four previously studied in Zheng et al. (2024b) bring the number of periodic orbit branches computed so far to 30, all solutions to the fully non-linear three-dimensional Navier--Stokes equations. These new invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects in rolls. These interesting and important fluid mechanical processes in a small flow unit are shown to 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. These orbits are shown to be able to reconstruct statistically the core part of the attractor, and these results may pave the way to quantitatively describing transitional fluid turbulence using periodic orbit theory.

physics.flu-dyn↗

arXiv2025

Periodic orbits underlying spatiotemporal chaos in the Lugiato-Lefever model

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.

nlin.PS↗

arXiv2025

Nonlinear periodic orbit solutions and their bifurcation structure at the origin of soliton hopping in coupled microresonators

Microresonator frequency combs, essential for future integrated optical systems, rely on dissipative Kerr solitons generated in a single microresonator to achieve coherent frequency comb generation. Recent advances in the nanofabrication of low-loss integrated nonlinear microresonators have paved the way for the exploration of coupled-resonator systems. These systems provide significant technological advantages, including higher conversion efficiency and the generation of dual dispersive waves. Beyond their practical benefits, coupled-resonator systems also reveal novel emergent nonlinear phenomena, such as soliton hopping, a dynamic process in which solitons periodically transfer between coupled resonators. In this study, we employ a dynamical system approach and the corresponding well-established numerical techniques, extensively developed within the context of hydrodynamics and transitional turbulence, to investigate the bifurcation structure of periodic orbit solutions of the coupled Lugiato-Lefever equations that underlie soliton hopping in photonic dimers and trimers. Our main finding uncovers a fundamental difference in the origin of the hopping process in dimers and trimers. We demonstrate that in dimers, hopping emerges from a branch of stable soliton solutions, whereas in trimers, it originates from an unstable branch. This distinction leads to a significant difference in pump power requirements. We relate the bifurcation structure of the periodic orbits including their stability to the observed dynamics in simulated laser scans mimicking typical experimental investigations. Subcritical Hopf bifurcations of unstable equilibrium branches specifically explain observed hysteresis, the coexistence of multiple attractors at the same parameter values, and the importance of choosing a specific path in parameter space to reliably achieve a desired dynamical regime.

nlin.PS↗

arXiv2025

Data-driven guessing and gluing of unstable periodic orbits

Unstable periodic orbits (UPOs) are believed to be the underlying dynamical structures of spatio-temporal chaos and turbulence. Finding these UPOs is however notoriously difficult. Matrix-free loop convergence algorithms deform entire space-time fields (loops) until they satisfy the evolution equations. Initial guesses for these robust variational convergence algorithms are thus periodic space-time fields in a high-dimensional state space, rendering their generation highly challenging. Usually guesses are generated with recurrency methods, which are most suited to shorter and more stable periodic orbits. Here we propose an alternative, data-driven method for generating initial guesses, enabled by the periodic nature of the guesses for loop convergence algorithms: while the dimension of the space used to discretize fluid flows is prohibitively large to construct suitable initial guesses, the dissipative dynamics will collapse onto a chaotic attractor of far lower dimension. We use an autoencoder to obtain a low-dimensional representation of the discretized physical space for the one-dimensional Kuramoto-Sivashinksy equation, in chaotic and hyperchaotic regimes. In this low-dimensional latent space, we construct loops based on the latent POD modes with random periodic coefficients, which are then decoded to physical space and used as initial guesses. These loops are found to be realistic initial guesses and, together with variational convergence algorithms, these guesses help us to quickly converge to UPOs. We further attempt to `glue' known UPOs in the latent space to create guesses for longer ones. This gluing procedure is successful and points towards a hierarchy of UPOs where longer UPOs shadow sequences of shorter ones.

