Higher-order Coupling Terms Research Articles

Cyclops states in repulsive theta-neuron networks

Networks of phase oscillators have become a widely established paradigmatic model for studying emergent collective behavior across several real-world systems, including neuronal networks, populations of chemical oscillators, and power grids. The Kuramoto model, involving one-dimensional or two-dimensional phase oscillators, demonstrates the potential for networks to showcase exceptional collective dynamics. This encompasses various outcomes such as full, partial, explosive, and asymmetry-induced synchronization, clusters, chimeras, solitary states, and generalized splay states. Notably, increasing all-to-all coupling in the classical Kuramoto model induces full synchronization as the most probable outcome and dominant rhythm. Kuramoto networks with repulsive coupling usually display splay, generalized, and cluster splay states, but the conditions under which a certain rhythm can arise and prevail are not entirely understood. Equally important for connecting Kuramoto networks to practical physical systems is understanding the function of higher-order coupling terms. These terms display a Fourier decomposition of a general 2π-periodic interaction function [1]. Previous studies have demonstrated that the inclusion of higher-order terms in the classical Kuramoto model of oscillators with all-to-all attractive coupling can lead to multiple synchronous states and switching between synchronization clusters. However, the impact of higher-order coupling modes on rhythm generation in repulsive networks remains unexplored. In this work, we present significant progress in addressing the critical issue related to repulsive Kuramoto–Sakaguchi networks of phase oscillators with phase-lagged first-order and higher-order coupling. We demonstrate that weakly repulsive networks of even and odd numbers of oscillators with first-order coupling are dominated by two-cluster and three-cluster splay states, respectively. The three-cluster splay states consist of two distinct coherent clusters and one solitary oscillator. These tripod states can be considered a fusion of a two-body chimera and a solitary state. We have dubbed these patterns of three oscillators as “Cyclops states” in reference to the Greek mythological giant with a single eye. The solitary oscillator and synchronous clusters respectively represent the Cyclops’ eye and shoulders. We present a remarkable discovery that the inclusion of higher-order coupling modes leads to worldwide stability of cyclops states across almost the entire range of the phase-lag parameter controlling repulsion [2]. Beyond the Kuramoto oscillators, we demonstrate the robust presence of this effect in networks of canonical theta-neurons with adaptive coupling. Furthermore, our results provide insight into identifying dominant rhythms within repulsive physical and biological networks.

Adaptive Intelligent Controller Design-Based ISS Modular Approach for Uncertain Nonlinear Systems With Time-Varying Full-State Constraints

The present article focuses on an input-to-state stability (ISS) problem of nonlinear strict-feedback system with time-varying full-state constraints. The adaptive neural controllers are designed by using backstepping, barrier Lyapunov function (BLF), and ISS small-gain approach. BLF is proved to satisfy the small-gain condition; thus, the conception of input-to-state stable time-varying BLF (ISSTVBLF) is produced. Also, the ISSTVBLF is novel embedded into the back-stepping design for a subsystem, which can prevent the full-state constraints from being violated all the time and avoid the designed difficulties resulting from the requirement of the entire BLF for the time-varying constrained system. The neural network approximate technique is utilized to estimate the unknown nonlinear functions. Then, a systematic procedure is developed to derive new adaptive tracking controllers based on the small-gain theorem. It is proved that all the closed-loop signals are semiglobal, uniform, and ultimate boundedness and the tracking error is driven to a small domain around zero. Finally, simulation study results illustrate the effectiveness of the proposed control schemes. <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>Impact Statement</i>—There are constraints in various systems. In this paper a strict feedback system with time-varying full-state constraints is studied. Handling the time-varying unknown nonlinearities and high-order coupling terms is a difficult question. It tends to be conservative to choose transformation functions to transform the constrained system to an unconstrained one. The controllers introduced in this paper, which are constructed by applying back-stepping, BLF and ISS small-gain approach, can overcome the difficulty of designing an overall Lyapunov function. Checking ISS small-gain condition is another difficulty, we construct a novel class K function for BLF in the design process, then the subsystem of the imbedded BLF is proved to be ISS. Simulation study results illustrate the effectiveness of proposed control schemes.

