Unperturbed Case Research Articles

Dissipative frequency converter: From Lindblad dynamics to non-Hermitian topology

A topological frequency converter represents a dynamical counterpart of the integer quantum Hall effect, where a two-level system enacts a quantized time-averaged power transfer between two driving modes of incommensurate frequency. Here, we investigate as to what extent temporal coherence in the quantum dynamics of the two-level system is important for the topological quantization of the converter. To this end, we consider dissipative channels corresponding to spontaneous decay and dephasing in the instantaneous eigenbasis of the Hamiltonian as well as spontaneous decay in a fixed basis. The dissipation is modelled using both a full Lindblad and an effective non-Hermitian (NH) Hamiltonian description. For all three dissipation channels, we find a transition from the unperturbed dynamics to a quantum watchdog effect, which destroys any power transfer in the strong coupling limit. This is striking because the watchdog effect leads to perfectly adiabatic dynamics in the instantaneous eigenbasis, at first glance similar to the unperturbed case. Furthermore, it is found that dephasing immediately leads to an exponential decay of the power transfer in time due to loss of polarization in the mixed quantum state. Finally, we discuss the appearance in the effective NH trajectory description of nonadiabatic processes, which are suppressed in the full Lindblad dynamics. Published by the American Physical Society 2024

Discrete-time sliding mode control with inverse hyperbolic sine reaching law

Discrete-time sliding mode control (DTSMC) is a popular control strategy used in applications like drones, aerospace, and automotive due to its robustness against parameter uncertainties and external disturbances. One of the crucial components of DTSMC is the reaching law, which governs the motion of the system towards the sliding surface. In this paper, a novel reaching law based on the inverse hyperbolic sine function is proposed. Initially, we examine the unperturbed (nominal) system case, followed by a subsequent scenario involving unidentified disturbances i.e. for the perturbed system. It is observed that the proposed reaching law not only decreases the control effort but also results in a smoother control input characterized by reduced chattering within a low quasi-sliding band. The simulation results show that the proposed reaching law outperforms the existing reaching law (seminal work of Gao et al.) in terms of both lower control effort and quasi-sliding mode band. Additionally, the proposed reaching law is compared with a similar class of reaching law i.e. with the hyperbolic tangent reaching law and the inverse tangent reaching law. The comparison results demonstrate superior performance in terms of robustness, further demonstrating the effectiveness of the proposed reaching law.

Investigating pseudo parabolic dynamics through phase portraits, sensitivity, chaos and soliton behavior

This research examines pseudoparabolic nonlinear Oskolkov-Benjamin-Bona-Mahony-Burgers (OBBMB) equation, widely applicable in fields like optical fiber, soil consolidation, thermodynamics, nonlinear networks, wave propagation, and fluid flow in rock discontinuities. Wave transformation and the generalized Kudryashov method is utilized to derive ordinary differential equations (ODE) and obtain analytical solutions, including bright, anti-kink, dark, and kink solitons. The system of ODE, has been then examined by means of bifurcation analysis at the equilibrium points taking parameter variation into account. Furthermore, in order to get insight into the influence of some external force perturbation theory has been employed. For this purpose, a variety of chaos detecting techniques, for instance poincaré diagram, time series profile, 3D phase portraits, multistability investigation, lyapounov exponents and bifurcation diagram are implemented to identify the quasi periodic and chaotic motions of the perturbed dynamical model. These techniques enabled to analyze how perturbed dynamical system behaves chaotically and departs from regular patterns. Moreover, it is observed that the underlying model is quite sensitivity, as it changing dramatically even with slight changes to the initial condition. The findings are intriguing, novel and theoretically useful in mathematical and physical models. These provide a valuable mechanism to scientists and researchers to investigate how these perturbations influence the system’s behavior and the extent to which it deviates from the unperturbed case.

