Dynamics of a time-delayed relay system.

We study the dynamics of a piecewise-linear second-order delay differential equation that is representative of feedback systems with relays (switches) that actuate after a fixed delay. The system under study exhibits strong multirhythmicity, the coexistence of many stable periodic solutions for the same values of the parameters. We present a detailed study of these periodic solutions and their bifurcations. Starting from an integrodifferential model, we show how to reduce the system to a set of finite-dimensional maps. We then demonstrate that the parameter regions of existence of periodic solutions can be understood in terms of discontinuity-induced bifurcations and their stability is determined by smooth bifurcations. Using this technique, we are able to show that slowly oscillating solutions are always stable if they exist. We also demonstrate the coexistence of stable periodic solutions with quasiperiodic solutions.

Seeing double with a multifunctional reservoir computer.

Multifunctional biological neural networks exploit multistability in order to perform multiple tasks without changing any network properties. Enabling artificial neural networks (ANNs) to obtain certain multistabilities in order to perform several tasks, where each task is related to a particular attractor in the network's state space, naturally has many benefits from a machine learning perspective. Given the association to multistability, in this paper, we explore how the relationship between different attractors influences the ability of a reservoir computer (RC), which is a dynamical system in the form of an ANN, to achieve multifunctionality. We construct the "seeing double" problem in order to systematically study how a RC reconstructs a coexistence of attractors when there is an overlap between them. As the amount of overlap increases, we discover that for multifunctionality to occur, there is a critical dependence on a suitable choice of the spectral radius for the RC's internal network connections. A bifurcation analysis reveals how multifunctionality emerges and is destroyed as the RC enters a chaotic regime that can lead to chaotic itinerancy.

Transitional cluster dynamics in a model for delay-coupled chemical oscillators.

Cluster synchronization is a fundamental phenomenon in systems of coupled oscillators. Here, we investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. A voltage parameter in the experimental setup controls the onset of oscillations via a Hopf bifurcation. For a smaller voltage, the oscillators exhibit simple, so-called primary, clustering patterns, where all phase differences between each set of coupled oscillators are identical. However, upon increasing the voltage, secondary states, where phase differences differ, are detected, in addition to the primary states. Previous work on this system saw the development of a mathematical model that explained how the existence, stability, and common frequency of the experimentally observed cluster states could be accurately controlled by the delay time of the coupling. In this study, we revisit the mathematical model of the electrochemical oscillators in order to address open questions by means of bifurcation analysis. Our analysis reveals how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types. The analysis further reveals complex interconnectedness between branches of different cluster types. We find that each secondary state provides a continuous transition between certain primary states. These connections are explained by studying the phase space and parameter symmetries of the respective states. Furthermore, we show that it is only for a larger value of the voltage parameter that the branches of secondary states develop intervals of stability. For a smaller voltage, all the branches of secondary states are completely unstable and are, therefore, hidden to experimentalists.

Origin of the Unusual Ground-State Spin S = 9 in a Cr10 Single-Molecule Magnet.

The molecular wheel [Cr10(OMe)20(O2CCMe3)10], abbreviated {Cr10}, with an unusual intermediate total spin S = 9 and non-negligible cluster anisotropy, D/kB = -0.045(2) K, is a rare case among wheels based on an even number of 3d-metals, which usually present an antiferromagnetic (AF) ground state (S = 0). Herein, we unveil the origin of such a behavior. Angular magnetometry measurements performed on a single crystal confirmed the axial anisotropic behavior of {Cr10}. For powder samples, the temperature dependence of the susceptibility plotted as χT(T) showed an overall ferromagnetic (FM) behavior down to 1.8 K, whereas the magnetization curve M(H) did not saturate at the expected 30 µB/fu for 10 FM coupled 3/2 spin Cr3+ ions, but to a much lower value, corresponding to S = 9. In addition, the X-ray magnetic circular dichroism (XMCD) measured at high magnetic field (170 kOe) and 7.5 K showed the polarization of the cluster moment up to 23 µB/fu. The magnetic results can be rationalized within a model, including the cluster anisotropy, in which the {Cr10} wheel is formed by two semiwheels, each with four Cr3+ spins FM coupled (JFM/kB = 2.0 K), separated by two Cr3+ ions AF coupled asymmetrically (J23/kB = J78/kB = -2.0 K; J34/kB = J89/kB = -0.25 K). Inelastic neutron scattering and heat capacity allowed us to confirm this model leading to the S = 9 ground state and first excited S = 8. Single-molecule magnet behavior with an activation energy of U/kB = 4.0(5) K in the absence of applied field was observed through ac susceptibility measurements down to 0.1 K. The intriguing magnetic behavior of {Cr10} arises from the detailed asymmetry in the molecule interactions produced by small-angle distortions in the angles of the Cr-O-Cr alkoxy bridges coupling the Cr3+ ions, as demonstrated by ab initio and density functional theory calculations, while the cluster anisotropy can be correlated to the single-ion anisotropies calculated for each Cr3+ ion in the wheel.

