Equivariant bifurcation theory and symmetry breaking

A study is made of the failure of the Maximal Isotropy Subgroup Conjecture for the Weyl group seriesW(D) _k . As part of the investigation, a general genericity and stability theorem is proved for bifurcation diagrams in equivariant bifurcation theory. As well, a concept of determinacy for equivariant bifurcation theory is introduced and it is shown that, for all compact Lie groupsG and absolutely irreducibleG-representationsV, G-equivariant bifurcation problems onV are finitely determined.     

Destabilization paradox due to breaking the Hamiltonian and reversible symmetry

Stability of a linear autonomous non-conservative system in the presence of potential, gyroscopic, dissipative, and non-conservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one are examined. It is known that marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present paper shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. In case of two degrees of freedom, approximations of the stability boundary near the singularities are found in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping.     

Actio-reactio symmetry breaking model fitted to the new tower-mine gravity experiments

Summary In the present paper the gravity acceleration in the neighbourhood of the earth-atmosphere interface is analyzed. The solution of the actio-reactio equations (see [7, 8]) are compared with the data measured by Holding et al. [4] in mines and Eckehardt et al. [6] on a tower.

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Anpassung eines die actio/reactio-symmetrie verletzenden modells an gravitationsexperimente im bergwerk und turm

Übersicht Analysiert wird die Gravitationsbeschleunigung in der Nähe der Grenze Erde/Atmosphäre. Die Lösung der Actio/Reactio-Gleichungen (vgl. [7, 8]) wird mit den Daten, die von Holding et al. [4] und von Eckehardt et al. [6] im Bergwerk bzw. auf einem Turm gemessen wurden, verglichen.

     

Inhomogeneous elastostatic problem solutions constructed from stress-associated homogeneous solutions

In this paper, we present a theorem that provides solutions for anisotropic and inhomogeneous elastostatic problems by using the known solution of an associated anisotropic and homogeneous problem if the associated problem has a stress state with a zero eigenvalue everywhere in the domain of the problem. The fundamental property on which this stress-associated solution (SAS) theorem is built is the coaxiality of the eigenvector associated with the zero stress eigenvalue in the homogeneous problem and the gradient of the scalar function ? characterizing the inhomogeneous character of the inhomogeneous problem. It is shown that most of the solutions of anisotropic elastic problems presented in the literature have this property and, therefore, it is possible to use the SAS theorem to construct new exact solutions for inhomogeneous problems, as well as to find—using the SAS theorem—solutions for the shape intrinsic and angularly inhomogeneous problems.     

A nonlinear symmetry breaking effect in shear cracks

Shear cracks propagation is a basic dynamical process that mediates interfacial failure. We develop a general weakly nonlinear elastic theory of shear cracks and show that these experience tensile-mode crack tip deformation, including possibly opening displacements, in agreement with Stephenson's prediction. We quantify this nonlinear symmetry breaking effect, under two-dimensional deformation conditions, by an explicit inequality in terms of the first and second order elastic constants in the quasi-static regime and by semi-analytic calculations in the fully dynamic regime. Our general results are applied to various materials. Finally, we discuss related works in the literature and note the potential relevance of elastic nonlinearities for various problem, including frictional sliding.     

Two-plane automatic balancing: A symmetry breaking analysis

We present an analysis of a two-plane automatic balancing device for rotating machinery. The mechanism consists of a pair of races that contain balancing balls which move to eliminate imbalance due to rotor eccentricity or principal axis misalignment. A model is developed that includes the effect of support anisotropy and rotor acceleration. The symmetry of the imbalance is considered, and techniques from equivariant bifurcation theory are used to derive a necessary condition for the stability of balanced operation. The unfolding of the solution structure is explored and we investigate mechanical systems in which either the supports or the automatic ball balancer is asymmetric. Here it is shown that, provided the imbalance is small, the balanced state is robust to the considered asymmetries.     

Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

Nonlinear Dynamics - We theoretically and experimentally examine the effect of forcing and damping on systems that can be described by the nonlinear Schrödinger equation (NLSE), by making use...     