Nonlinear diffusion on networks: Perturbations and consensus dynamics

<p>In this paper, we study a class of equations representing nonlinear diffusion on networks. A particular instance of our model could be seen as a network equivalent of the porous-medium equation. We are interested in studying perturbations of such a system and describing the consensus dynamics. The nonlinearity of the equations gives rise to potentially intricate structures of equilibria that could intersect the consensus space, creating singularities. For the unperturbed case, we characterize the sets of equilibria by exploiting the symmetries under group transformations of the nonlinear vector field. Under small perturbations, we obtain a slow-fast system. Thus, we analyze the slow-fast dynamics near the singularities on the consensus space. The analysis at this stage is carried out for complete networks, allowing a detailed characterization of the system. We provide a linear approximation of the intersecting branches of equilibria at the singular points; as a consequence, we show that, generically, the singularities on the consensus space turn out to be <italic>transcritical</italic>. We prove under local assumptions the existence of canard solutions. For generic graph structures, assuming more strict conditions on the perturbation, we prove the existence of a maximal canard, which coincides with the consensus subspace. In addition, we validate by numerical simulations the principal findings of our main theory, extending the study to non-complete graphs. Moreover, we show how the delayed loss of stability associated with the canards induces transient spatio-temporal patterns.</p>

Formation of Inner Planets in the Presence of a Cold Jupiter: Orbital Evolution and Relative Velocities of Planetesimals

We investigate the orbital evolution of planetesimals in the inner disk in the presence of nebula gas and a (proto-) cold Jupiter. By varying the mass, eccentricity, and semimajor axis of the planet, we study the dependence of the relative velocities of the planetesimals on these parameters. For classic small planetesimals (1016–1020 g) whose mutual gravitational interaction is negligible, gas drag introduces a size-dependent alignment of orbits and keeps the relative velocity low for similar-sized bodies, while preventing orbital alignment for different-sized planetesimals. Regardless of the location and the mass ratio of the planetesimals, increasing the mass and eccentricity or decreasing the orbital distance of the planet always lead to higher relative velocities of the planetesimals. However, for massive planetesimals, the interplay of viscous stirring, gas damping, and secular perturbation results in the lower velocity dispersion of equal-sized planetesimals when the planet is more massive or when it is located on a closer or more eccentric orbit. The random velocities of such planetesimals remain almost unperturbed when the planet is located beyond Jupiter’s current orbit or when it is less massive or less eccentric than Jupiter. Unlike small planetesimals, such large planetesimals can grow in a runaway fashion, as in the unperturbed case. Our results imply that the presence of a cold Jupiter does not impede the formation of inner rocky planets through planetesimal accretion, provided that the planetesimals are initially large.

Three-dimensional modelling of the shock–turbulence interaction

ABSTRACTThe complex interaction between shocks and plasma turbulence is extremely important to address crucial features of energy conversion in a broad range of astrophysical systems. We study the interaction between a supercritical, perpendicular shock and pre-existing, fully developed plasma turbulence, employing a novel combination of magnetohydrodynamic and small-scale, hybrid-kinetic simulations where a shock is propagating through a turbulent medium. The variability of the shock front in the unperturbed case and for two levels of upstream fluctuations is addressed. We find that the behaviour of shock ripples, i.e. shock surface fluctuations with short (a few ion skin depths, di) wavelengths, is modified by the presence of pre-existing turbulence, which also induces strong corrugations of the shock front at larger scales. We link this complex behaviour of the shock front and the shock downstream structuring with the proton temperature anisotropies produced in the shock–turbulence system. Finally, we put our modelling effort in the context of spacecraft observations, elucidating the role of novel cross-scale, multispacecraft measurements in resolving shock front irregularities at different scales. These results are relevant for a broad range of astrophysical systems characterized by the presence of shock waves interacting with plasma turbulence.

Effects of head modeling errors on the spatial frequency representation of MEG

Objectives. We aim to investigate the effects of head model inaccuracies on signal and source reconstruction accuracies for various sensor array distances to the head. This allows for the assessment of the importance of head modeling for next-generation magnetoencephalography (MEG) sensors, optically-pumped magnetometers (OPM). Approach. A 1-shell boundary element method (BEM) spherical head model with 642 vertices of radius 9 cm and conductivity of 0.33 S m−1 was defined. The vertices were then randomly perturbed radially up to 2%, 4%, 6%, 8% and 10% of the radius. For each head perturbation case, the forward signal was calculated for dipolar sources located at 2 cm, 4 cm, 6 cm and 8 cm from the origin (center of the sphere), and for a 324 sensor array located at 10 cm to 15 cm from the origin. Equivalent current dipole (ECD) source localization was performed for each of these forward signals. The signal for each perturbed spherical head case was then analyzed in the spatial frequency domain, and the signal and ECD errors were quantified relative to the unperturbed case. Main results. In the noiseless and high signal-to-noise ratio (SNR) case of approximately ≥6 dB, inaccuracies in our spherical BEM head conductor models lead to increased signal and ECD inaccuracies when sensor arrays are placed closer to the head. This is true especially in the case of deep and superficial sources. In the noisy case however, the higher SNR for closer sensor arrays allows for an improved ECD fit and outweighs the effects of head geometry inaccuracies. Significance. OPMs may be placed directly on the head, as opposed to the more commonly used superconducting quantum interference device sensors which must be placed a few centimeters away from the head. OPMs thus allow for signals of higher spatial resolution to be captured, resulting in potentially more accurate source localizations. Our results suggest that an increased emphasis on accurate head modeling for OPMs may be necessary to fully realize its improved source localization potential.

