Multiple Stable Solutions Research Articles

Multiple stable postures of a falling object in fluids

We present evidence revealing that an object with specific properties can exhibit multiple stable falling postures at low Reynolds numbers. By scrutinizing the force equilibrium relationship of a fixed object at various attack angles and Reynolds numbers, we introduce a methodology that can obtain the stable falling postures of the object. This method saves computational resources and more intuitively presents the results in the full parameter domain. Our findings are substantiated by free-fall tests conducted through both physical experiments and numerical simulations, which validate the existence of multiple stable solutions in accordance with the interpolation results obtained with fixed objects. Additionally, we quantify the abundance and distribution patterns of stable falling postures for a diverse range of representative shapes. This discovery highlights the existence of multiple stable solutions that are universally present across objects of different shapes. The implications of this research extend to the design, stability control and trajectory prediction of all free and controlled flights in both air and water.

Data-Driven Bifurcation Analysis of Experimental Aeroelastic Systems Using Preflutter Measurements

Identification of flutter margins in modern aeroelastic systems is a challenging task due to increased nonlinearities in novel designs, which can result in instabilities occurring below the linear flutter speed. These instabilities pose a significant risk as they may involve multiple stable solutions, such as large-amplitude self-sustained oscillations. The lack of efficient nonlinear bifurcation analysis methods for experimental systems exacerbates the challenges associated with postflutter analysis. This paper presents a data-driven method for predicting flutter instabilities and bifurcation diagrams of an experimental nonlinear 2-degree-of-freedom (2-DOF) airfoil. The approach uses measurement data from the preflutter regime to forecast the postflutter dynamics, eliminating the need for computationally expensive models. This study is the first application of the recently introduced data-driven bifurcation forecasting method to experimental aeroelastic systems. The results show that the proposed method is accurate, with predictions matching the measured behavior of the system. The presented study provides valuable insights into the nonlinear stability and dynamics of experimental airfoils and demonstrates the potential for applicability of this approach in the analysis of experimental systems. The findings have significant implications for online monitoring and evaluation of the nonlinear dynamics of aeroelastic systems in the aerospace industry, where safety is of crucial importance.

Stable non-BPS black holes and black strings in five dimensions

In this paper we study black hole and black string solutions in five dimensional N=2 supergravity theories arising from the compactification of M-theory on Calabi-Yau manifolds. In particular, we consider explicit examples of three parameter Calabi-Yau manifolds which are obtained as hypersurfaces in toric varieties. Using the attractor mechanism, we obtain Bogomol’nyi-Prasad-Sommerfield (BPS) as well as non-BPS black holes in these compactified supergravity theories. We also consider the black string solutions in these models. We analyze the stability of these extremal black brane configurations by computing the recombination factor. We find multiple stable non-BPS attractor solutions in some of these models. Published by the American Physical Society 2024

Multistability in a coupled ocean–atmosphere reduced‐order model: Nonlinear temperature equations

AbstractMultistabilities in the ocean–atmosphere flow were found in a reduced‐order ocean–atmosphere coupled model, by solving the nonlinear temperature equations numerically. In this article, we explain how the full nonlinear Stefan–Bolzmann law was implemented numerically and the resulting change to the system dynamics was compared with the original model where these terms were linearised. Multiple stable solutions were found that display distinct ocean–atmosphere flows, as well as different Lyapunov stability properties. In addition, distinct low‐frequency variability (LFV) behaviour was observed in multiple attractors. We investigated the impact on these solutions of changing the magnitude of the ocean–atmospheric coupling, as well as the atmospheric emissivity, to simulate an increasing greenhouse effect. Where multistabilities exist for fixed parameters, the possibility for tipping between solutions was investigated, but tipping did not occur in this version of the model where there is a constant solar forcing. This study was undertaken using a reduced‐order coupled quasigeostrophic ocean–atmosphere model.

Hopf Bifurcation, Approximate Periodic Solutions and Multistability of Some Nonautonomous Delayed Differential Equations

Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields.