Equilibrium states in homogeneous turbulence with baroclinic instability

The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ?U _i /?x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise ${(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)}The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ∂U _i /∂x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise (?B/?x₂ = N_h² = -2WS){(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)} and vertical (?B/?x₃ = N_v²){(\partial B{/}\partial x_3\,{=}\,N_v^2)} directions. Computations based on the rapid distortion theory (RDT) are performed for several values of the rotation number R = 2Ω/S and the Richardson number R_i = N_v²/S² < 1{R_i\,{=}\,N_v^2/S^2 <1 }. It is shown that, during an initial phase, the energies and the buoyancy fluxes are sensitive to the effects of pressure and viscosity. At large time, the ratios of energies, as well as the normalized fluxes, evolve to an asymptotically constant value, while the pressure–strain correlation scaled with the product of the turbulent kinetic energy by the shear rate approaches zero. Accordingly, an analytical parametric study based on the “pressure-less” approach (PLA) is also presented. The analytical study indicates that, when R _i  < 1, there is an exponential instability and equilibrium states of turbulence, in agreement with RDT. The energies and the buoyancy fluxes grow exponentially for large times with the same rate (γ in St units). The asymptotic value of the ratios of energies yielded by RDT is well described by its PLA counterpart derived analytically. At R _i  = 0, the asymptotic value of γ increases with increasing R approaching 2 for high rotation rates. At low rotation rates, an important contribution to the kinetic energy comes from the streamwise kinetic energy, whereas, at high rotation rates, the contribution of the vertical kinetic energy is dominant. When 0 < R _i  < 1 and R 1 0{R\ne 0}, the asymptotic value of γ decreases as R _i increases so as it becomes zero at R _i  = 1.     

General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form

Within the framework of Lie group analysis of differential equations, a theorem is determined stating necessary and sufficient conditions allowing one to recover an invertible point transformation mapping a general dynamical system described by nonhomogeneous and nonautonomous first order quasilinear partial differential equations to homogeneous and autonomous form. The proof of the theorem is constructive and the new independent and dependent variables are obtained by determining the canonical variables associated to a suitable subalgebra of the Lie algebra of point symmetries admitted by the source system. The theorem is applied by considering some examples of physical interest arising from different contexts.     

Regularization and symmetry breaking for the thermistor problem with the Thomson effect

We examine a mathematical model of the thermistor taking into account the Thomson effect. We show that this system of equations is always uniquely solvable, in sharp contrast with the case where the Thomson effect is neglected. Another feature of the problem is breaking of the symmetry of the solutions in accordance with practical experiments.

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Sommario Si esamina un modello matematico per il dispositivo noto come termistore tenendo conto anche dell'effetto Thomson. Viene provato che il problema al contorno per il sistema di equazioni differenziali ordinario che regge il dispositivo ha sempre una e una sola soluzione a differenza di quanto accade quando l'effetto Thomson è transcurato. Si prova inoltre la perdita di simmetria della soluzione, fatto rilevato anche sperimentalmente.

     

Renormalization group symmetry method and gas dynamics

A new scenario in the renormalization group symmetry method is introduced to solve an initial value problem for a system of partial differential equations. As a specific example, we give an exact solution for the adiabatic perfect gas dynamics, which describes a contracting and expanding localized mass of gas.     

辛体系下含对称破缺因素动力学系统的近似守恒律

自冯康先生创立Hamilton系统辛几何算法以来,诸如辛结构和能量守恒等守恒律逐渐成为动力学系统数值分析方法有效性的检验标准之一。然而,诸如阻尼耗散、外部激励与控制和变参数等对称破缺因素是实际力学系统本质特征,影响着系统的对称性与守恒量。因此,本文在辛体系下讨论含有对称破缺因素的动力学系统的近似守恒律。针对有限维随机激励Hamilton系统,讨论其辛结构;针对无限维非保守动力学系统、无限维变参数动力学系统、Hamilton函数时空依赖的无限维动力学系统和无限维随机激励动力学系统,重点讨论了对称破缺因素对系统局部动量耗散的影响。上述结果为含有对称破缺因素的动力学系统的辛分析方法奠定数学基础。     