Beyond linear coupling in microwave optomechanics

We explore the nonlinear dynamics of a cavity optomechanical system. Our realization consisting of a drumhead nano-electro-mechanical resonator (NEMS) coupled to a microwave cavity, allows for a nearly ideal platform to study the nonlinearities arising purely due to radiation-pressure physics. Experiments are performed under a strong microwave Stokes pumping which triggers mechanical self-sustained oscillations. We analyze the results in the framework of an extended nonlinear optomechanical theory, and demonstrate that quadratic and cubic coupling terms in the opto-mechanical Hamiltonian have to be considered. Quantitative agreement with the measurements is obtained considering only genuine geometrical nonlinearities: no thermo-optical instabilities are observed, in contrast with laser-driven systems. Based on these results, we describe a method to quantify nonlinear properties of microwave optomechanical devices. Such a technique, available now in the quantum electro-mechanics toolbox, but completely generic, is mandatory for the development of new schemes where higher-order coupling terms are proposed as a new resource, like Quantum Non-Demolition measurements, or in the search for new fundamental quantum signatures, like Quantum Gravity. We also find that the motion imprints a wide comb of extremely narrow peaks in the microwave output field, which could also be exploited in specific microwave-based measurements, potentially limited only by the quantum noise of the optical and the mechanical fields for a ground-state cooled NEMS device.

Multiple-Scale Analysis of a Tunable Bi-Stable Piezoelectric Energy Harvester

Abstract This paper presents the theoretical modeling and multiple-scale analysis of a novel piezoelectric energy harvester composed of a metal cantilever beam, piezoelectric films, and an axial preload spring at the moveable end. The harvester experiences mono- and bi-stable regimes as the stiffness of preload spring increases. The governing equations are derived with two high-order coupling terms induced by the axial motion. The literature shows that these high-order coupling terms lead to tedious calculations in the stability analysis of solutions. This work introduces an analytical strategy and the implementation of the multiple-scale method for the harvester in either the mono- or bi-stable status. Numerical simulations are performed to verify the analytical solutions. The influence of the electrical resistance, excitation level, and the spring pre-deformation on the voltage outputs and dynamics are investigated. The spring pre-deformation has a slight influence on the energy harvesting performance of the mono-stable system, but a large effect on that of the bi-stable system.

Stabilization of bianisotropic fluid models for time domain simulation

Willis fluids are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This effective dynamic response coupling may arise due to microstructural asymmetry, long range order, or time-varying material properties and has been shown to be analogous to electromagnetic bianisotropy [Phys. Rev. B 96, 104303 (2017)]. Recent work has shown that, when wavevectors exceed a critical magnitude, the currently employed model for passive Willis fluids possesses a linear instability stemming from the truncation of higher-order coupling terms in the derivation of the constitutive relationships [J. Acoust. Soc. Am. 144(3), 1832 (2018)]. While this instability can be avoided when using frequency domain methods, it poses significant problems for time domain simulations, as short-wavelength numerical noise may grow exponentially. In the present work, we report on methods of stabilizing the Willis fluid model. Specifically, we consider mathematical manipulations of the wave equation derived from the lowest-order constitutive equations, similar to methods that have been explored in continuum approximations of nonlinear spring-mass chains, as well as higher-order coupling terms. Physical interpretations and implications for time domain finite difference and finite element simulations are discussed. [Work supported by the postdoctoral program at ARL:UT.]Willis fluids are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This effective dynamic response coupling may arise due to microstructural asymmetry, long range order, or time-varying material properties and has been shown to be analogous to electromagnetic bianisotropy [Phys. Rev. B 96, 104303 (2017)]. Recent work has shown that, when wavevectors exceed a critical magnitude, the currently employed model for passive Willis fluids possesses a linear instability stemming from the truncation of higher-order coupling terms in the derivation of the constitutive relationships [J. Acoust. Soc. Am. 144(3), 1832 (2018)]. While this instability can be avoided when using frequency domain methods, it poses significant problems for time domain simulations, as short-wavelength numerical noise may grow exponentially. In the present work, we report on methods of stabilizing the Willis fluid model. Specifically, we consider mathema...

Stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity

Scalar-tensor theories of gravity where a new scalar degree of freedom couples to the Gauss-Bonnet invariant can exhibit the phenomenon of spontaneous black hole scalarization. These theories admit both the classic black holes predicted by general relativity as well as novel hairy black hole solutions. The stability of the hairy black holes is strongly dependent on the precise form of the scalar-gravity coupling. A radial stability investigation revealed that all scalarized black hole solutions are unstable when the coupling between the scalar field and the Gauss-Bonnet invariant is quadratic in the scalar, whereas stable solutions exist for exponential couplings. Here we elucidate this behavior. We demonstrate that, while the quadratic term controls the onset of the tachyonic instability that gives rise to the black hole hair, the higher-order coupling terms control the nonlinearities that quench that instability, and hence also control the stability of the hairy black hole solutions.