A new vibro-impact bistable oscillator with an adjustable rigid wall

In this paper, a new vibro-impact bistable oscillator under vicious damping and external harmonic excitation is proposed by adding an adjustable rigid wall to one side of the archetypal smooth and discontinuous (SD) oscillator and an impact mapping is given to describe energy loss during collisions. In the unperturbed ideal case, there exist rich and perfect geometrical structure including grazing phenomena such as periodic orbits and homoclinic orbits tangent to the wall due to its adjustability as a constraint. Perturbed dynamics is studied to show the influence of the constraint on the original SD oscillator through the Lyapunov exponent diagrams, bifurcation diagrams, and the corresponding phase diagrams and attractors. Some new non-smooth chaotic attractors and periodic orbits are found to have grazing or transversal interaction with the unilateral rigid wall. Due to the effect of the introduced impact mapping and different locations of the rigid wall, the finger-like chaotic attractors of the SD oscillator will be suppressed and changed into different non-smooth periodic motions by varying the non-dimensional parameter which reflects the degree of the springs compressed, while the periodic motions of the SD oscillator will become new non-smooth chaotic attractors or different non-smooth periodic motions.

Reaction moments matter when designing lower-extremity robots for tripping recovery.

Balance recovery after tripping often requires an active adaptation of foot placement. Thus far, few attempts have been made to actively assist forward foot placement for balance recovery employing wearable devices. This study aims to explore the possibilities of active forward foot placement through two paradigms of actuation: assistive moments exerted with the reaction moments either internal or external to the human body, namely 'joint' moments and 'free' moments, respectively. Both paradigms can be applied to manipulate the motion of segments of the body (e.g., the shank or thigh), but joint actuators also exert opposing reaction moments on neighbouring body segments, altering posture and potentially inhibiting tripping recovery. We therefore hypothesised that a free moment paradigm is more effective in assisting balance recovery following tripping. The simulation software SCONE was used to simulate gait and tripping over various ground-fixed obstacles during the early swing phase. To aid forward foot placement, joint moments and free moments were applied either on the thigh to augment hip flexion or on the shank to augment knee extension. Two realizations of joint moments on the hip were simulated, with the reaction moment applied to either the pelvis or the contralateral thigh. The simulation results show that assisting hip flexion with either actuation paradigm on the thigh can result in full recovery of gait with a margin of stability and leg kinematics closely matching the unperturbed case. However, when assisting knee extension with moments on the shank, free moment effectively assist balance but joint moments with the reaction moment on the thigh do not. For joint moments assisting hip flexion, placement of the reaction moment on the contralateral thigh was more effective in achieving the desired limb dynamics than placing the reaction on the pelvis. Poor choice of placement of reaction moments may therefore have detrimental consequences for balance recovery, and removing them entirely (i.e., free moment) could be a more effective and reliable alternative. These results challenge conventional assumptions and may inform the design and development of a new generation of minimalistic wearable devices to promote balance during gait.

Multi-fidelity vortex simulations of rotor flows: Validation against detailed wake measurements

Two flow models with different fidelity of the DTU vortex solver MIRAS have been used to simulate the wake generated by a model wind turbine with various levels of asymmetry. Predictions are validated against experimental Particle Image Velocimetry measurements and dye visualizations. The experiments were conducted in a recirculating free-surface water channel with an immersed two-bladed rotor mounted on a shaft. The blades were designed to approximate a Joukowsky rotor. A detailed comparison between the measurements and the simulations is first performed for an unperturbed baseline case at different tip speed ratios. The analysis consists of a tip vortex characterization, including the vortex core profile, and a comparison of the instantaneous and mean velocity and vorticity fields, along with the mean wake profile at different downstream locations. Overall, good agreement is obtained between measurements and simulations, especially with the higher-fidelity particle-mesh model, which is capable of very closely predicting many of the flow features observed in the experiments. Rotor asymmetry triggers a vortex instability, commonly known as leapfrogging, which accelerates the breakdown of tip vortices, enhancing the mixing of wake structures and promoting a faster wake recovery. The prediction accuracy of this instability by the solvers is analyzed for different tip speed ratios and perturbation amplitudes. This work aims at setting the groundwork for future flow instability studies with the MIRAS solver.