Spiral patterns of color symmetry from dynamics

This paper explores the esthetics of a kind of logarithmic spiral tilings that has not been investigated. It possesses the form similar to the structure of spiral galaxies, which globally displays the cyclic symmetry. The paper first studies the symmetry group associated with the spiral tiling. Then, using the generators of the group, the construction method of such tilings is analyzed in detail. To create colorful patterns on spiral tilings, a special dynamical system which is compatible with its symmetry group is designed. To promote the esthetic appeal of spiral patterns, a simple but practical strategy for generating patterns of color symmetry is presented. Based on the resulting patterns, several interesting methods are proposed to construct more types of derived spiral patterns.     

Application of joint coordinates and homogeneous transformations to modeling of vehicle dynamics

The paper presents a method of modeling dynamics of multibody systems with open and closed kinematic chains. The joint coordinates and homogeneous transformations are applied in order to formulate the equations of motion of a rigid body. In this method, constraint equations are introduced only in the case when closed subchains are considered or when the joint reactions have to be calculated. This allows the number of generalized coordinates in the system to be reduced in comparison to the case when absolute coordinates are applied. It is shown how the method can be applied to modeling of vehicle dynamics. The calculation results are compared with those obtained when the ADAMS/Car package is used. Experimental verification has been performed and is reported in the paper, as well. In both cases, a good correspondence of results has been achieved. Final remarks concerning advantages and disadvantages of the method are formulated at the end of the paper.     

A dynamical study of the evolution of pressure waves propagating through a semi-infinite region of homogeneous gas combustion subject to a time-harmonic signal at the boundary

In this paper, we study the evolution of pressure waves propagating in a region of gas combustion subject to a time-harmonic signal at the boundary. The problem is modeled by a non-linear, hyperbolic partial differential equation. Steady-state behavior is investigated using the perturbation method to ensure that enough time has passed for transient effects to have dissipated. The zeroth-order and first two approximations are obtained. Furthermore, the behavior of the following quantities is investigated, with particular attention paid to the low and high-frequency limits: the location of the peak of the first-order approximation, dispersion relations and phase speeds. Additionally, a maximum value of the perturbation parameter is determined ensuring boundedness of the solution. Next, approximate solutions are obtained in the low and high frequency limits and a comparison is made with the corresponding perturbation solution. Finally, the solution obtained from the perturbation method is compared with the long-time solution obtained by a non-standard finite-difference scheme.     

Exact solutions to drift-flux multiphase flow models through Lie group symmetry analysis

In the present paper, Lie group symmetry method is used to obtain some exact solutions for a hyperbolic system of partial differential equations (PDEs), which governs an isothermal no-slip drift-flux model for multiphase flow problem. Those symmetries are used for the governing system of equations to obtain infinitesimal transformations, which consequently reduces the governing system of PDEs to a system of ODEs. Further, the solutions of the system of ODEs which in turn produces some exact solutions for the PDEs are presented. Finally, the evolutionary behavior of weak discontinuity is discussed.     

Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents

Nonlinear Dynamics - We study nonlinear dynamics in a system of two coupled oscillators, describing the motion of two interacting microbubble contrast agents. In the case of identical bubbles, the...     

Non-linear buckling and symmetry breaking of a soft elastic sheet sliding on a cylindrical substrate

We consider the axial compression of a thin sheet wrapped around a rigid cylindrical substrate. In contrast to the wrinkling-to-fold transitions exhibited in similar systems, we find that the sheet always buckles into a single symmetric fold, while periodic solutions are unstable. Upon further compression, the solution breaks symmetry and stabilizes into a recumbent fold. Using linear analysis and numerics, we theoretically predict the buckling force and energy as a function of the compressive displacement. We compare our theory to experiments employing cylindrical neoprene sheets and find remarkably good agreement.