We study a stochastic version of the one-dimensional discrete nonlinear Schr & ouml;dinger equation (DNSE), which is derived from first principles, and thus possesses all the properties required by statistical mechanics, such as detailed balance and the H-theorem. The stochastic version shows disordered and localised dynamics, and displays a corresponding phase transition at a finite temperature value. The phase transition can be captured in a quantitative way by a mean-field type approach. The corresponding coarsening dynamics shows an unexpected dependence on the noise strength, which is reminiscent of stochastic resonance. The phase transition is linked with negative temperature phase transitions, which have been reported recently for the Hamiltonian dynamics of the DNSE. Our approach gives a clue to how these negative temperature phase transitions can be implemented in experimental setups, which are inevitably coupled to a positive temperature heat bath.

We present a noninvasive feedback control scheme to access unstable periodic orbits in dynamicmode atomic force microscopy. By utilizing this method we are able to explore the complete nonlinear frequency-response curves, including unstable branches between saddle-node bifurcations. The overall scheme is designed to be fast and derivative-free, which allows for simple experimental implementations. The control is closely related to phase-locked loop testing, a truncated Fourier discretization of periodic orbits, and an adaptive filter for online signal demodulation. We demonstrate the feedback control on various contact models for dynamic-mode atomic force microscopy and hope to motivate laboratory experiments by elucidating software and corresponding hardware implementations in detail.

Extended dynamic mode decomposition (EDMD) is a data-driven algorithm for approximating spectral data of the Koopman operator associated to a dynamical system, combining a Galerkin method of order N and collocation method of order M. Spectral convergence of this method subtly depends on appropriate choice of the space of observables. For chaotic analytic full branch maps of the interval, we derive a constraint between M and N guaranteeing spectral convergence of EDMD.

We present a data-driven feedback control scheme, which stabilizes unstable periodic orbits of dynamical systems. Model equations of the underlying system are assumed to be unavailable. Our approach relies only on previously measured input-state trajectories of a Poincare'\ map of the system. The success of a stabilizing feedback control in experimental systems typically depends on an adequate choice of control gains. The presented scheme computes stabilizing control gains by solving an optimization problem. No explicit model is identified and the control gains are computed directly from the measured trajectories. As a demonstration, we apply the scheme to a control-based continuation problem and obtain the corresponding bifurcation diagram.

We revisit aspects of dynamics and stability of localised states in the deterministic and stochastic discrete nonlinear Schr & ouml;dinger equation. By a combination of analytic and numerical techniques, we show that for deterministic motion localised initial conditions disperse if the strength of the nonlinear part drops below a threshold and that localised states are unstable in a noisy environment. As expected, the constants of motion in the nonlinear Schr & ouml;dinger equation play a crucial role. An infinite temperature state emerges when multiplicative noise is applied, while additive noise yields unbounded dynamics since conservation of normalisation is violated.

The nonlinear interaction between the tip of a scanning probe microscope (SPM) and a sample is manifested in the emergence of bifurcations and unstable branches in the frequency response of a driven cantilever. While extensively investigated theoretically, exploring the unstable branch in an actual SPM experiment is lacking so far, reflecting the broader challenge in studying mechanical nanojunction oscillators under strongly varying external forces. Here we demonstrate experimental tracking of unstable periodic orbits between two saddle-node bifurcation points in the attractive regime, revealing the full set of stationary oscillatory states. This is achieved by a minimally invasive control scheme based on fast adaptive phase extraction and Fourier discretisation of the tip dynamics. Stabilization of unstable branches of oscillating AFM cantilevers opens avenues for novel experimental modes, potentially enabling ultrasensitive surface detection at considerably large amplitudes with minimal tip-surface interaction, new insights in tip-surface interaction mechanisms, as well as new AFM modes enabling arbitrary setpoint choice while inherently avoiding discontinuities.