Generic 3D formulation for sliding isolators with variable curvature and its experimental verification

Sliding isolators with variable curvature (SIVCs), whose isolation stiffness can be continuously varied along with isolator displacement, allow a passive isolation system to adapt to seismic excitations. The variable stiffness also has the advantage of mitigating the long-period resonance effect when the isolation system is subjected to ground motion with strong long-period components. However, most previous studies on SIVCs were theoretical and adopted simplified models that omit the tri-directional coupled motion due to the geometric effect of the concave sliding surface. In order to provide a more precise analytical tool, this study developed a set of three-dimensional formulas based on Lagrange’s equation of motion to predict the complete dynamic forces of an SIVC system transmitted to the superstructure. The formulas, which are applicable to all types of axially symmetric SIVCs, were then verified experimentally using a shaking table test that involved an SIVC system consisting of four polynomial friction pendulum isolators (PFPIs). The derived formulas show that the horizontal and vertical forces of the SIVC include two high-order nonlinear terms related to the curvature and slope of the SIVC sliding surface. In conjunction with the three-dimensional sliding motion of the isolator, these high-order terms cause the tri-directional coupling effect. The experimental results of the shaking table test verified the presence of this dynamic coupling effect and validated the proposed analytical model. Using the isolator parametrical values obtained from the test, the peak responses of the PFPI system subjected to 165 sets of bidirectional horizontal ground motions with various intensity levels were simulated. The simulation results indicate that the high-order coupling terms have a limited influence on isolator drift, but have a notable effect on the shear and vertical forces of the PFPIs when the motion of the isolation system is enlarged by increased seismic intensity.

Cooperative Couplings between Octahedral Rotations and Ferroelectricity in Perovskites and Related Materials.

The structure of ABO_{3} perovskites is dominated by two types of unstable modes, namely, the oxygen octahedral rotation (AFD) and ferroelectric (FE) mode. It is generally believed that such AFD and FE modes tend to compete and suppress each other. Here we use first-principles methods to show that a dual nature of the FE-AFD coupling, which turns from competitive to cooperative as the AFD mode strengthens, occurs in numerous perovskite oxides. We provide a unified model of such a dual interaction by introducing novel high-order coupling terms and explain the atomistic origin of the resulting new form of ferroelectricity in terms of universal steric mechanisms. We also predict that such a novel form of ferroelectricity leads to atypical behaviors, such as an enhancement of all the three Cartesian components of the electric polarization under hydrostatic pressure and compressive epitaxial strain.

Tracking control for strict-feedback nonlinear systems with time-varying full state constraints

This paper focuses on the tracking control problem for strict-feedback nonlinear systems subject to asymmetric time-varying full state constraints. Time-varying asymmetric barrier Lyapunov functions are employed to ensure time-varying constraint satisfaction. By allowing the barriers to vary with the desired trajectory in time, the initial condition requirements are relaxed. High-order coupling terms caused by backstepping are cancelled through a novel variable substitution for the first time. Besides the normal case, where the full knowledge of the system is available, we also handle scenarios of parametric uncertainties. Asymptotic tracking is achieved without violation of any constraints, and all signals in the closed-loop system are ultimately bounded. State-constrained systems with input saturation and bounded disturbances are also considered; the tracking error converges to a bounded set around zero. The performance of the asymmetric-barrier-Lyapunov-function-based control is illustrated through a numerical example.

Evolution of magnetic and dielectric properties in Sr-substituted high-temperature multiferroic YBaCuFeO5

We report the evolution of structural, magnetic and dielectric properties due to the partial substitution of Ba by Sr in the high-temperature multiferroic YBaCuFeO5. This compound exhibits ferroelectric and antiferromagnetic transitions around 200 K and these two phenomena are presumed to be coupled with each other. Our studies on magnetic and dielectric properties of the (, 0.25 and 0.5) show that the substitution of Sr shifts the magnetic transition towards higher temperature, whereas the dielectric transition to lower temperature. These results indicate that the magnetic and dielectric transitions might not be interrelated in the Sr-substituted YBaCuFeO5. The nature of magnetodielectric coupling changes across the series with the presence of higher-order coupling terms. Additionally, in these compounds glassy dynamics of electric dipoles is observed at low temperatures.

