Solution Branches Research Articles

Effect of dry friction on a parametric nonlinear oscillator

Parametrically excited oscillators are used in several domains, in particular to improve the dynamical behaviour of systems like in the case of the parametric amplification or parametric energy harvesting. Although dry friction is often omitted during system modelling due to the complexity of its nonsmooth nature, it is sometimes necessary to account for this kind of damping to adequately represent the system motion. In this paper, it is proposed to investigate the effect of dry friction on the dynamical behaviour of a nonlinear parametric oscillator. Using the pendulum case as example, the problem is formulated according to a Mathieu-Duffing equation. Semi-analytical developments using the harmonic balance method and the method of varying amplitudes are used to find the solutions of this equation and their stability. These results are validated thanks to a comparison with time integration simulations. Effects of initial conditions on the basins of attractions of the solutions are also studied using these simulations. It is found that trivial and nontrivial solutions of the oscillator including dry friction are not connected, giving birth to isolated periodic solutions branches. Thus, both initial displacement and phase between the excitation and the oscillator displacement must be carefully chosen to reach periodic solutions. Finally, a method based on the energy principle is used to find the critical forcing amplitude and frequency needed to obtain the birth of nontrivial solutions for the nonlinear parametric oscillator including dry friction.

Scalarized non-topological neutron stars in multi-scalar Gauss\u2013Bonnet gravity

In the present paper we construct novel non-topological, spontaneously scalarized neutron stars in multi-scalar Gauss–Bonnet gravity with maximally symmetric target space, and nontrivial map varphi :spacetime rightarrow target space. The theory is characterized by the fact that for some classes of coupling functions the field equations allow solutions with trivial scalar field, which coincide with the general relativistic ones. For a certain range of parameters those solutions lose stability and new branches of solutions with nontrivial scalar field bifurcate from the trivial branch. For a given set of parameters, those branches are characterized by the number of zeros of the scalar field, and they are energetically more favorable than the general relativistic ones.

A heuristic search method for detecting multiple period solution branches of nonlinear rotor bearing systems

A heuristic search method for detecting multiple period solution branches of nonlinear rotor bearing systems

Nonlinear responses of a dual-rotor system with rub-impact fault subject to interval uncertain parameters

This paper aims to study the nonlinear steady-state response of a dual-rotor system with rub-impact fault subject to unknown-but-bounded (UBB) uncertainties. Mathematical modelling of the non-linear dynamical system is carried out based on the Lagrangian formulation. The nonlinear dynamic response of the rubbing dual-rotor system without uncertainty is solved by using the multi-dimensional harmonic balance method coupled with the alternating frequency/time technique. The arc-length continuation is used to track the solution branches. To predict the response range subject to uncertainty, a non-intrusive surrogate model in conjunction with the polar angle interpolation (PAI) with efficiency enhancement is developed to track the propagations of parametric variabilities. The PAI is dedicated to dealing with collocations where the responses have multiple solutions. Effects of UBB variables in the physical model and fault-related parameters are investigated comprehensively. Different features of the variabilities in the steady-state responses are found under the typical uncertain degrees. The interval scanning method is used to validate the computation accuracy of the whole procedure. Moreover, the working mechanism of the PAI method is demonstrated via examples in detail. The results obtained in simulations can provide useful guidance for the nonlinear dynamic investigations and rub-impact fault diagnosis of dual-rotor systems under the UBB uncertainties. The proposed non-intrusive uncertainty quantification framework based on the PAI will also be beneficial to other nonlinear vibration problems where multiple solutions are involved.

Global stability analysis of flexible channel flow with a hyperelastic wall

We consider the stability of flux-driven flow through a long planar rigid channel, where a segment of one wall is replaced by a pre-tensioned hyperelastic (neo-Hookean) solid of finite thickness and subject to a uniform external pressure. We construct the steady configuration of the nonlinear system using Newton's method with spectral collocation and high-order finite differences. In agreement with previous studies, which use an asymptotically thin wall, we show that the thick-walled system always has at least one stable steady configuration, while for large Reynolds numbers the system exhibits three co-existing steady states for a range of external pressures. Two of these steady configurations are stable to non-oscillatory perturbations, one where the flexible wall is inflated (the upper branch) and one where the flexible wall is collapsed (the lower branch), connected by an unstable intermediate branch. We test the stability of these steady configurations to oscillatory perturbations using both a global eigensolver (constructed based on an analytical domain mapping technique) and also fully nonlinear simulations. We find that both the lower and upper branches of steady solutions can become unstable to self-excited oscillations, where the oscillating wall profile has two extrema. In the absence of wall inertia, increasing wall thickness partially stabilises the onset of oscillations, but the effect remains weak until the wall thickness becomes comparable to the width of the undeformed channel. However, with finite wall inertia and a relatively thick wall, higher-frequency modes of oscillation dominate the primary global instability for large Reynolds numbers.