Stability and Sommerfeld effect in a multi-resonant types vibrating system with isolated rigid frame driven by four exciters

The previous studies on the vibrating system with multiple exciters less considered the Sommerfeld effect, which were mainly focused on synchronization and stability problems of the system in the whole resonant region. A dynamical model is proposed in this paper to study the synchronization, stability and the Sommerfeld effect of four exciters in the multi-resonant types vibrating system (MRTVS). In order to improve the power of the system, the vibrating system with two rigid frames is driven by four especially distributed exciters. The motion differential equations of the system are given by Lagrange’s equations, and the dimensionless coupling equations and synchronization criterion are constructed as well by the average method. The theory condition for stability of four exciters in synchronous states is derived from the Hamilton principle. The coupling dynamic characteristics, stability and Sommerfeld effect of the system are discussed numerically. The whole resonant region of the system is divided into three segments by natural frequencies, and the synchronous and stable states in the different resonant types are analyzed respectively. The diversity phenomenon of the nonlinear system is observed, in other words, the system exists multiple stable equilibrium solutions at a certain particular resonant region. Then the Sommerfeld effect around the natural frequencies (NFs) is further revealed. The correctness of theoretical analyses is examined by simulation and the experiment. It’s shown that the reasonable working point in engineering should be selected in the sub-resonant region with respect to the natural frequency of the main vibrating system. The present work can provide theoretical guidance for designing some new types of vibrating machines with high driving power.

Microwave-Assisted Thermoelectricity in S - I - S′ Tunnel Junctions

Asymmetric superconducting tunnel junctions with gaps ${\mathrm{\ensuremath{\Delta}}}_{1}>{\mathrm{\ensuremath{\Delta}}}_{2}$ have been proven to show a peculiar nonlinear bipolar thermoelectric effect. This arises due to the spontaneous breaking of electron-hole symmetry in the system, and it is maximized at the matching-peak bias $|V|={V}_{p}=({\mathrm{\ensuremath{\Delta}}}_{1}\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{2})/e$. In this paper, we investigate the interplay of photon-assisted tunneling (PAT) and bipolar thermoelectric generation. In particular, we show how thermoelectricity, at the matching peak, is supported by photon absorption and emission processes at the frequency-shifted sidebands $V=\ifmmode\pm\else\textpm\fi{}{V}_{p}+n\ensuremath{\hbar}\ensuremath{\omega}$, $n\ensuremath{\in}\mathbb{Z}$. This represents a sort of microwave-assisted thermoelectricity. We show the existence of multiple stable solutions, being associated with different photon sidebands, when a load is connected to the junction. Finally, we discuss how the nonlinear cooling effects are modified by the PAT. The proposed device can detect millimeter-wavelength signals by converting a temperature gradient into a thermoelectric current or voltage.

Coupled resonance of FGM nanotubes transporting super-critical high-speed pulsatile flow under forced vibration: size-dependence and bifurcation topology

The nano electromechanical system (NEMS) with functionally graded (FG) nanotubes as the core component is usually in the multi field environment leading to multi-source excitation resonance. As a typical and basic excitation form, the nonlinear principal resonance of FG nanotubes under the supercritical pulsatile flow and forced excitation is studied. Geometric nonlinearity originates from the tensile hardening of immovable boundary modeled​ by von Kármán’s nonlinear geometric relationship. Size-dependencies for both nanofluids and nanosolids are characterized by slip-flow model and nonlocal strain gradient model coupled with surface effects, respectively. Under the framework of Zhang–Fu’s beam theory, the governing equations for coupled resonance based on the initial post-buckling equilibrium state are established. A combination of perturbation-incremental harmonic balance method (IHBM) is established and employed in bifurcation analysis of such a strongly nonlinear coupled resonance system for the first time The softening-type nonlinear behavior and multiple-stable solutions are found and two dominant factors controlling the bifurcation topology are revealed. It is suggested that the size effects of both nanoflow and nanosolid are dual and may change the bifurcation topology through two mechanisms. One is the change of two excitations from strong interaction to weak interaction, and the other is the change of multi-solution threshold of the two-excitation amplitude ratio. The results can guide the nonlinear design and adjustment of the NEMS to obtain available working frequency band maintaining stable motion state.

