Infinitesimal Stability Research Articles

Some Aspects of Differential Topology of Subcartesian Spaces

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

Unstability problem of real analytic maps

AbstractAs well known, the stability of proper maps is characterized by the infinitesimal stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal stability does not imply stability; for instance, a Whitney umbrella is not stable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.

Stability of a Groucho-Style Bounding Run in the Sagittal Plane

This paper develops a three-degree-of-freedom sagittal-plane hybrid dynamical systems model of a Groucho-style bounding quadrupedal run. Simple within-stance controls using a modular architecture yield a closed-form expression for a family of hybrid limit cycles that represent bounding behavior over a range of user-selected fore-aft speeds as a function of the model’s kinematic and dynamical parameters. Controls acting on the hybrid transitions are structured so as to achieve a cascade composition of in-place bounding driving the fore-aft degree of freedom, thereby decoupling the linearized dynamics of an approximation to the stride map. Careful selection of the feedback channels used to implement these controls affords infinitesimal deadbeat stability, which is relatively robust against parameter mismatch. Experiments with a physical quadruped reasonably closely match the bounding behavior predicted by the hybrid limit cycle and its stable linearized approximation.

On the Ellipticity of Static Equations of Strain Gradient Elasticity and Infinitesimal Stability

On the Ellipticity of Static Equations of Strain Gradient Elasticity and Infinitesimal Stability

Strong Ellipticity and Infinitesimal Stability within Nth-Order Gradient Elasticity

We formulate a series of strong ellipticity inequalities for equilibrium equations of the gradient elasticity up to the Nth order. Within this model of a continuum, there exists a deformation energy introduced as an objective function of deformation gradients up to the Nth order. As a result, the equilibrium equations constitute a system of 2N-order nonlinear partial differential equations (PDEs). Using these inequalities for a boundary-value problem with the Dirichlet boundary conditions, we prove the positive definiteness of the second variation of the functional of the total energy. In other words, we establish sufficient conditions for infinitesimal instability. Here, we restrict ourselves to a particular class of deformations which includes affine deformations.

On ellipticity of static equations of strain gradient elasticity and infinitesimal stability

Within the framework of strain gradient elasticity under finite deformations we formulate the strong ellipticity conditions of equilibrium equations. Within the model a strain energy density is a function of the first and second deformation gradients. Ellipticity involves certain constraints on the tangent elastic moduli. It is also closely related to infinitesimal stability which is defined as the positive definiteness of the second variation of the potential energy functional. Here we consider the first boundary-value problem, that is with Dirichlettype boundary conditions. For one-dimensional deformations we determine necessary and sufficient conditions of infinitesimal instability. The latter constitute two inequalities for elastic moduli.

On Strong Ellipticity and Infinitesimal Stability in Third-Order Nonlinear Strain Gradient Elasticity

On Strong Ellipticity and Infinitesimal Stability in Third-Order Nonlinear Strain Gradient Elasticity

Strong ellipticity conditions and infinitesimal stability within nonlinear strain gradient elasticity

We discuss connections between the strong ellipticity condition and the infinitesimal instability within the nonlinear strain gradient elasticity. The strong ellipticity (SE) condition describes the property of equations of statics whereas the infinitesimal stability is introduced as the positive definiteness of the second variation of an energy functional. Here we establish few implications which simplify the further analysis of stability using formulated SE conditions. The results could be useful for the analysis of solutions of homogenized models of beam-lattice materials at different scales.

Study of rotating Bénard-Brinkman convection of Newtonian liquids and nanoliquids in enclosures

Taylor-Bénard convection of water and water-based nanoliquids confined in three different types of high porosity rectangular enclosures, viz., shallow, square and tall, is studied analytically using both infinitesimal and finite amplitude stability analyses. We make use of the modified-Buongiorno-Brinkman model(MBBM) for the governing equations concerning nanoliquid-saturated porous enclosures bounded by rigid-rigid boundaries and obtain analytical results. Among three types of enclosures, maximum and minimum heat transfers are observed in tall and shallow enclosures respectively. Water well dispersed with a dilute concentration of single-walled carbon nanotubes(SWCNTs) is considered as a working medium. The water-SWCNTs is able to flow in the porous medium because the medium is loosely-packed with porosity in the range 0.5 ≤ ϕ ≤ 1. In addition to this, the maximum volume fraction of nanoparticles considered in the system is 6% and thus this does not alter the fluidity of the system. We found from the study that the presence of low concentration(volume fraction-0.06) of SWCNTs in a water-saturated porous medium effectively improves the heat transport of the system due to its high thermal conductivity and large surface area. Due to the presence of a porous medium, however, the onset of convection gets delayed and heat transport in nanoliquids gets substantially reduced in a Bénard-Brinkman configuration resulting from the weak thermal conductivity of the porous medium. Thus the porous medium acts as the heat storage system. Also, in a rotating frame of reference the heat transport gets reduced and rotation serves as an external mechanism of regulating heat transport in the system. The nonlinear dynamics of the system is studied using the 6-mode Lorenz model. Chaotic motion in the system is studied using the maximum Lyapunov exponent(MLE). The Hofp-bifurcation point of the system along with the MLE is used to investigate periodic, nearly periodic and mildly chaotic behaviors of the system.