Local well-posedness to inhomogeneous Ericksen–Leslie system with general Leslie stress tensor

In this paper, we establish the local well-posedness of strong solutions to the Cauchy problem of the density-dependent liquid crystal system, with general Leslie stress tensor, in \(\mathbb {R}^3\). By using a biharmonic regularization argument, and making full use of the property of the intrinsic cancellation of the regularized system, we are able to overcome the difficulties caused by the high-order coupling terms, establish some appropriate a priori estimates, which are independent of the regularized parameter, and consequently obtain a strong solution to the original liquid crystal system, by letting the regularization parameter go to zero. The main advantage of the biharmonic regularization is that it does not destroy the property of the intrinsic cancellation of the original system.

Local well-posedness of strong solutions to density-dependent liquid crystal system

In this paper, we study the Cauchy problem to the density-dependent liquid crystal system in R3. We establish the local existence and uniqueness of strong solutions to this system. In order to overcome the difficulties caused by the high order coupling terms, a biharmonic regularization of the system, as an auxiliary system, is introduced, and we make full use of the intrinsic cancellation properties between the high order coupling terms.

On the Non Linear Behaviour of Nematic Liquid Single Crystal Elastomers (NLSCE) Under Biaxial Mechanic and Electrical Force Fields

A continuum theory has been developed to give account of the three characteristic zones appearing in the “semisoft” behaviour of nematic single crystal elastomers under the effect of biaxial mechanic and electrical force fields considering higher order coupling terms. The analysis is based on a model which takes into account two coexistent coupled components showing a preferred direction. Two cases are analysed according to the type of input: stress and displacement. Bifurcation phenomena are predicted in the first case. Sress-strain curves show the characteristic features experimentally observed. The case where shear terms in the strain are considered is approximately analysed.

Visual Servoing Tracking Control of a Ball and Plate System: Design, Implementation and Experimental Validation

This paper presents the design, implementation and validation of real-time visual servoing tracking control for a ball and plate system. The position of the ball is measured with a machine vision system. The image processing algorithms of the machine vision system are pipelined and implemented on a field programmable gate array (FPGA) device to meet real-time constraints. A detailed dynamic model of the system is derived for the simulation study. By neglecting the high-order coupling terms, the ball and plate system model is simplified into two decoupled ball and beam systems, and an approximate input-output feedback linearization approach is then used to design the controller for trajectory tracking. The designed control law is implemented on a digital signal processor (DSP). The validity of the performance of the developed control system is investigated through simulation and experimental studies. Experimental results show that the designed system functions well with reasonable agreement with simulations.

Towards a higher-order description of Jahn–Teller coupling effects in molecular spectroscopy: The [formula omitted] state of NO 3

Static and dynamic Jahn–Teller coupling effects in the 2E″ excited electronic state of the nitrate radical (NO 3) are analyzed theoretically. An earlier vibronic coupling model proposed by us [S. Mahapatra, W. Eisfeld, H. Köppel, Chem. Phys. Lett. 441 (2007) 7] is extended here and the importance of higher-order coupling terms established. More specifically, the earlier analysis rests on a quadratic-coupling approach where terms up to second order in the nuclear displacements are retained. In the present analysis we find coupling terms of 3rd and 4th order to be important. This is related to low-energy dissociation thresholds in the system which render the (diabatic) potential energy surfaces anharmonic already for low excitation energies. The results are compared with the experimental photodetachment spectrum of Neumark and coworkers [A. Weaver, D.W. Arnold, S.E. Bradforth, D.M. Neumark, J. Chem. Phys. 94 (1991) 1740], and the agreement is much improved when the additional coupling terms are taken into account.

Pseudorotation in fullerene anions

Jahn-Teller (JT) problems are often characterised by an adiabatic potential energy surface (APES) containing either a set of isoenergetic wells or a trough of equivalent-energy points, which may be warped by higher-order coupling terms or anisotropic effects. In all three cases, the JT effect will be dynamic. Either tunnelling between the wells or rotation (of a distortion) around the trough will restore the original symmetry of the system. This motion is referred to as pseudorotation. It should be possible to observe a JT system in a distorted geometry if measurements are made on a sufficiently short timescale. In various cubic systems, this timescale has been calculated to be the order of picoseconds. Such timescales are accessible using modern methods of ultrafast spectroscopy. Measurements of pseudorotation rates can lead to important information on the strength and nature of the JT coupling present. We will present analytical calculations that allow the rate of pseudorotation to be determined in terms of the vibronic coupling parameters. We will show how these results can be applied to E ⊗ e systems and then to the more complicated system applicable to C 60 - anions. This is of particular interest because of the high icosahedral symmetry of fullerene ions and also because of the many potential uses of materials containing these ions. We conclude by outlining experiments that should be capable of measuring pseudorotation in C 60 anions.