One-to-one internal resonance in a symmetric MEMS micromirror

Resonant modal interaction is a nonlinear dynamic phenomenon observed in structures excited at large vibration amplitudes. In the present work, we report the experimental evidence of a 1:1 internal resonance in a symmetric resonant micromirror. The experiments are complemented with a reduced model obtained from the 3D finite element discretization of the device by parametrizing the system motion along a low dimensional invariant set of the phase space. The presence of coupling monomials in the governing equations makes the resulting dynamics non-linearizable. Both model and experimental data show the existence of a complex pattern of multiple stable solutions for a given value of the excitation frequency.

State-space renormalization group theory of nonequilibrium reaction networks: Exact solutions for hypercubic lattices in arbitrary dimensions.

Nonequilibrium reaction networks (NRNs) underlie most biological functions. Despite their diverse dynamic properties, NRNs share the signature characteristics of persistent probability fluxes and continuous energy dissipation even in the steady state. Dynamics of NRNs can be described at different coarse-grained levels. Our previous work showed that the apparent energy dissipation rate at a coarse-grained level follows an inverse power-law dependence on the scale of coarse-graining. The scaling exponent is determined by the network structure and correlation of stationary probability fluxes. However, it remains unclear whether and how the (renormalized) flux correlation varies with coarse-graining. Following Kadanoff's real space renormalization group (RG) approach for critical phenomena, we address this question by developing a state-space renormalization group theory for NRNs, which leads to an iterative RG equationfor the flux correlation function. In square and hypercubic lattices, we solve the RG equationexactly and find two types of fixed point solutions. There is a family of nontrivial fixed points where the correlation exhibits power-law decay, characterized by a power exponent that can take any value within a continuous range. There is also a trivial fixed point where the correlation vanishes beyond the nearest neighbors. The power-law fixed point is stable if and only if the power exponent is less than the lattice dimension n. Consequently, the correlation function converges to the power-law fixed point only when the correlation in the fine-grained network decays slower than r^{-n} and to the trivial fixed point otherwise. If the flux correlation in the fine-grained network contains multiple stable solutions with different exponents, the RG iteration dynamics select the fixed point solution with the smallest exponent. The analytical results are supported by numerical simulations. We also discuss a possible connection between the RG flows of flux correlation with those of the Kosterlitz-Thouless transition.

Analytical detection of stationary turing pattern in a predator-prey system with generalist predator

A prey-predator model with Holling type-II functional response and a generalist predator exhibits complex dynamics in response to parameter variation. Generalist predators implicitly exploiting multiple food resources reduce predation pressure on their focal prey species that causes it to become more stable compared to a prey-predator system with specialist predator. In the temporal system, bistability and tristability are observed along with various global and local bifurcations. Existence of homogeneous and heterogeneous positive steady state solutions are shown to exist for suitable ranges of parameter values in the corresponding spatio-temporal diffusive system. Weakly nonlinear analysis, using multi-scale perturbation technique, is employed to derive amplitude equation for the stationary patterns near the Turing bifurcation threshold. The analytical results of the amplitude equations are validated using exhaustive numerical simulations. We also identify bifurcation of multiple stable stationary patch solutions as well as dynamic pattern solution for parameter values in the Turing and Turing-Hopf regions.

Viral infection dynamics with mitosis, intracellular delays and immune response.

In this paper, we propose a delayed viral infection model with mitosis of uninfected target cells, two infection modes (virus-to-cell transmission and cell-to-cell transmission), and immune response. The model involves intracellular delays during the processes of viral infection, viral production, and CTLs recruitment. We verify that the threshold dynamics are determined by the basic reproduction number $ R_0 $ for infection and the basic reproduction number $ R_{IM} $ for immune response. The model dynamics become very rich when $ R_{IM} > 1 $. In this case, we use the CTLs recruitment delay $ \tau_3 $ as the bifurcation parameter to obtain stability switches on the positive equilibrium and global Hopf bifurcation diagrams for the model system. This allows us to show that $ \tau_3 $ can lead to multiple stability switches, the coexistence of multiple stable periodic solutions, and even chaos. A brief simulation of two-parameter bifurcation analysis indicates that both the CTLs recruitment delay $ \tau_3 $ and the mitosis rate $ r $ have a strong impact on the viral dynamics, but they do behave differently.