Nucleation of creases and folds in hyperelastic solids is not a local bifurcation

We consider creases and folds in compressed hyperelastic solids from the point of view of bifurcation theory. They refer to highly localized surface deformations that occur at compressive loads significantly below the value of the well-known Biot instability. Much work from the literature attempts to make the case that this phenomenon corresponds to a “local bifurcation” distinct from the Biot instability. A local bifurcation is a path of equilibrium solutions emanating from a (bifurcation) point on the trivial solution branch that exists in all sufficiently small neighborhoods of that bifurcation point. The inference is usually made by first introducing a small surface imperfection; a solution curve is then obtained that is seemingly close to a perfect bifurcation diagram. However, imperfection theory is valid only in some sufficiently small neighborhood of a bifurcation point. Thus, in the absence of an equilibrium path connecting these solutions to the trivial one, there is no justification for concluding that creasing and folding are local bifurcations of the perfect system.In this work, we directly address the nucleation of these solutions in the perfect, imperfection-free case. We demonstrate that surface instabilities in functionally graded and bilayer elastic halfspaces, corresponding to local bifurcations from the homogeneous state, are necessarily smooth and oscillatory; creases/folds eventually do develop along the global bifurcating solution branches, albeit “far” from the trivial solution, as evidenced by the corresponding bifurcation diagrams. In addition, we find that their stable realization occurs at load levels well below that of the initial surface instability. Moreover, we obtain such results for the perfect homogeneous halfspace, by switching the continuation parameter from macroscopic lateral strain to the film-to-substrate shear modulus ratio. When this ratio reaches unity, we obtain the desired localized deformation solution, avoiding the need for analysis near the highly degenerate homogeneous state at the Biot instability.

Simulation of time-dependent radiative heat motion over a stretching/shrinking sheet of hybrid Nanofluid: Stability analysis for dual solutions

The heat transfer characteristics for the flow of a time-dependent hybrid nanofluid with thermal radiation and source/sink over a stretching/shrinking sheet are examined in the current investigation. We have transformed the governing equations of the presented study into the similarity equations utilizing similarity variables. However, a numerical solution is obtained by using in-build MATLAB code bvp5c. The mass and energy profiles for diverse values of thermophysical parameters are studied together with their physical quantities. It is observed that dual solutions exist, that is, one is upper, and the other is lower branch solution for a definite choice of the unsteadiness parameter. Also, stability analysis is executed to determine the long-term stability of dual solutions, indicating that out of the two, only one is stable and the other is unstable. It is revealed that comparatively, the first solution shows stability, while the second solution shows instability. There is a considerable influence of second-order slip on the problem’s respective flow and heat transfer characteristics. Further, major outcomes also show the dimensionless frictional stress and the magnitude of conventional heat transfer enhancement with growing suction parameter values.

О решении интегральных уравнений Гаммерштейна с нагрузками и бифуркационными параметрами

The Hammerstein integral equation with loads on the desired solution is considered. The equation contains a parameter for any value of which the equation has a trivial solution. Necessary and sufficient conditions are obtained for the coefficients of the equation and those values of the parameter (bifurcation points) in its neighborhood the equation has a nontrivial real solutions. The leading terms of the asymptotics of such branches of solutions are constructed. Examples are given illustrating the proven existence theorems

Parameter identification of a physical model of brass instruments by constrained continuation

Numerical continuation using the Asymptotic Numerical Method (ANM), together with the Harmonic Balance Method (HBM), makes it possible to follow the periodic solutions of non-linear dynamical systems such as physical models of wind instruments. This has been recently applied to practical problems such as the categorization of musical instruments from the calculated bifurcation diagrams [V. Fréour et al. Journal of the Acoustical Society of America 148 (2020) https://doi.org/10.1121/10.0001603]. Nevertheless, one problem often encountered concerns the uncertainty on some parameters of the model (reed parameters in particular), the values of which are set almost arbitrarily because they are too difficult to measure experimentally. In this work we propose a novel approach where constraints, defined from experimental measurements, are added to the system. This operation allows uncertain parameters of the model to be relaxed and the continuation of the periodic solution with constraints to be performed. It is thus possible to quantify the variations of the relaxed parameters along the solution branch. The application of this technique to a physical model of a trumpet is presented in this paper, with constraints derived from experimental measurements on a trumpet player.