Logarithmic forms and singular projective foliations

In this article we describe formulas for regular projective logarithmic $q$-forms and characterize those that define singular foliations of codimension $q$ in a projective space. The main result is the proof of their infinitesimal stability when $q=2$ with some extra degree assumptions. This work allows us to determine new irreducible components of the corresponding moduli space of codimension two singular projective foliations of a fixed degree, which are also generically reduced according to its scheme structure. We also set the background to extend these results to the case $2 \le q\le n-2$.

Robust [formula omitted] filtering and control for a class of linear systems with fractional stochastic noise

In this paper, a class of linear stochastic systems driven by fractional Brownian motion are investigated. The fractional infinitesimal operator and stability criterion based on the Lyapunov approach for the systems with fractional stochastic noise are employed, which are different from the results of classical stochastic systems. Firstly, the robust H∞ filtering problem is studied, and the stochastic stability and H∞ performance of the filtering error system are guaranteed by the feasibility of linear matrix inequalities. Secondly, robust H∞ control problem is investigated, and the closed-loop system driven by a fractional Brownian motion is stochastically stable and has H∞ performance if some linear matrix inequalities are feasible under the designed controller. Finally, two numerical examples show the effectiveness and correctness of the proposed methods.

Infinitesimal deformations and stability of rods made of nonlocal elastic materials

Aim of the paper is the formulation of a criterion of infinitesimal stability for a class of rods made of nonlocal elastic materials. To that end, the nonlinear equilibrium equations of naturally straight, inextensible rods subject to terminal loads are written, and the constitutive equation assumed to represent the material response in rods of finite length is discussed. Then, the equations describing the infinitesimal deformations superimposed upon a finite one are deduced. The expression of the work done by the increments of the external loads associated with an infinitesimal deformation is employed to formulate, for the considered rods, the criterion of infinitesimal stability which does not require the existence of a stored-energy function. The criterion is applied to study the stability of simply supported rods subject to axial forces, of rods with one end clamped and the other one constrained to have the tangent parallel to the undeformed rod axis, subject to axial forces and twisting couples, and of annular rods formed from naturally straight rods with the addition of twist. The results show that rods made of nonlocal elastic materials exhibit a reduction in rigidity with respect to rods having the same geometry and made of usual elastic materials with the same tensile and shear moduli.

Stability Properties of the Value Function in an Infinite Horizon Optimal Control Problem

Properties of the value function are examined in an infinite horizon optimal control problem with an integrand index appearing in the quality functional with a discount factor. The properties are analyzed in the case when the payoff functional of the control system includes a quality index represented by an unbounded function. An upper estimate is given for the growth rate of the value function. Necessary and sufficient conditions are obtained to ensure that the value function satisfies the infinitesimal stability properties. The question of coincidence of the value function with the generalized minimax solution of the Hamilton–Jacobi–Bellman–Isaacs equation is discussed. The uniqueness of the corresponding minimax solution is shown. The growth asymptotic behavior of the value function is described for the logarithmic, power, and exponential quality functionals, which arise in economic and financial modeling. The obtained results can be used to construct grid approximation methods for the value function as the generalized minimax solution of the Hamilton–Jacobi–Bellman–Isaacs equation. These methods are effective tools in the modeling of economic growth processes.

Stability aspects of the Hayes delay differential equation with scalable hysteresis

Hysteresis models are strongly nonlinear, with slope discontinuities at every load reversal. Stability analysis of hysteretically damped systems is therefore challenging. Recently, a scalable hysteresis model has been reported, motivated by materials with distributed microscopic frictional cracks. In a scalable system, if x(t) is a solution, then so is $$\alpha x(t)$$ for any $$\alpha > 0$$ . Scalability in the hysteretic damping allows us to study interesting stability aspects of otherwise linear dynamic systems. In this paper, we study the first-order Hayes delay differential equation with scalable hysteresis. Stability of this system can be examined on a two-dimensional parameter plane. We use Galerkin projection to convert the Hayes delay differential equation with hysteresis to a system of ODEs. We then use numerically obtained Lyapunov-like exponents for stability analysis. Some stability boundaries in the parameter plane contain periodic solutions, which we compute numerically by continuation and also approximately using harmonic balance and related approximations. There is an extended region in the parameter plane for which nonzero equilibria exist (like sticking solutions in scalar dry friction), and infinitesimal stability analysis thereof leads to a pseudolinear delay differential equation. On the stability boundary of the pseudolinear equation, there are solutions with linear drift in one state and periodicity in all other states. Stability regions of the original and the pseudolinearized equation overlap, but are not identical. The reason is explained in terms of differences in sets of initial conditions used for computing Lyapunov-like exponents.

