Topological chaos of universal elementary cellular automata?rule

The dynamical behaviors of elementary cellular automata rule 110 are analyzed from the viewpoint of symbolic dynamics in the space of bi-infinite symbolic sequences. This paper conducts a rigorous analysis of the relationship between rules 110, 170 and 240 by applying blocking transformation and releasing transformation. Based on this result, the topological chaos of T ₁₁₀ induced by rule 110 is evaluated; that is, $T_{110}^{9}$ and $T_{110}^{16}$ are topologically mixing and possess the positive topological entropies on their respective subsystems. Therefore, it is natural to argue that the intrinsic complexity of rule 110 is high according to the usual measure of complexity organized around the symbolic dynamics of stationary symbol sequences. Finally, it is worth mentioning that the method presented in this paper is also applicable to other blocking transformation equivalences therein.     

A Topological Proof of Stability of N-Front Solutions of the FitzHugh–Nagumo Equations

Consideration is devoted to traveling N-front wave solutions of the FitzHugh–Nagumo equations of the bistable type. Especially, stability of the N-front wave is proven. In the proof, the eigenvalue problem for the N-front wave bifurcating from coexisting simple front and back waves is regarded as a bifurcation problem for projectivised eigenvalue equations, and a topological index is employed to detect eigenvalues.     

Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to the depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg–Landau framework.     

On the Existence of Periodic Solutions of the Navier–Stokes Equations in a Thin Domain Using the Topological Degree

We use the method of the topological degree, the theory of fractional powers of positive operators, and the Grisvard formula together with results proved by G. Raugel and G. R. Sell to study the periodic solutions of the incompressible Navier–Stokes equations in a thin three-dimensional domain.     

Topological and non-topological solitons of the Klein–Gordon equations in 1+2 dimensions

This paper obtains the topological and non-topological 1-soliton solution of the Klein–Gordon equation in 1+2 dimensions. There are five various forms of this equation that will be studied. The solitary wave ansatz will be used to carry out the integration.     

Topological horseshoe analysis and circuit realization for a fractional-order Lü system

The paper first discusses a newly reported fractional-order Lü system of order as low as 2.7 and shows its chaotic characteristics by numerical simulations. Then by using the topological horseshoe theory and computer-assisted proof, the existence of chaos in the system is verified theoretically. Finally, an analog hardware circuit is made for the fractional-order system, and the observed results demonstrate that the fractional-order Lü system is chaotic in physical experiment.     

Crack tortuousity in the nacreous layer – Topological dependence and biomimetic design guideline

The nacreous layer in seashells is known for two phenomenal aspects: light-weightiness and superior fracture toughness. Of a multitude of toughening mechanisms, the highly meandering nature of the crack path through its staggered architecture has been reported to contribute approximately a third of its overall toughness. In the current article, we are trying to establish the scientific rationale associated with the influence of overlap length on the crack-tip driving force from a local perspective via development of a simplified analytical model. Characteristic overlap lengths computed showed reasonable agreement with the values reported in the nacreous layer and previously published experimental data. Biomimetic design guideline obtained from the current investigation would thereby lead to development of synthetic staggered architecture materials with improved stiffness, load-transfer and toughness.     

Topological classification of affine mappings from ℝ<Superscript>2</Superscript> into ℝ<Superscript>2</Superscript>

We investigate affine mappings from ℝ² into ℝ² and establish necessary and sufficient conditions for the topological conjugacy of these mappings. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 472–480, October–December, 2008.     

Investigation of the Stability Problems of Elastic Bodies Using the Method of Mathematical Theory of Elasticity

In this paper,using the equilibrium equations and boundary conditionsof elastic stability problem of Новожилов and the method of mathematicaltheory of elasticity,we solve some elastic stability problems,which werestudied byищлинскииandвоицеховская,and obtained more reason-able results than theirs.     

Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity

Denoting by the stress tensor, by the linearized strain tensor, by A the elasticity tensor, and assuming that is a convex potential, the inclusion accounts for nonlinear viscoelasticity, and encompasses both the linear Kelvin–Voigt model of solid-type viscoelasticity and the Prager model of rigid plasticity with linear kinematic strain-hardening. This relation is assumed to represent the constitutive behavior of a space-distributed system, and is here coupled with the dynamical equation. An initial- and boundary-value problem is formulated, and the existence and uniqueness of the solution are proved via classical techniques based on compactness and monotonicity. A composite material is then considered, in which the function and the tensor A rapidly oscillate in space. A two-scale model is derived via Nguetseng’s notion of two-scale convergence. This provides a detailed account of the mesoscopic state of the system. Any dependence on the fine-scale variable is then eliminated, and the existence of a solution of a new single-scale macroscopic model is proved. The final outcome is at variance with the nonlinear extension of the generalized Kelvin–Voigt model, which is based on an apparently unjustified mean-field-type hypothesis.     