nlin.CD↗

arXiv2024

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 chaotic attractor in a partial differential equation (PDE) is less than three, even though that attractor is embedded within an infinite-dimensional space. Here we study the Kuramoto-Sivashinsky PDE at the onset of chaos. We use two different dimensionality-reduction techniques -- proper orthogonal decomposition and an autoencoder neural network -- to find two different mappings of the chaotic attractor into three dimensions. By finding the image of the attractor's UPOs in these reduced spaces and examining their linking numbers, we construct templates for the branched manifold which encodes the topological properties of the attractor. The templates obtained using two different dimensionality reduction methods are equivalent. The organization of the periodic orbits is identical and consistent symbolic sequences for low-period UPOs are derived. While this is not a formal mathematical proof, this agreement is strong evidence that the dimensional reduction is robust, in this case, and that an accurate topological characterization of the chaotic attractor of the chaotic PDE has been achieved.

nlin.CD↗

arXiv2024

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

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-dimensional domain extended in both the vertical and the transverse dimensions, Gao et al. (2018) have observed a variety of convection patterns which are not described by linear stability analysis. We investigate the fully non-linear dynamics of vertical convection using a dynamical-systems approach based on the Oberbeck-Boussinesq equations. We compute the invariant solutions of these equations and the bifurcations that are responsible for the creation and termination of various branches. We map out a sequence of local bifurcations from the laminar base state, including simultaneous bifurcations involving patterned steady states with different symmetries. This atypical phenomenon of multiple branches simultaneously bifurcating from a single parent branch is explained by the role of D4 symmetry. In addition, two global bifurcations are identified: first, a homoclinic cycle from modulated transverse rolls and second, a heteroclinic cycle linking two symmetry-related diamond-roll patterns. These are confirmed by phase space projections as well as the functional form of the divergence of the period close to the bifurcation points. The heteroclinic orbit is shown to be robust and to result from a 1:2 mode interaction. The intricacy of this bifurcation diagram highlights the essential role played by dynamical systems theory and computation in hydrodynamic configurations.

physics.flu-dyn↗

arXiv2024

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

In a vertical channel driven by an imposed horizontal temperature gradient, numerical simulations have previously shown steady, time-periodic and chaotic dynamics. We explore the observed dynamics by constructing invariant solutions of the three-dimensional Oberbeck-Boussinesq equations, characterizing the stability of these equilibria and periodic orbits, and following the bifurcation structure of the solution branches under parametric continuation in Rayleigh number. We find that in a narrow vertically-periodic domain of aspect ratio ten, the flow is dominated by the competition between three and four co-rotating rolls. We demonstrate that branches of three and four-roll equilibria are connected and can be understood in terms of their discrete symmetries. Specifically, the D4 symmetry of the four-roll branch dictates the existence of qualitatively different intermediate branches that themselves connect to the three-roll branch in a transcritical bifurcation due to D3 symmetry. The physical appearance, disappearance, merging and splitting of rolls along the connecting branch provide a physical and phenomenological illustration of the equivariant theory of D3-D4 mode interaction. We observe other manifestations of the competition between three and four rolls, in which the symmetry in time or in the transverse direction is broken, leading to limit cycles or wavy rolls, respectively. Our work highlights the interest of combining numerical simulations, bifurcation theory, and group theory, in order to understand the transitions between and origin of flow patterns.

physics.flu-dyn↗

arXiv2024

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

A two-dimensional extension of a recently developed formalism for slow-fast quasilinear (QL) systems subject to fast instabilities is derived. Prior work has demonstrated that the emergent dynamics of these systems is characterized by a slow evolution of mean fields coupled to marginally stable, fast fluctuation fields. By exploiting this emergent behavior, an efficient fast-eigenvalue/slow-initial-value solution algorithm can be developed in which the amplitude of the fast fluctuations is slaved to the slowly evolving mean fields to ensure marginal stability (and temporal scale separation) is maintained. For 2D systems that are spatially-extended in one direction, the fluctuation eigenfunctions are labeled by their wavenumbers characterizing spatial variability in that direction, and the marginal mode(s) also must coincide with the fastest-growing mode(s) over all admissible wavenumbers. Here, we introduce two equivalent procedures for deriving an ordinary differential equation governing the slow evolution of the wavenumber of the fastest-growing fluctuation mode that simultaneously must be slaved to the mean dynamics to ensure the mode has zero growth rate. We illustrate the procedure in the context of a 2D model partial differential equation that shares certain attributes with the equations governing strongly stratified shear flows. The slaved evolution follows one or more marginal stability manifolds, which constitute select state-space structures that are not invariant under the full flow dynamics yet capture quasi-coherent states in physical space in a manner analogous to invariant solutions identified in, e.g., transitionally-turbulent shear flows. Accordingly, we propose that marginal stability manifolds are central organizing structures in a dynamical systems description of certain classes of multiscale flows where scale separation justifies a QL approximation of the dynamics.