Turbulence generation by shock interaction with a highly nonuniform medium.

An initially planar shock wave propagating into a medium of nonuniform density will be perturbed, leading to the generation of postshock velocity perturbations. Using numerical simulations we study this phenomenon in the case of highly nonuniform density (order-unity normalized variance, σ_{ρ}/ρ[over ¯]∼1) and strong shocks (shock Mach numbers M[over ¯]_{s}≳10). This leads to a highly disrupted shock and a turbulent postshock flow. We simulate this interaction for a range of shock drives and initial density configurations meant to mimic those which might be presently achieved in experiments. Theoretical considerations lead to scaling relations, which are found to reasonably predict the postshock turbulence properties. The turbulent velocity dispersion and turbulent Mach number are found to depend on the preshock density dispersion and shock speed in a manner consistent with the linear Richtymer-Meshkov instability prediction. We also show a dependence of the turbulence generation on the scale of density perturbations. The postshock pressure and density, which can be substantially reduced relative to the unperturbed case, are found to be reasonably predicted by a simplified analysis that treats the extended shock transition region as a single normal shock.

Stability of the planar synchronous full two-body problem—The approach of periodic orbits

We investigate the dynamical stability of the synchronous state of the planar full two-body problem (PF2BP) by employing the approach of periodic orbit in the primary’s body-fixed frame. Our results indicate that the traditional model to the spin–orbit resonances by neglecting the rotational motion’s influence on the orbital motion is inappropriate in the binary asteroid system because the two asteroids are close to each other, leading to strong coupling between the orbits and rotations. Focusing on the high-order spin–orbit resonance, the family genealogy of periodic orbits in the unperturbed case is broken apart by some resonances in the perturbed case. In the case of no spin–orbit resonances, the periodic orbit is near-circular and is generally but not always stable. In the case of spin–orbit resonances, the periodic orbit can be elliptic, and one branch of the periodic orbits is stable. In contrast, the other branch is unstable for small to moderate orbit eccentricities.

Robust-Exact-Differentiator-Inspired Discrete-Time Differentiation

This article proposes a discrete-time differentiation algorithm of arbitrary order inspired by the continuous-time uniform robust exact differentiator and the continuous-time arbitrary-order robust exact differentiator. As the well-known explicit Euler method is not suitable for discretizing algorithms with the fixed-time convergence property, a semi-implicit approach is proposed. The discrete-time differentiators of orders 2 and 3 are studied in detail, and it is proven that the estimation errors vanish independent of their initial condition in the unperturbed case. In the presence of perturbations, it is shown that the origin of the estimation errors is surrounded by an attractive set. Furthermore, the performance of the proposed algorithm is evaluated via simulation studies.

CR embeddability of quotients of the Rossi sphere via spectral theory

We look at the action of finite subgroups of [Formula: see text] on [Formula: see text], viewed as a CR manifold, both with the standard CR structure as the unit sphere in [Formula: see text] and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of [Formula: see text] to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum.

On Ambarzumian type theorems for tree domains

It is known that the spectrum of the spectral Sturm-Liouville problem on an equilateral tree with (generalized) Neumann's conditions at all vertices uniquely determines the potentials on the edges in the unperturbed case, i.e. case of the zero potentials on the edges (Ambarzumian's theorem). This case is exceptional, and in general case (when the Dirichlet conditions are imposed at some of the pendant vertices) even two spectra of spectral problems do not determine uniquely the potentials on the edges. We consider the spectral Sturm-Liouville problem on an equilateral tree rooted at its pendant vertex with (generalized) Neumann conditions at all vertices except of the root and the Dirichlet condition at the root. In this case Ambarzumian's theorem can't be applied. We show that if the spectrum of this problem is unperturbed, the spectrum of the Neumann-Dirichlet problem on the root edge is also unperturbed and the spectra of the problems on the complimentary subtrees with (generalized) Neumann conditions at all vertices except the subtrees' roots and the Dirichlet condition at the subtrees' roots are unperturbed then the potential on each edge of the tree is 0 almost everywhere.