Improved theoretical analysis and design guidelines of a two-degree-of-freedom galloping piezoelectric energy harvester

This paper presents an improved analysis of a two-degree-of-freedom (2DOF) galloping-based piezoelectric energy harvester (GPEH). First, an overview of the 2DOF GPEH, together with some discussions on its physical implementation, are presented. The theoretical model of the 2DOF GPEH is then developed. The analytical solutions are derived using the harmonic balance method. The dynamic behaviour of the 2DOF GPEH is predicted according to the solution characteristics. Moreover, the mode activation mechanism of the 2DOF GPEH is theoretically unveiled: depending on the system parameters; there may exist a single or multiple stable solutions which correspond to different vibration modes of the 2DOF GPEH. Subsequently, an equivalent circuit model of the 2DOF GPEH is established. Circuit simulations are performed to verify the analytical solutions. Case studies through detailed theoretical analysis and circuit simulation give in-depth insights into the dynamic behaviour of the 2DOF GPEH. It is demonstrated that by tuning the stiffness of the auxiliary oscillator, either the first or the second mode vibration of the 2DOF GPEH can be activated, resulting in completely different dynamic behaviours and energy harvesting performance. Finally, from the perspectives of reducing the cut-in wind speed and improving the voltage output, several design guidelines are provided.

Self-excited vibrations due to viscoelastic interactions

Self-excited vibrations represent a big concern in engineering, particularly in automotive, railway and aeronautic industry. Many lumped models have been proposed over the years to analyze the stability of such systems. Among the instability mechanisms a falling characteristic of the friction law and mode coupling have been shown to give friction-excited oscillations. The mass-on-moving-belt system has been studied extensively in Literature, very often adopting a prescribed form of the friction law and linearizing the contact stiffness. Instead, in this work, the case of a spherical oscillator excited by a moving viscoelastic halfspace is considered. The friction law and the nonlinear normal contact stiffness have been computed via boundary element numerical simulations for varying substrate velocity and indentation depth, then they have been adopted in time-marching dynamical numerical simulations. It is shown that the horizontal and vertical dynamics of the oscillator are tightly connected each other through the viscoelastic substrate. Furthermore, for certain normal forces/substrate velocity, the system has multiple stable solutions, which may include lift-off of the oscillator, which are selected based solely on the system initial conditions.

Rich dynamics in a delayed HTLV-I infection model: Stability switch, multiple stable cycles, and torus

In this paper, we investigate the impact of time delay in CTL immune response on a HTLV-I infection model. By defining basic reproduction number for viral infection R0 and basic reproduction number for CTL response RCTL, we characterize the model dynamics according to whether these two threshold values are greater than one. Especially, we obtain the global dynamics if R0≤1 or RCTL≤1<R0, as well as infection persistent result when R0>1. However, the model dynamics become much richer when RCTL<1. In this case, we use the time delay as a bifurcation parameter to obtain stability switch result on the positive equilibrium and global bifurcation diagrams for the model system. We also conduct higher-order normal form analysis and apply center manifold theory to classify the rich model dynamics near the double Hopf bifurcation points. Our analysis indicates that time delay in CTL immune response can induce not only Hopf bifurcation and double Hopf bifurcation, but also quasi-periodic orbits (torus) and coexistence of multiple stable periodic solutions.