Model of Self‐Organization of an Ensemble of Cracks Radiating Sound

A model of selforganization of cracks arising in a rock specimen (granite) compressed by a press is proposed. The model is based on the assumption of acoustic wave interaction between the cracks. To construct the model of selforganization of cracks, solutions of the Fokker–Planck equation are used. The experimentally observed spontaneous increase in the activity of acoustic emission, spatial and temporal clusterization, and formation of a fractal structure in rock specimens under constant and slowly varying loads are explained.     

Estimation of overall properties of random polycrystals with the use of invariant decompositions of Hooke’s tensor

In the paper the theoretical analysis of bounds and self-consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented. In the study two invariant decompositions of Hooke’s tensors are used. The applied method enables derivation of novel expressions for estimates of the bulk and shear moduli, which depend on invariants of local stiffness tensor. With use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained.     

Modeling of the Front of Melting and Destruction of a Melt Film During Gas‐Laser Cutting of Metals

The products of laser cutting of metals on an automated laser setup are investigated. Results of model experiments are presented, where soft wax was used instead of metal transforming into the melt; soft wax filled a narrow flat slot between two glass plates and was removed by a heated air stream. The physical processes of melting of the liquidwax film, its destruction, and entrainment by the gas jet being assumed to be analogous to the processes of metalmelt spraying inside the cut in fullscale experiments, the characteristic size of drops formed thereby is evaluated. The modeling results are in qualitative agreement with the results of fullscale experiments. It is shown that the quality of laser cutting of metals directly depends on the character of spraying of the liquid melt and the process of its removal.     

Table of Contents of Volume 35 – 2000

Volume Contents

Table of Contents of Volume 35 – 2000     

Table of Contents of Volume 36 – 2001

Volume Contents

Table of Contents of Volume 36 – 2001     

Group foliation of the Lamé equations of the classical dynamical theory of elasticity

We perform the group foliation of the system of Lamé equations of the classical dynamical theory of elasticity for an infinite subgroup contained in a normal divisor of the main group. The resolving system of this foliation includes the following two classical systems of mathematical physics: the system of equations of vortex-free acoustics and the system of Maxwell equations, which allows one to use wider groups to obtain exact solutions of the Lamé equations. We obtain a first-order conformal-invariant system, which describes shear waves in a three-dimensional elastic medium. We also give examples of partially invariant solutions.     

Influence of small harmonic terms on eigenvalues of monodromy matrix of piecewise-linear oscillators

In this paper we consider the problem which can appear at the determination of the dynamical stability of the responses of oscillators with discontinuous or steep derivative of the restoring characteristic obtained in the frequency domain. For that purpose, a simple one degree-of-freedom system with piecewise-linear force-displacement relationship subjected to a harmonic excitation is analysed. Stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the Floquet-Liapounov theorem. Confirmation of the results obtained in the frequency domain is done by comparing with the results obtained in the time domain by the method of piecing the exact solutions. Determination of the dynamical stability can be made more reliable by using the proposed plot of maximum modulus of the eigenvalues of the monodromy matrix in dependence of non-dimensional frequency and the number of harmonics included in the supposed approximate solution.     

The exact theory of bending of prismatic shells — an application of transfer matrices

Conclusion In this paper a new application of transfer matrices has been made in connection with the exact theory of bending of prismatic shells. It is shown that use of transfer matrices reduces the number of unknowns from 8 n to four, where n is the total number of walls, for a given integer m. This simplification is specially applicable to structures with open or simply connected closed sections.     

Reliability Analysis of Structural Members Composed of Several Random Components,Making Use of Theory of Random Processes∗

This paper presents a method of reliability analysis of structural members, considering both the load and the resistance to be composed of several random components. The analysis is based on the theory of linear statistical approximation, extended to random processes. To illustrate application of the present theory, numerical calculations are carried out on a singly reinforced concrete beam with rectangular cross section and on a steel compression member with random initial deflections     

Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.