Integral state-feedback control of linear time-varying systems: A performance preserving approach

An integral extension of state-feedback controllers for linear time-varying plants is proposed, which preserves performance of the nominal controller in the unperturbed case. Similar to time-invariant state feedback with integral action, the controller achieves complete rejection of disturbances whose effective action on the plant is constant with respect to the control input. Moreover, bounded-input bounded-state stability with respect to arbitrary disturbances is shown. A modification for preventing controller windup as well as tuning guidelines are discussed. The efficacy of the proposed technique is demonstrated in simulation for a two-tank system that is linearized along a time-varying reference trajectory.

Passive structural health monitoring using an acoustic digital twin

This work presents simulation results on passive structural health monitoring of a vibrating elastic structure for defect localization. We consider an object that consists of a shell with vibration sensors surrounding a complex, vibrating structure such that they are coupled at a finite number of points. The non-defective model of the external shell is called an acoustic digital twin (ADT). We use a combination of the concept of an acoustic digital twin (ADT) together with an adjoint-based high resolution array processing approach to detect a defect. The “dataset” in the presence of a defect was obtained from simulations and was “measured” on a linear array placed on the surface of a candidate ADT. A set of sensors placed at the connecting points provides the time histories of the excitation forces, which are used as inputs to the ADT in order to obtain the fields at the surface in the unperturbed case (no defect). The residual field (difference between defect and non-defect case) was used in the Matched Field Processing (MFP) algorithm in order to successfully locate the defect on the surface of the object.

Impact of three-dimensional magnetic perturbations on turbulence in tokamak edge plasmas

The impact of resonant magnetic perturbations (RMPs) on the plasma edge equilibrium and on the turbulence is investigated in a circular limited configuration. The study is based on a Braginski-based isothermal fluid model. The flow response of an unperturbed case to a small amplitude three-dimensional single mode RMP is studied and a scan in amplitude and poloidal and toroidal mode number is performed. Special attention is given when magnetic islands appear in the simulation domain on flux surfaces of rational safety factor. Results show an impact of magnetic perturbations (MPs) on both the plasma equilibrium and on the turbulence properties, with a deviation to the reference solution which depends on the MPs amplitude and on their wavenumbers. The impact of MPs on turbulence is however globally weaker than on the plasma equilibrium, suggesting a stabilizing effect of the MP on turbulent transport. Experimental trends are recovered such as the density pump-out and the increase of the radial electric field as well as the reorganization of the parallel velocity. The ballooning of the transport is modified under the effect of the perturbations, with a shift of the peaked poloidal region from the upper to the lower outer midplane. In the present model, the SOL width is observed decreasing in the presence of MPs. Turbulence properties are also impacted with the density fluctuations level decreasing in perturbed solutions and the intermittency is globally weakened.

Tomographic particle image velocimetry flow structures downstream of a dynamic cylindrical element in a turbulent boundary layer by multi-scale proper orthogonal decomposition

This study reports the modification of large and small scales in a turbulent boundary layer (TBL) perturbed by a dynamic cylindrical element (DCE). Tomographic particle image velocimetry (Tomo-PIV) was utilized to measure the flow fields downstream of the dynamic perturbation. By the approach of multi-scale proper orthogonal decomposition (mPOD), the coherent modes relevant to the predefined frequency bands were extracted from the Tomo-PIV dataset. Then, a method was developed to construct the large- and small-scale structures and the DCE-perturbed structure based on the mPOD modes. The DCE impact on the large- and small-scale structures was elaborated by comparing with the unperturbed TBL case. The two-point correlation analysis indicated that large-scale structures appear downstream of the DCE perturbation in a short streamwise length scale. More importantly, the scale rearrangements were further examined by presenting the modulation coefficients between the large scales and small-scale energy. It revealed that even though the DCE perturbation alters the level of correlation, three different types of interaction scenario can still be observed. In the near-wall region, the large-scale structures have an amplitude modulation effect on the small-scale energy with the lower positive coefficients. The reversal scale arrangement was observed at the wall-normal height around the DCE amplitude, which could be attributed to the fluid exchange caused by the new-generated turbulent structures. In the log region, it confirmed that the inclined shear layer resides along the low-speed regions, which supported the robustness of the conceptual model of hairpin packets in the current DCE-perturbed TBL.

Stability of the Enhanced Area Law of the Entanglement Entropy

We consider a multi-dimensional continuum Schrödinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper bound and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically enhanced area law as in the unperturbed case of the free Fermi gas. The central idea for the upper bound is to use a limiting absorption principle for such kinds of Schrödinger operators.