physics.flu-dyn↗

arXiv2023

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

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 solutions inaccessible and thus hindering progress towards a dynamical description of turbulence in terms of invariant solutions. We compute equilibria of three-dimensional wall-bounded shear flows using an adjoint-based matrix-free variational approach. To address the challenge of computing pressure in the presence of solid walls, we develop a formulation that circumvents the explicit construction of pressure and instead employs the influence matrix method. Together with a data-driven convergence acceleration technique based on dynamic mode decomposition, this yields a practically feasible alternative to state-of-the-art Newton methods for converging equilibrium solutions. We compute multiple equilibria of plane Couette flow starting from inaccurate guesses extracted from a turbulent time series. The variational method outperforms Newton(-hookstep) iterations in successfully converging from poor initial guesses, suggesting a larger convergence radius.

physics.flu-dyn↗

arXiv2023

Jacobian-Free Variational Method for Constructing Connecting Orbits in Nonlinear Dynamical Systems

In a dynamical systems description of spatiotemporally chaotic PDEs including those describing turbulence, chaos is viewed as a trajectory evolving within a network of non-chaotic, dynamically unstable, time-invariant solutions embedded in the chaotic attractor of the system. While equilibria, periodic orbits and invariant tori can be constructed using existing methods, computations of heteroclinic and homoclinic connections mediating the evolution between the former invariant solutions remain challenging. We propose a robust matrix-free variational method for computing connecting orbits between equilibrium solutions of a dynamical system that can be applied to high-dimensional problems. Instead of a common shooting-based approach, we define a minimization problem in the space of smooth state space curves that connect the two equilibria with a cost function measuring the deviation of a connecting curve from an integral curve of the vector field. Minimization deforms a trial curve until, at a global minimum, a connecting orbit is obtained. The method is robust, has no limitation on the dimension of the unstable manifold at the origin equilibrium, and does not suffer from exponential error amplification associated with time-marching a chaotic system. Owing to adjoint-based minimization techniques, no Jacobian matrices need to be constructed and the memory requirement scales linearly with the size of the problem. The robustness of the method is demonstrated for the one-dimensional Kuramoto-Sivashinsky equation.

nlin.CD↗

arXiv2023

Predicting chaotic statistics with unstable invariant tori

It has recently been speculated that statistical properties of chaos may be captured by weighted sums over unstable invariant tori embedded in the chaotic attractor of hyperchaotic dissipative systems; analogous to sums over periodic orbits formalized within periodic orbit theory. Using a novel numerical method for converging unstable invariant 2-tori in a chaotic PDE, we identify many quasiperiodic, unstable, invariant 2-torus solutions of a modified Kuramoto-Sivashinsky equation exhibiting hyperchaotic dynamics with two positive Lyapunov exponents. The set of tori covers significant parts of the chaotic attractor and weighted averages of the properties of the tori -- with weights computed based on their respective stability eigenvalues -- approximate statistics for the chaotic dynamics. These results are a step towards including higher-dimensional invariant sets in a generalized periodic orbit theory for hyperchaotic systems including spatio-temporally chaotic PDEs.