On branching of periodic solutions of quasilinear systems of ordinary differential equations

On branching of periodic solutions of quasilinear systems of ordinary differential equations

Non-synchronized solutions to nonlinear elliptic Schrödinger systems on a closed Riemannian manifold

<p style='text-indent:20px;'>On a smooth, closed Riemannian manifold, we study the question of proportionality of components, also called synchronization, of vector-valued solutions to nonlinear elliptic Schrödinger systems with constant coefficients. In particular, we obtain bifurcation results showing the existence of branches of non-synchronized solutions emanating from the constant solutions.</p>

Local analysis of a two phase free boundary problem concerning mean curvature

We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the additional property that its normal derivative $\partial_n u$ is a multiple of the radius of curvature at each point on the boundary. When the coefficients satisfy some non-criticality condition, we construct nontrivial solutions to this overdetermined problem employing a perturbation argument relying on shape derivatives and the implicit function theorem. Moreover, in the critical case, we employ the use of the Crandall-Rabinowitz theorem to show the existence of a branch of symmetry breaking solutions bifurcating from trivial ones. Finally, some remarks on the one phase case and a similar overdetermined problem of Serrin type are given.

Duel Solutions in Hiemenz Flow of an Electro-Conductive Viscous Nanofluid Containing Elliptic Single-/Multi-Wall Carbon Nanotubes With Magnetic Induction Effects

Abstract Modern magnetic nanomaterials are increasingly embracing new technologies including smart coatings, intelligent lubricants, and functional working fluids in energy systems. Motivated by studying the manufacturing magnetofluid dynamics of electroconductive viscous nanofluids, in this work, we analyzed the magnetohydrodynamics (MHD) convection flow and heat transfer of an incompressible viscous nanofluid containing carbon nanotubes (CNTs) past a stretching sheet. Magnetic induction effects are included. Similarity solutions are derived where possible in addition to dual branch solutions. Both single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs) are considered taking water and kerosene oil as base fluids. The governing continuity, momentum, magnetic induction, and heat conservation partial differential equations are converted to coupled, nonlinear systems of ordinary differential equations via similarity transformations. The emerging control parameters are shown to be Prandtl number (Pr), nanoparticle volume fraction parameter (φ), inverse magnetic Prandtl number (λ), magnetic body force parameter (β) and stretching rate parameter (A), and the type of carbon nanotube. Numerical solutions to the ordinary differential boundary value problem are conducted with the efficient bvp4c solver in matlab. Validation with earlier studies is included. Computations of reduced skin friction and reduced wall heat transfer rate (Nusselt number) are also comprised in order to identify the critical parameter values for the existence of dual solutions (upper and lower branch) for velocity, temperature, and induced magnetic field functions. Dual solutions are shown to exist for some cases studied. The simulations indicate that when the stretching rate ratio parameter is less than 1, SWCNT nanofluids exhibit higher velocity than MWCNT nanofluids with increasing magnetic parameters for water- and kerosene-oil-based CNT nanofluids. Generally, SWCNT nanofluids achieve enhanced heat transfer performance compared to MWCNT nanofluids. Water-based CNT nanofluids also attain greater flow acceleration compared with kerosene-oil-based CNT nanofluids.

On the global bifurcation diagram of the Gelfand problem

For domains of first kind [7,13] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel'fand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of non-minimal solutions which is not just concerned with radial solutions [28] and/or with symmetric domains [23]. Toward our goal we parametrize the branch not by the $L^{\infty}(\Omega)$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a carefully modified spectral analysis of mean field type equations.