Photoionization-induced dynamics of ammonia: Ab initio potential energy surfaces and time-dependent wave packet calculations for the ammonia cation

An analytical anharmonic six-dimensional three-sheeted potential energy surface for the ground and first excited states of the ammonia cation has been developed which is tailored to model the ultrafast photoinduced dynamics. Selected ab initio cuts, obtained by multireference configuration interaction calculations, have been used to determine the parameters of a diabatic representation for this Jahn-Teller and pseudo-Jahn-Teller system. The model includes higher-order coupling terms both for the Jahn-Teller and for the pseudo-Jahn-Teller matrix elements. The relaxation to the ground state is possible via dynamical pseudo-Jahn-Teller couplings involving the asymmetric bending and stretching coordinates. The photoelectron spectrum of NH3 and the internal conversion dynamics of NH3+ have been determined by wave packet propagation calculations employing the multiconfigurational time-dependent Hartree method. Three different time scales are found in the dynamics calculations for the second absorption band. The ultrafast Jahn-Teller dynamics of the two excited states occurs on a 5 fs time scale. The major part of the internal conversion to the ground state takes place within a short time scale of 20 fs. This fast internal conversion is, however, incomplete and the remaining excited state population does not decay completely even within 100 fs.

Temperature Dependence of Vibrational Energy Transfer in a Protein Molecule

The anharmonic dynamics of a protein molecule was studied by molecular dynamics simulations of the intramolecular vibrational energy transfer in myoglobin. A small excess kinetic energy was added to a specified normal mode, and the process of the energy transfer to other modes was observed. It was found that the vibrational energy was transferred by two distinct mechanisms depending on temperature. Near zero temperature, the vibrational energy is transferred as a process of the Fermi resonance mostly through the third-order coupling terms from one mode to only a limited number of modes, satisfying the resonance condition. As the temperature increases, the resonance-type transfer is dominated by the off-resonance energy transfer through various mode-coupling terms. Near room temperature, the energy transfer involves higher-order coupling terms and indirect processes through intermediate modes in addition to the transfers through the lower-order couplings. In the short-time limit immediately after starting ...

Coupled Bending-Bending-Torsion Vibration of a Pre-Twisted Beam with Aerofoil Cross-Section by the Finite Element Method

The present study deals with a finite element model for coupled bending-bending-torsion vibration analysis of a pretwisted Timoshenko beam with varying aerofoil cross-section. The element derived in this paper has two nodes, with seven degrees of freedom at each node. The nodal variables are transverse displacements, cross-section rotations and the shear angles in two planes and torsional displacement. The advantage of the present element is the exclusion of unnecessary derivatives of fundamental nodal variables, which were included to obtain invertable square matrix by other researchers, by choosing proper displacement functions and using relationship between cross-sectional rotation and the shear deformation. Element stiffness and mass matrices are developed from strain and kinetic energy expressions by assigning proper order polynomial expressions for cross-section properties and considering higher order coupling coefficients. The correctness of the present model is confirmed by the experimental results available in the literature. Comparison of the proposed model results with those in the literature indicates that a faster convergence is obtained. The results presented also provide some insights in the formulation by clearly indicating that higher order coupling terms have considerable influence on the natural frequencies.

Mechanisms of exchange bias: partial wall pinning, and fluctuations

Exchange bias is a phenomena associated with magnetization and hysteresis in ferromagnet/antiferromagnet bilayers. Prospects for control and enhancement of exchange bias effects make it important for applications to several spin electronic devices, and requires a clear understanding of underlying mechanisms. A general theoretical formulation of the bias problem in terms of partial magnetic domain wall formation and pinning is outlined. A key result is the existence of higher order coupling terms when more than one sublattice of the antiferromagnet is present at the interface. Results from calculations of finite temperature effects on bias and coercivity are described, and the concept of viscosity in the antiferromagnet is discussed.