nlin.CD↗

arXiv2020

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

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 yet been demonstrated. Here we quantitatively analyze virion accumulation by accounting for aerosolized virion emission and destabilization. Reported superspreading events analyzed within this framework point towards aerosol mediated transmission of SARS-CoV-2. Virion exposure calculated for these events is found to trace out a single value, suggesting a universal minimum infective dose (MID) via aerosol that is comparable to the MIDs measured for other respiratory viruses; thus, the consistent infectious exposure levels and their commensurability to known aerosol-MIDs establishes the plausibility of aerosol transmission of SARS-CoV-2. Using filtration at a rate exceeding the destabilization rate of aerosolized SARS-CoV-2 can reduce exposure below this infective dose.

q-bio.PE↗

arXiv2020

Nondestructive Prediction of the Buckling Load of Imperfect Shells

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 and can not be predicted, which challenges the reliable design of such structures. Consequently, probabilistic descriptions in terms of empirical design rules are used and reliable design requires to be very conservative. We introduce a nonlinear description where finite-amplitude perturbations trigger buckling. Drawing from the analogy between imperfect shells which buckle and imperfect pipe flow which becomes turbulent, we experimentally show that lateral probing of cylindrical shells reveals their strength non-destructively. A new ridge-tracking method is applied to commercial cylinders with a hole showing that when the location where buckling is nucleated is known we can accurately predict the buckling load of each individual shell, within $\pm 5\%$. Our study provides a new promising framework to understand shell buckling, and more generally, imperfection-sensitive instabilities.

physics.app-ph↗

arXiv2020

Interaction and breakup of droplet pairs in a microchannel Y-junction

We combine theory, numerical simulation and experiments to investigate the breakup of two identical droplets entering a Y-junction with controlled spatial offset by which the second droplet trails the first. Based on fully resolved 3D simulations, we describe the flow physics leading to breakup. Scaling arguments, numerical simulation and experiments consistently show that for small initial offset, breakup always occurs with the droplet fragment volume depending linearly on the offset. Above a critical offset, which increases with the capillary number, the droplets enter the constriction sequentially without breakup. Our results are relevant for understanding the breakup behavior in a dense emulsion flowing through a linearly converging channel leading to a constriction. Such geometry is commonly used for the serial interrogation of droplet content in droplet microfluidic applications, where droplet breakup can limit the maximum throughput for such process. For capillary numbers up to $Ca\simeq10^{-2}$, the results from the two-droplet system in a Y-junction are consistent with breakup observations in dense emulsions flowing through a linearly converging channel. The deterministic relation between initial offset and resulting breakup in the two-droplet system suggests that the stochasticity that is observed in the breakup of a dense emulsion arises from multi-droplet interactions. The numerical value of the prefactor in the linear relation between initial offset and droplet fragment volume determined from experiments differs slightly from the one extracted from fully resolved numerical simulations. This discrepancy suggests that even at very high bulk surfactant concentrations, the rate-limiting surfactant adsorption kinetics allows for Marangoni stresses to develop and modify the droplet dynamics.

physics.flu-dyn↗

arXiv2020

Adjoint-based variational method for constructing periodic orbits of high-dimensional chaotic systems

Chaotic dynamics in systems ranging from low-dimensional nonlinear differential equations to high-dimensional spatio-temporal systems including fluid turbulence is supported by non-chaotic, exactly recurring time-periodic solutions of the governing equations. These unstable periodic orbits capture key features of the turbulent dynamics and sufficiently large sets of orbits promise a framework to predict the statistics of the chaotic flow. Computing periodic orbits for high-dimensional spatio-temporally chaotic systems remains challenging as known methods either show poor convergence properties because they are based on time-marching of a chaotic system causing exponential error amplification; or they require constructing Jacobian matrices which is prohibitively expensive. We propose a new matrix-free method that is unaffected by exponential error amplification, is globally convergent and can be applied to high-dimensional systems. The adjoint-based variational method constructs an initial value problem in the space of closed loops such that periodic orbits are attracting fixed points for the loop-dynamics. We introduce the method for general autonomous systems. An implementation for the one-dimensional Kuramoto-Sivashinsky equation demonstrates the robust convergence of periodic orbits underlying spatio-temporal chaos. Convergence does not require accurate initial guesses and is independent of the period of the respective orbit.

nlin.CD↗