Symmetry kills the square in a multifunctional reservoir computer.

The learning capabilities of a reservoir computer (RC) can be stifled due to symmetry in its design. Including quadratic terms in the training of a RC produces a "square readout matrix" that breaks the symmetry to quell the influence of "mirror-attractors," which are inverted copies of the RC's solutions in state space. In this paper, we prove analytically that certain symmetries in the training data forbid the square readout matrix to exist. These analytical results are explored numerically from the perspective of "multifunctionality," by training the RC to specifically reconstruct a coexistence of the Lorenz attractor and its mirror-attractor. We demonstrate that the square readout matrix emerges when the position of one attractor is slightly altered, even if there are overlapping regions between the attractors or if there is a second pair of attractors. We also find that at large spectral radius values of the RC's internal connections, the square readout matrix reappears prior to the RC crossing the edge of chaos.

Multifunctionality in a reservoir computer.

Multifunctionality is a well observed phenomenological feature of biological neural networks and considered to be of fundamental importance to the survival of certain species over time. These multifunctional neural networks are capable of performing more than one task without changing any network connections. In this paper, we investigate how this neurological idiosyncrasy can be achieved in an artificial setting with a modern machine learning paradigm known as "reservoir computing." A training technique is designed to enable a reservoir computer to perform tasks of a multifunctional nature. We explore the critical effects that changes in certain parameters can have on the reservoir computers' ability to express multifunctionality. We also expose the existence of several "untrained attractors"; attractors that dwell within the prediction state space of the reservoir computer were not part of the training. We conduct a bifurcation analysis of these untrained attractors and discuss the implications of our results.

Border-collision bifurcations in a driven time-delay system.

We show that a simple piecewise-linear system with time delay and periodic forcing gives rise to a rich bifurcation structure of torus bifurcations and Arnold tongues, as well as multistability across a significant portion of the parameter space. The simplicity of our model enables us to study the dynamical features analytically. Specifically, these features are explained in terms of border-collision bifurcations of an associated Poincaré map. Given that time delay and periodic forcing are common ingredients in mathematical models, this analysis provides widely applicable insight.

Dynamics of two identical mutually delay-coupled semiconductor lasers in photonic integrated circuits.

We theoretically investigate a system of two mutually delay-coupled semiconductor lasers, in a face to face configuration for integration in a photonic integrated circuit. This system is described by single-mode rate equations, which are a system of delay differential equations with one fixed delay. Several bifurcation scenarios involving multistabilities are presented, followed by a comprehensive frequency analysis of the symmetric and symmetry-broken, one-color and two-color states.

Asymmetric Pentagonal Metal Meshes for Flexible Transparent Electrodes and Heaters.

Metal meshes have emerged as an important class of flexible transparent electrodes. We report on the characteristics of a new class of asymmetric meshes, tiled using a recently discovered family of pentagons. Micron-scale meshes were fabricated on flexible polyethylene terephthalate substrates via optical lithography, metal evaporation (Ti 10 nm, Pt 50 nm), and lift-off. Three different designs were assessed, each with the same tessellation pattern and line width (5 µm), but with different sizes of the fundamental pentagonal unit. Good mechanical stability was observed for both tensile strain and compressive strain. After 1000 bending cycles, devices subjected to tensile strain showed fractional resistance increases in the range of 8-17%, while devices subjected to compressive strain showed fractional resistance increases in the range of 0-7%. The performance of the pentagonal metal mesh devices as visible transparent heaters via Joule heating was also assessed. Rapid response times (∼15 s) at low bias voltage (≤5 V) and good thermal resistance characteristics (213-258 °C cm2/W) were found using measured thermal imaging data. Deicing of an ice-bearing glass coupon on top of the transparent heater was also successfully demonstrated.