Building on tools that have been successfully used in the study of rational billiards, such as induced maps and interval exchange transformations, we provide a construction of a one-parameter family of isosceles triangles exhibiting non-periodic trajectories that are not everywhere dense. This provides, by elementary means, a definitive negative answer to a long-standing open question on the density of non-periodic trajectories in triangular billiards.

We show that spectral data of the Koopman operator arising from an analytic expanding circle map τ can be effectively calculated using an EDMD-type algorithm combining a collocation method of order m with a Galerkin method of order n. The main result is that if m≥δn, where δ is an explicitly given positive number quantifying by how much τ expands concentric annuli containing the unit circle, then the method converges and approximates the spectrum of the Koopman operator, taken to be acting on a space of analytic hyperfunctions, exponentially fast in n. Additionally, these results extend to more general expansive maps on suitable annuli containing the unit circle.

Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.

The striatum as part of the basal ganglia is central to both motor, and cognitive functions. Here, we propose a large-scale biophysical network for this part of the brain, using modified Hodgkin-Huxley dynamics to model neurons, and a connectivity informed by a detailed human atlas. The model shows different spatio-temporal activity patterns corresponding to lower (presumably normal) and increased cortico-striatal activation (as found in, e.g., obsessive-compulsive disorder), depending on the intensity of the cortical inputs. By applying equation-free methods, we are able to perform a macroscopic network analysis directly from microscale simulations. We identify the mean synaptic activity as the macroscopic variable of the system, which shows similarity with local field potentials. The equation-free approach results in a numerical bifurcation and stability analysis of the macroscopic dynamics of the striatal network. The different macroscopic states can be assigned to normal/healthy and pathological conditions, as known from neurological disorders. Finally, guided by the equation-free bifurcation analysis, we propose a therapeutic close loop control scheme for the striatal network.

Numerical continuation tools are nowadays standard to analyse nonlinear dynamical systems by numerical means. These powerful methods are unfortunately not available in real experiments without having access to an accurate mathematical model. Implementing such a concept in real world experiments using control and data processing to track unstable states and their bifurcations, requires robust control techniques with large basins and good global properties. Here we propose design principles for control techniques for periodic states which lead to large basins and which are robust, without the need to have access to a detailed mathematical model. Our analytic considerations for the control design will be based on weakly nonlinear analysis of periodically driven oscillator systems. We then demonstrate by numerical means that in strong nonlinear regimes successful control with large basins of attraction can be achieved when only plain time series data are available.

Ray-tracing is a well established approach for modelling wave propagation at high frequencies, in which the ray trajectories are defined by a Hamiltonian system of ODEs. An approximation of the wave amplitude is then derived from estimating the density of rays in the neighbourhood of a given evaluation point. An alternative approach is to formulate the ray-tracing model directly in terms of the ray density in phase-space using the Liouville equation. The solutions may then be expressed in integral form using the Frobenius-Perron (F-P) operator, which is a transfer operator transporting the ray density along the trajectories. The classical approach for discretising such operators dates back to 1960 and the work of Stanislaw Ulam. The convergence of the Ulam method has been established in some cases, typically in low dimensional settings with continuous densities and hyperbolic dynamics. In this chapter, we outline some recent work investigating the convergence of the Ulam method for ray tracing in triangular billiards, where the dynamics are parabolic and the flow map contains jump discontinuities.

This paper presents a framework to perform bifurcation analysis in laboratory experiments or simulations. We employ control-based continuation to study the dynamics of a macroscopic variable of a microscopically defined model, exploring the potential viability of the underlying feedback control techniques in an experiment. In contrast to previous experimental studies that used iterative root-finding methods on the feedback control targets, we propose a feedback control law that is inherently non-invasive. That is, the control discovers the location of equilibria and stabilizes them simultaneously. We call the proposed control zero-in-equilibrium feedback control and we prove that it is able to stabilize branches of equilibria, except at singularities of codimension n+1, where n is the number of state space dimensions the feedback can depend on. We apply the method to a simulated evacuation scenario were pedestrians have to reach an exit after maneuvering left or right around an obstacle. The scenario shows a hysteresis phenomenon with bistability and tipping between two possible steady pedestrian flows in microscopic simulations. We demonstrate for the evacuation scenario that the proposed control law is able to uniformly discover and stabilize steady flows along the entire branch, including points where other non-invasive approaches to feedback control become singular.