Effect of geometric imperfections and circumferential symmetry on the internal resonances of cylindrical shells

In the present work the resonant response of an imperfect cylindrical shell is investigated using Donnell’s nonlinear shallow shell theory. For this, a reduced order multi degree of freedom model is obtained by discretizing the partial differential equation of motion using the Galerkin procedure and a consistent modal expansion which captures the nonlinear modal interactions and couplings between two asymmetric vibration modes with near commensurable natural frequencies in a 1:2 ratio. As a result of the circumferential symmetry each mode exhibits a 1:1 internal resonance, leading to a possible 1:1:2:2 multiple internal resonances. These modes are coupled through quadratic and cubic nonlinearities arising from the shell curvature and nonlinear strain–displacement relation. The existence and stability of solutions and their bifurcations are investigated using numerical continuation methods for bifurcation analysis and their stability are studied using Floquet theory. It is known that geometric imperfections have a strong influence on the response of thin shell structures. Here, a detailed parametric analysis shows the influence of different forms of geometric imperfections on the shell natural frequencies and bifurcations in the main resonance region. Several branches of solutions due to multiple bifurcations are detected leading to dynamic jumps under increasing and decreasing frequency sweep. Steady-state harmonic and quasi-periodic responses resulting from Neimark–Sacker bifurcations are detected. The reduced order model demonstrates the influence of the geometric imperfection shape and magnitude on the bifurcation scenario and the energy transfer among the four interacting modes.

Experimental study of the flows in a non-axisymmetric ellipsoid under precession

Precession driven flows are of great interest for both, industrial and geophysical applications. While cylindrical, spherical and spheroidal geometries have been investigated in great detail, the numerically and theoretically more challenging case of a non-axisymmetric cavity has received less attention. We report experimental results on the flows in a precessing triaxial ellipsoid, with a focus on the base flow of uniform vorticity, which we show to be in good agreement with existing theoretical models. As predicted, the uniform vorticity component exhibits two branches of solutions leading to a hysteresis cycle as a function of the Poincaré number. The first branch is observed at low forcing and characterized by large amplitude of the total fluid rotation and a moderate tilt angle of the fluid rotation axis. In contrast, the second branch displays only a moderate fluid rotation and a large tilt angle of the fluid rotation axis, which tends to align with the precession axis. In addition, we observe the occurrence of parametric instabilities early in the first branch, which saturate in the second branch, where we observe the same order of the kinetic energy in the base flow and instabilities.

Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE

We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto–Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton–Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.

Effect of thermal radiation and MHD on hybrid Ag–TiO2/H2O nanofluid past a permeable porous medium with heat generation

The major objective of this study is to identify the behaviour of flow and heat transfer on mixed convection with the appearance of hybrid nanofluid in a porous medium, heat generation, suction/injection, thermal radiation and magnetohydrodynamics (MHD). Two distinct fluids of Ag (silver) and TiO2 (titanium dioxide) are used in this model to represent the hybrid condition and a model of non-linear ODEs is obtained by conducting a technique of similarity transformation. The findings and solutions are gained with the aid of the shooting method in Maple and the bvp4c solver in Matlab. The outcomes of dimensionless velocity and temperature profiles are exemplified quantitively through respective figures with performing parameters such as suction, mixed convection, thermal radiation and volume fraction of nanoparticle. Through our comparison analysis, the application of added hybrid nanoparticles revealed a better performance of thermal transmittance compared to the traditional mono-nanoparticle. We also identified two different branches of solutions in the occurrence of λ < 0 (opposing flow). In addition, a stability analysis was conducted throughout this research due to the existence of two levels of solutions, and the first branch is perceived to be the most solid and recognizable solution.

Hopf bifurcations for an oversteer vehicle – the influence of wheel load changes

AbstractIn [1] the occurence of a Hopf bifurcation at steady‐state cornering for a two‐wheel oversteering vehicle with a brush model for the tyre forces was found and following the bifurcating branch of periodic solutions we observed a Canard phenomenon and relaxation oscillations, which are typical for singularly perturbed systems. In that article we assumed for simplicity, that the car's center of mass lies at street level. In this presentation we investigate the influence of an elevated height on the stability of the car's steady motion and the postcritical behaviour.

Generic properties of free boundary problems in plasma physics* *DB has been partially supported by Beyond Borders Project 2019 (sponsored by Univ. of Rome ‘Tor Vergata’) ‘Variational Approaches to PDE’s’, MIUR Excellence Department Project awarded to the Department of Mathematics, Univ. of Rome Tor Vergata, CUP E83C18000100006.

We are concerned with the global bifurcation analysis of positive solutions to free boundary problems arising in plasma physics. We show that in general, in the sense of domain variations, the following alternative holds: either the shape of the branch of solutions resembles the monotone one of the model case of the two-dimensional disk, or it is a continuous simple curve without bifurcation points which ends up at a point where the boundary density vanishes. On the other hand, we deduce a general criterion ensuring the existence of a free boundary in the interior of the domain. Application to a classic nonlinear eigenvalue problem is also discussed.