Nonlinear dynamic modeling and analysis of borehole propagation for directional drilling

Boreholes with complex trajectories are drilled with the help of downhole rotary steerable systems. These robotic actuators, which are embedded in the drillstring, are used to steer the bit in the desired direction. This paper presents a dynamic non-smooth borehole propagation model for planar directional drilling. Essential nonlinearities, induced by the saturation of the bit tilt and by non-ideal (undergauged) stabilizers, are modeled using complementarity conditions, leading to a closed-form analytical description of the model in terms of a so-called delay complementarity system. The analytical form of the model allows for a comprehensive dynamic and parametric analysis. Firstly, (quasi-)stationary solutions generated by constant actuator forces are analyzed parametrically as a function of the actuation force. Secondly, an analysis of the local stability of these solutions shows the coexistence of multiple (stable and unstable) solutions and their dependency on key system parameters, such as the weight-on-bit and bit characteristics. Thirdly, a numerical simulation study shows the existence of steady-state oscillations, which are a consequence of the non-smooth characteristics of the bit tilt saturation and the stabilizers. Such limit cycles represent borehole rippling, which is the planar equivalent of the highly detrimental borehole spiraling observed in practice. The constructed model and the pursued analysis provide essential insights in the effects causing undesired borehole rippling. Herewith, the presented results can be used to support improved directional drilling system design and to form the basis for further work on automation techniques for the downhole robotic actuator to mitigate spiraled boreholes.

Complex spatiotemporal behavior and coherent excitations in critically-coupled chains of neural circuits.

We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the "synaptic matrix" embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a "critical" synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dynamics of the units is dictated by a chaotic map of the interval, and the interactions are given by allowing the critical coupling to act for a finite period , thus given by a unitary matrix . It is now explicit that such critical couplings are volume-preserving in the sense of Liouville's theorem. We show that this map is also capable of producing a variety of complex spatiotemporal patterns including gliders, like our original chain of neural circuits. Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.

Phase models and clustering in networks of oscillators with delayed coupling

We consider a general model for a network of oscillators with time delayed coupling where the coupling matrix is circulant. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. Our results extend previous work to systems with time delay and a more general coupling matrix. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies.

Coupled lateral and axial soil-pipe interaction and lateral buckling Part II: Solutions

This paper presents techniques for constructing numerical solutions for some of the problems formulated in Part I. An algorithm to search for periodic solutions is developed and implemented in software. It is shown that there exist a finite number of discrete well defined periodic solutions for the lateral buckling problem of a straight pipeline under constant pressure and temperature. It is also shown that periodic solutions are unstable. A second numerical algorithm is developed for solving lateral buckling problem for an initially straight semi-infinite pipeline terminating at a sliding manifold under constant or varying temperature profiles. The resulting boundary value problem requires special numerical methods due to the stiffness of the governing differential equations. Algorithm is applied to find the solutions of problems for three different size pipelines; an 8.625 inch, a 16 inch, and a 32 inch diameter. Multiple stable chaotic solutions corresponding to distinct but extremely close initial conditions are revealed. The characteristics of these solutions match with the characteristics of the periodic solutions. This suggests that unstable periodic solutions act as separatrix between narrow zones containing stable chaotic solutions. It is seen that chaotic solutions can relieve a significant portion of the thermal and pressure loads without inducing excessive pipeline displacements. The results show that the pipeline end expansion is significantly less than the value given by common engineering methods. It is also shown that pipeline operating conditions and soil-pipe interaction parameters influence the stability and bifurcations of solutions.

Influence of the temperature-dependent viscosity on convective flow in the radial force field.

The numerical investigation of convective flows in the radial force field caused by an oscillating electric field between spherical surfaces has been performed. A temperature difference (T_{1}>T_{2}) as well as a radial force field triggers a fluid flow similar to the Rayleigh-Bénard convection. The onset of convective flow has been studied by means of the linear stability analysis as a function of the radius ratio η=R_{1}/R_{2}. The influence of the temperature-dependent viscosity has been investigated in detail. We found that a varying viscosity contrast β=ν(T_{2})/ν(T_{1}) between β=1 (constant viscosity) and β=50 decreases the critical Rayleigh number by a factor of 6. Additionally, we perform a bifurcation analysis based on numerical simulations which have been calculated using a modified pseudospectral code. Numerical results have been compared with the GeoFlow experiment which is located on the International Space Station (ISS). Nonturbulent three-dimensional structures are found in the numerically predicted parameter regime. Furthermore, we observed multiple stable solutions in both experiments and numerical simulations, respectively.