Influence of combined fundamental potentials in a nonlinear vibration energy harvester.

Ambient mechanical vibrations have emerged as a viable energy source for low-power wireless sensor nodes aiming the upcoming era of the 'Internet of Things'. Recently, purposefully induced dynamical nonlinearities have been exploited to widen the frequency spectrum of vibration energy harvesters. Here we investigate some critical inconsistencies between the theoretical formulation and applications of the bistable Duffing nonlinearity in vibration energy harvesting. A novel nonlinear vibration energy harvesting device with the capability to switch amidst individually tunable bistable-quadratic, monostable-quartic and bistable-quartic potentials has been designed and characterized. Our study highlights the fundamentally different large deflection behaviors of the theoretical bistable-quartic Duffing oscillator and the experimentally adapted bistable-quadratic systems, and underlines their implications in the respective spectral responses. The results suggest enhanced performance in the bistable-quartic potential in comparison to others, primarily due to lower potential barrier and higher restoring forces facilitating large amplitude inter-well motion at relatively lower accelerations.

Does Laser Surgery Interfere with Optical Nerve Identification in Maxillofacial Hard and Soft Tissue?--An Experimental Ex Vivo Study.

The protection of sensitive structures (e.g., nerves) from iatrogenic damage is of major importance when performing laser surgical procedures. Especially in the head and neck area both function and esthetics can be affected to a great extent. Despite its many benefits, the surgical utilization of a laser is therefore still limited to superficial tissue ablation. A remote feedback system which guides the laser in a tissue-specific way would provide a remedy. In this context, it has been shown that nerval structures can be specifically recognized by their optical diffuse reflectance spectra both before and after laser ablation. However, for a translation of these findings to the actual laser ablation process, a nerve protection within the laser pulse is of utmost significance. Thus, it was the aim of the study to evaluate, if the process of Er:YAG laser surgery--which comes with spray water cooling, angulation of the probe (60°) and optical process emissions--interferes with optical tissue differentiation. For the first time, no stable conditions but the ongoing process of laser tissue ablation was examined. Therefore, six different tissue types (nerve, skin, muscle, fat, cortical and cancellous bone) were acquired from 15 pig heads. Measurements were performed during Er:YAG laser ablation. Diffuse reflectance spectra (4500, wavelength range: 350-650 nm) where acquired. Principal component analysis (PCA) and quadratic discriminant analysis (QDA) were calculated for classification purposes. The clinical highly relevant differentiation between nerve and bone was performed correctly with an AUC of 95.3% (cortial bone) respectively 92.4% (cancellous bone). The identification of nerve tissue against the biological very similar fat tissue yielded good results with an AUC value of 83.4% (sensitivity: 72.3%, specificity: of 82.3%). This clearly demonstrates that nerve identification by diffuse reflectance spectroscopy works reliably in the ongoing process of laser ablation in spite of the laser beam, spray water cooling and the tissue alterations entailed by tissue laser ablation. This is an essential step towards a clinical utilization.

Absence of evidence ≠ evidence of absence: statistical analysis of inclusions in multiferroic thin films.

Assertions that a new material may offer particularly advantageous properties should always be subjected to careful critical evaluation, especially when those properties can be affected by the presence of inclusions at trace level. This is particularly important for claims relating to new multiferroic compounds, which can easily be confounded by unobserved second phase magnetic inclusions. We demonstrate an original methodology for the detection, localization and quantification of second phase inclusions in thin Aurivillius type films. Additionally, we develop a dedicated statistical model and demonstrate its application to the analysis of Bi(6)Ti(2.8)Fe(1.52)Mn(0.68)O18 (B6TFMO) thin films, that makes it possible to put a high, defined confidence level (e.g. 99.5%) to the statement of 'new single phase multiferroic materials'. While our methodology has been specifically developed for magnetic inclusions, it can easily be adapted to any other material system that can be affected by low level inclusions.

Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers.

We theoretically investigate the dynamics of two mutually coupled, identical single-mode semi-conductor lasers. For small separation and large coupling between the lasers, symmetry-broken one-color states are shown to be stable. In this case the light outputs of the lasers have significantly different intensities while at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable symmetry-broken two-color states, where both lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. Using a five-dimensional model, we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetry-broken one-color and two-color states. Several of these states give rise to multistabilities and therefore allow for the design of all-optical memory elements on the basis of two coupled single-mode lasers. The switching performance of selected designs of optical memory elements is studied numerically.

An odd-number limitation of extended time-delayed feedback control in autonomous systems.

We propose a necessary condition for the successful stabilization of a periodic orbit, using the extended version of time-delayed feedback control. This condition depends on the number of real Floquet multipliers larger than unity and is therefore related to the well-known odd-number limitation in non-autonomous systems. We show that the period of the orbit that is induced by mismatching the delay time of the control scheme and the period of the uncontrolled orbit plays an important role in the formulation of the odd-number limitation in the autonomous case.

Analytical limitation for time-delayed feedback control in autonomous systems.

We prove an analytical limitation on the use of time-delayed feedback control for the stabilization of periodic orbits in autonomous systems. This limitation depends on the number of real Floquet multipliers larger than unity, and is therefore similar to the well-known odd number limitation of time-delayed feedback control. Recently, a two-dimensional example has been found, which explicitly demonstrates that the unmodified odd number limitation does not apply in the case of autonomous systems. We show that our limitation correctly predicts the stability boundaries in this case.

Nonlinear dynamics of non-equilibrium holes in p-type modulation-doped GaInNAs/GaAs quantum wells.

Nonlinear charge transport parallel to the layers of p-modulation-doped GaInNAs/GaAs quantum wells (QWs) is studied both theoretically and experimentally. Experimental results show that at low temperature, T = 13 K, the presence of an applied electric field of about 6 kV/cm leads to the heating of the high mobility holes in the GaInNAs QWs, and their real-space transfer (RST) into the low-mobility GaAs barriers. This results in a negative differential mobility and self-generated oscillatory instabilities in the RST regime. We developed an analytical model based upon the coupled nonlinear dynamics of the real-space hole transfer and of the interface potential barrier controlled by space-charge in the doped GaAs layer. Our simulation results predict dc bias-dependent self-generated current oscillations with frequencies in the high microwave range.

Dynamics of light propagation in spatiotemporal dielectric structures.

Propagation, transmission and reflection properties of linearly polarized plane waves and arbitrarily short electromagnetic pulses in one-dimensional dispersionless dielectric media possessing an arbitrary space-time dependence of the refractive index are studied by using a two-component, highly symmetric version of Maxwell's equations. The use of any slow varying amplitude approximation is avoided. Transfer matrices of sharp nonstationary interfaces are calculated explicitly, together with the amplitudes of all secondary waves produced in the scattering. Time-varying multilayer structures and spatiotemporal lenses in various configurations are investigated analytically and numerically in a unified approach. Several effects are reported, such as pulse compression, broadening and spectral manipulation of pulses by a spatiotemporal lens, and the closure of the forbidden frequency gaps with the subsequent opening of wave number band gaps in a generalized Bragg reflector.

Improvement of time-delayed feedback control by periodic modulation: analytical theory of Floquet mode control scheme.

We investigate time-delayed feedback control schemes which are based on the unstable modes of the target state, to stabilize unstable periodic orbits. The periodic time dependence of these modes introduces an external time scale in the control process. Phase shifts that develop between these modes and the controlled periodic orbit may lead to a huge increase of the control performance. We illustrate such a feature on a nonlinear reaction diffusion system with global coupling and give a detailed investigation for the Rössler model. In addition we provide the analytical explanation for the observed control features.

Giant improvement of time-delayed feedback control by spatio-temporal filtering.

Control of spatio-temporal chaos by the time-delay autosynchronization method is improved by several orders of magnitude. Unstable time periodic patterns are efficiently stabilized if one employs filters and couplings which originate from the Floquet eigenvalue problem of the unstable orbit. We illustrate our scheme by an application to a globally coupled reaction-diffusion model which describes charge transport in semiconductor devices.