Numerical continuation tools are nowadays standard methods for the bifurcation analysis of dynamical systems. Unfortunately, the full power of these methods is still unavailable in experiments, in particular, if no underlying mathematical model is at hand. We here aim to narrow this gap by providing control based continuation of periodic states which can be ultimately implemented in real-world experimental set-ups. Taking inspiration from atomic force microscopy, we develop experimentally relevant control and tracking tools for time periodic solutions in driven nonlinear oscillator systems based on stroboscopic maps.

We identify a symmetry induced mechanism which dominates the long time behaviour in symmetric triangular billiards. We rigorously prove the existence of invariant sets in symmetric irrational billiards on which the dynamics is governed by an interval exchange transformation. Counterintuitively, this property of symmetric irrational billiards is analogous to the case of general rational billiards, and it highlights the non-trivial impact of symmetries in non-hyperbolic dynamical systems. Our findings provide an explanation for the logarithmic subdiffusive relaxation processes observed in certain triangular billiards. In addition we are able to settle a long standing conjecture about the existence of non-periodic and not everywhere dense trajectories in triangular billiards.

The motion of pedestrians is a paradigmatic phenomenon to study collective human behavior. We propose a model-free approach to analyze the movement of pedestrians in experiments and get a quantitative understanding of crowd dynamics. Using concepts from control and analysis of dynamical systems, we set up a scheme which allows us to identify dynamical unstable signatures in pedestrian flows. These signatures are the building blocks for crowd control and soft management of people and thus result in a fundamental understanding of collective human behavior. Our approach is entirely data driven, and we provide a proof of concept using field and laboratory experiments. In addition, this methodology provides, based on experimental observations, quantitative benchmarks to judge the quality of mathematical models for pedestrian motion.

Polygonal billiards constitute some of the simplest yet counterintuitive dynamical systems in physics. Even basic features of the dynamics, such as ergodicity of the microcanonical distribution or the decay of correlations have not been settled in general. In this Letter, we will highlight the importance of symmetries of the billiard table for the resulting dynamics. Although typical triangular billiards appear to show correlation decay, symmetric billiards may not even be ergodic with respect to the uniform distribution in phase space.

A complete description of resonances for rational toral Anosov diffeomorphisms preserving certain Reinhardt domains is presented. As a consequence it is shown that every homotopy class of two-dimensional Anosov diffeomorphisms contains maps with the sequence of resonances decaying stretched-exponentially. This is achieved by introducing a certain group of rational toral diffeomorphisms and computing the resonances of the respective composition operator considered on suitable anisotropic spaces of hyperfunctions. The class of examples is sufficiently rich to also include non-linear Anosov maps with trivial resonances, or resonances decaying exponentially, as well as with or without area-preservation or reversing symmetries.

Euler solved the problem of the collapse of tall thin columns under unexpectedly small loads in 1744. The analogous problem of the collapse of circular elastic rings or tubes under external pressure was mathematically intractable and only fully solved recently. In the context of carbon nanotubes, an additional phenomenon was found experimentally and in atomistic simulations but not explained: the collapse pressure of smaller diameter tubes deviates below the continuum mechanics solution [Torres-Dias et al., Carbon 123, 145 (2017)]. Here, this deviation is shown to occur in discretized straight columns and it is fully explained in terms of the phonon dispersion curve. This reveals an unexpected link between the static mechanical properties of discrete systems and their dynamics described through dispersion curves.