On the Asymptotic Behaviour of the Solutions of the Equations of Linear Elastodynamics in Unbounded Domains

We prove, among other things, that if the acoustic tensor satisfies a suitable growth condition at infinity (the hyperbolicity condition) and the total initial energy is summable with a suitable weight, then the solution to the initial boundary value problem of linear elastodynamics in unbounded domains decays at infinity, at every instant, with a rate depending on the weight. Moreover, we show that the hyperbolicity condition is necessary and sufficient for the equipartition in mean of the total energy.     

Asymptotic Analysis of Three-Scale Model of pH-Dependent Flows in 1:1 Clays with Danckwerts’ Boundary Conditions

The electroremediation process is an efficient method for removing pollutants from clayey soils. We model this process in kaolinite clays considering three scales—nano, micro and macro—under the assumption of a stratified geometry in conjunction with more realistic Danckwerts’ boundary conditions imposed at the electrodes. The resulting multiscale model is a coupled system of nonlinear partial differential equations. We derive analytical solutions of the macroscopic equations considering the asymptotic behavior of strongly convective and diffusive regimes. We perform numerical simulations of different scenarios for the electroremediation using the Galerkin finite element method together with a staggered algorithm and the Newton–Raphson method. We validate the accuracy of the proposed algorithm by comparing the discrete solutions to the analytical ones. Finally, we explore and discuss optimal scenarios for the electroremediation process depending on the input values of pH, electrical current, and mass inflow using dimensionless numbers defined from the analytical solutions.     

Asymptotic Behavior of Global Classical Solutions to the Mixed Initial-Boundary Value Problem for Quasilinear Hyperbolic Systems with Small BV Data

In this paper, we investigate the asymptotic behavior of global classical solutions to the mixed initial-boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t,x)|t≥0,x≥0}. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C ¹ traveling wave solutions, provided that the C ¹ norm of the initial and boundary data is bounded and the BV norm of the initial and boundary data is sufficiently small. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space-time R ¹⁺ⁿ , are also given.     

A Universal Expression for Bounds on the Torsional Rigidity of Cylindrical Shafts Containing Fibers with Arbitrary Transverse Cross-Sections

We derive upper and lower bounds for the torsional rigidity of host shafts containing a number of cylindrical fibers. The transverse cross-sections of the host shaft and the fibers are simply connected, but could be arbitrary in shape. Utilizing the fact that the torsion solution of a homogeneous host shaft with simply connected cross-section can be known, we propose a method to construct statically and kinematically admissible fields interior to the matrix and to the fibers. Previous developments on bounding the torsional rigidity of composite shaft so far are confined to circular fibers. Here we try to simulate fibers with non-circular cross-section and incorporate the interactions of the cross-sectional shapes of the host shaft and the fibers at the same time. Proceeding from extremal principles of elasticity, together with propositions of some domain integration procedures, we provide a universal expression for bounds on the torsional rigidity of the composite shaft. The exact expressions depend on the constituent information of the fibers and the host shaft, which could offer useful information to tailor the shape and the arrangement of the constituents to achieve an optimal value.     

The Theory of Thin-Walled Rods of Open Profile as a Result of Asymptotic Splitting in the Problem of Deformation of a Noncircular Cylindrical Shell

A new asymptotic approach to the theory of thin-walled rods of open profile is suggested. For the problem of linear static deformation of a noncircular cylindrical shell we consider solutions, which are slowly varying along the axial coordinate. A small parameter is introduced in the equations of the modern theory of shells. Conditions of compatibility for the shell strain measures are employed. The principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. We conclude the procedure with the subsequent solution for the field of displacements. The analysis shows that the known equations of thin-walled rods, which were previously obtained with some approximate methods using hypotheses and approximations of displacements, are asymptotically exact. The presented semi-numerical analysis of the shell equations allows us to estimate the accuracy of the obtained solution. The results of the paper constitute a sound basis to the equations of the theory of thin-walled rods and provide trustworthy information concerning the distribution of stresses in the cross-section.     