Analysis of Economic Growth Models via Value Function Design

Abstract Properties of the value function are examined in an infinite horizon optimal control problem with an unlimited integrand index appearing in the quality functional with a discount factor. Optimal control problems of such type describe solutions in models of economic growth. Necessary and sufficient conditions are derived to ensure that the value function satisfies the infinitesimal stability properties. It is proved that value function coincides with the minimax solution of the Hamilton Jacobi equation. Description of the growth asymptotic behavior for the value function is provided for the logarithmic, power and exponential quality functionals. An example is given to illustrate construction of the value function in economic growth models.

The effect of damage on small-amplitude radial oscillations of an incompressible isotropic tube under pressure

Abstract This study investigates the small-amplitude radial oscillations of a damaged incompressible isotropic homogeneous limited elastic cylindrical Gent tube under constant inflation pressure. The proposed work focuses on incorporating damage into an infinitesimal stability analysis, mainly for a class of materials characterized by their limited molecular-chain extensibility. These materials mimic the responses of soft biological tissues. A stability analysis of a damaged radially oscillating cylindrical tube is necessary for studying angioplasties, surgical procedures used to correct some inflammation-based cardiac disorders. Stability theory is inferred both from the mathematical formulas presented and the graphical results obtained for pressure–stretch and frequency–stretch curves. Finally, a stability analysis for an incompressible damaged Gent cylindrical membrane is presented, and universal relations are obtained. Infinitesimal stability of a long, damaged cylindrical Gent cavity is further explored, and all results are illustrated graphically.

Infinitesimal CR automorphisms and stability groups of infinite-type models in C2

The purpose of this paper is to give explicit descriptions for stability groups of real rigid hypersurfaces of infinite type in $\mathbb C^2$. The decompositions of infinitesimal CR automorphisms are also given.

A lower bound estimate of the critical load in bifurcation analysis for incompressible elastic solids

A procedure for obtaining a lower bound estimate of the critical load for arbitrary incompressible hyperelastic solids is presented. By considering a lower bound estimate for the Hadamard functional based on the Korn inequality, we establish sufficient conditions for the infinitesimal stability of a distorted configuration. We then determine an optimal lower bound estimate of the critical load in a monotonic loading process and specialize our procedure to the case of homogeneous deformations of incompressible, hyperelastic bodies. We apply our procedure to some representative dead-load boundary value problems for Mooney–Rivlin elastic solids and discuss its effectiveness and handiness for applications by comparing our results to other estimates.

A novel strategy to identify the critical conditions for growth-induced instabilities

Geometric instabilities in living structures can be critical for healthy biological function, and abnormal buckling, folding, or wrinkling patterns are often important indicators of disease. Mathematical models typically attribute these instabilities to differential growth, and characterize them using the concept of fictitious configurations. This kinematic approach toward growth-induced instabilities is based on the multiplicative decomposition of the total deformation gradient into a reversible elastic part and an irreversible growth part. While this generic concept is generally accepted and well established today, the critical conditions for the formation of growth-induced instabilities remain elusive and poorly understood. Here we propose a novel strategy for the stability analysis of growing structures motivated by the idea of replacing growth by prestress. Conceptually speaking, we kinematically map the stress-free grown configuration onto a prestressed initial configuration. This allows us to adopt a classical infinitesimal stability analysis to identify critical material parameter ranges beyond which growth-induced instabilities may occur. We illustrate the proposed concept by a series of numerical examples using the finite element method. Understanding the critical conditions for growth-induced instabilities may have immediate applications in plastic and reconstructive surgery, asthma, obstructive sleep apnoea, and brain development.

Structural stability of attractor–repellor endomorphisms with singularities

AbstractWe prove a theorem on the structural stability of smooth attractor–repellor endomorphisms of compact manifolds, with singularities. By attractor–repellor, we mean that the non-wandering set of the dynamics f is the disjoint union of an expanding compact subset with a hyperbolic attractor on which f acts bijectively. The statement of this result is both infinitesimal and dynamical. To our knowledge, this is the first in this hybrid direction. Our results also generalize Mather’s theorem in singularity theory, which states that infinitesimal stability implies structural stability for composed mappings to the larger category of laminations.