Analysis of a frictional oblique impact observed in skew bridges

The oblique contact/impact of skew bridges triggers a unique rotational mechanism which earthquake reconnaissance reports correlate with deck unseating of such bridges. Building on the work of other researchers, the present study adopts a fully non-smooth rigid body approach and set-valued force laws, in order to analyze in depth this oblique multi-impact phenomenon. A linear complementarity formulation is proposed which yields a great variety of (multi-) impact states, depending on the initial (pre-impact) conditions, such as “slip” or “stick” at one corner (single-impact) or two corners (double-impact) of the body. The pertinent existential conditions of those impact states reveal a complex dynamic behavior. With respect to the rotational mechanism associated with double-impact, the physically feasible impact states as well as, counter-intuitive exceptions are recognized. The study proves that double oblique impact, both frictionless and frictional, may or may not produce rotation of the body and proposes criteria that distinguish each case. Most importantly, it is shown that the tendency of skew bridges to rotate (and hence unseat) after deck-abutment collisions is not a factor of the skew angle alone, but rather of the overall geometry in-plan, plus the impact parameters (coefficient of restitution and coefficient of friction). The study also provides a theoretical justification of the observed tendency of skew bridges to jam at the obtuse corner and rotate in such a way that the skew angle increases. Finally, counter-intuitive trends hidden in the response are unveiled which indicate that, due to friction, a skew bridge may also rotate so that the skew angle decreases.     

Electrical Characteristics of Three‐Component Dielectric Composites with Close‐Packed Inclusions

The electric field and effective permittivity are calculated for a twodimensional threecomponent dielectric material reinforced by cylindrical fibers. A composite material with a square close packing of inclusions is considered. The field in the periodic system is investigated using the exact solution of the model problem of interaction of two dissimilar cylindrical inclusions in an external homogeneous electric field. A diagram of the relative effective permittivity is constructed.     

Analysis of nonlinear dynamic response and dynamic buckling for laminated shallow spherical thick shells with damage

Based on the nonlinear theory of shallow spherical thick shells and the damage mechanics, a set of nonlinear equations of motion for the laminated shallow spherical thick shells with damage subjected to a normal concentrated load on the top are established. According to Hertz law, the contact force acted upon the shells is determined due to the impact of a mass, and it is related to the mass and initial velocity of the striking object, the geometrical and physical character of the shell. By using the finite difference method and the time increment procedure, the nonlinear equations are resolved. In the numerical examples, the effects of the damage, the initial velocity, and mass of the striking object, the shells’ geometrical parameters on the dynamic responses and dynamic buckling of the laminated shallow spherical thick shells are discussed. Research of Y. Fu, Z. Gao and F. Zhu was supported by National Natural Science Foundation of China (No. 10572049).     

Constrained controllability of semilinear systems with delays

In the paper, finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point time-variable delays in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local relative controllability are formulated and proved. It is generally assumed that the values of admissible controls are in a convex and closed cone with the vertex at zero. The special case of constant multiple point delays is also discussed. Moreover, some remarks and comments on the existing results for controllability of nonlinear and semilinear dynamical systems are also presented.     

Time optimal motions of mechanical system with a prescribed trajectory

The problem of time minimization of a holonomic scleronomic mechanical system on a prescribed trajectory between two specified positions in configuration space is solved. The generalized force with restricted coordinates is taken as the controlling force. The application of the Green theorem (the well-known Miele method in flight mechanics) has shown that at every instant at least one control is at its boundary and possesses controlling functions with interruptions. It is assumed that at least one generalized coordinate exists that is monotonous during the interval of movement. An algorithm for numerical computation is presented for assessing the boundary of the admissible domain in the state space, thus, solving the problem of finding the optimal control as a function of time. Numerical integration is, therefore, carried out forward from the start point and backward from the end point by the use of the Runge-Kutta method. The mentioned procedure is illustrated in the example of time minimization for a manipulator which has its tip moving in a straight line. The application of the presented method simplifies solving of this type of problem compared to other methods, for instance, dynamic programming.     

Representation of discrete sequences with N-dimensional iterated function systems in tensor form

Iterated Function System (IFS) models have been used to represent discrete sequences where the attractor of the IFS is self-affine or piecewise self-affine in R ² or R ³ (R is the set of real numbers). In this paper, the piecewise hidden-variable fractal model is extended from R ³ to R ⁿ (n is an integer greater than 3), which is called the multi-dimensional piecewise hidden variable fractal model. This new model uses a “mapping partial derivative” and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the hidden variables. Therefore the result is very general. Moreover, the piecewise hidden-variable fractal model in tensor form is more terse than in the usual matrix form.     

Bifurcation and chaos of biochemical reaction model with impulsive perturbations

In this paper, a biochemical model with the impulsive perturbations is considered. By using the Floquet theorem for the impulsive equation and small-amplitude perturbation skills, we see that the boundary-periodic solution ([(x)\tilde](t),0)(\tilde{x}(t),0) is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. By numerical simulation, we can show that the system presents rich dynamics, including periodic solutions, quasi-periodic oscillations, period doubling cascades, periodic halving cascades, symmetry bifurcations, and chaos.     

Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos

This letter proposes a new 3D quadratic autonomous chaotic system which displays an extremely complicated dynamical behavior over a large range of parameters. The new chaotic system has five real equilibrium points. Interestingly, this system can generate one-wing, two-wing, three-wing and four-wing chaotic attractors and periodic motion with variation of only one parameter. Besides, this new system can generate two coexisting one-wing and two coexisting two-wing attractors with different initial conditions. Furthermore, the transient chaos phenomenon happens in the system. Some basic dynamical behaviors of the proposed chaotic system are studied. Furthermore, the bifurcation diagram, Lyapunov exponents and Poincaré mapping are investigated. Numerical simulations are carried out in order to demonstrate the obtained analytical results. The interesting findings clearly show that this is a special strange new chaotic system, which deserves further detailed investigation.     

Limit Analysis of Structures with Stochastic Strength Variations∗

On the basis of a probabilistic fomulation of the fundamental theorems of “limit analysis,” a procedure is developed which allows, with a very limited amount of computing work, the determination of a domain containing the probability distribution curve of the collapse load factor of any structure that satisfies the usual conditions for validity of the limit analysis, but has randomly distributed limit strengths.

Further improvements of the bounds thus obtained can be achieved by the equivalent of either the equilibrium or the kinematic methods of limit analysis.     

Analysis of Stability on Elastic Plates with Initial Imperfections

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Asymptotic properties of a stochastic Lotka–Volterra cooperative system with impulsive perturbations

A stochastic Lotka–Volterra cooperative system with impulsive effects is proposed and concerned. The existence and uniqueness of the global positive solution are investigated. The \(p\) th moment and the asymptotic pathwise properties are estimated. Finally, sufficient conditions for extinction and stability in the mean are presented. Our results show that the impulse does not affect the properties if the impulsive perturbations are bounded. However, if the impulsive perturbations are unbounded, then some properties could be changed significantly.     

Determination of the global responses characteristics of a piecewise smooth dynamical system with contact

In this paper the global response characteristics of a piecewise smooth dynamical system with contact, which is specifically used to describe the rotor/stator rubbing systems, is studied analytically. A method to derive the global response characteristics of the model is proposed by studying each piece of the equations corresponding to different phases of the rotor motion, i.e., the phase without rubbing, the phase with rubbing and the phase of self-excited backward whirl. After solving the typical responses in each phase and deriving the corresponding existence boundaries in the parameter space, an overall picture of the global response characteristics of the model is obtained. As is shown, five types of the coexistences of the different rotor responses and deep insights into the interactive effect of parameters on the dynamic behavior of the model are gained.     

Asymptotic Properties of a Stochastic SIR Epidemic Model with Beddington–DeAngelis Incidence Rate

In this paper, the stochastic SIR epidemic model with Beddington–DeAngelis incidence rate is investigated. We classify the model by introducing a threshold value \(\lambda \). To be more specific, we show that if \(\lambda <0\) then the disease-free is globally asymptotic stable i.e., the disease will eventually disappear while the epidemic is persistence provided that \(\lambda >0\). In this case, we derive that the model under consideration has a unique invariant probability measure. We also depict the support of invariant probability measure and prove the convergence in total variation norm of transition probabilities to the invariant measure. Some of numerical examples are given to illustrate our results.     

Lag synchronization of a class of chaotic systems with unknown parameters

Research on chaos synchronization of dynamical systems has been largely reported in literature. However, synchronization of different structure—uncertain dynamical systems—has received less attention. This paper addresses synchronization of a class of time-delay chaotic systems containing uncertain parameters. A unified scheme is established for synchronization between two strictly different time-delay uncertain chaotic systems. The synchronization is successfully achieved by designing an adaptive controller with the estimates of the unknown parameters and the nonlinear feedback gain. The result is rigorously proved by the Lyapunov stability theorem. Moreover, we illustrate the application of the proposed scheme by numerical simulation, which demonstrates the effectiveness and feasibility of the proposed synchronization method.     

Analysis of a Belyakov homoclinic connection with ?2-symmetry

We study the existence, in a two-parameter plane, of double- and triple-pulse homoclinic orbits in a ?₂-symmetric three-dimensional system, in the vicinity of a Belyakov point (a?point where the involved equilibrium in the homoclinic connection changes from saddle to saddle-focus) in the Shil??nikov zone. The first-order computation of these global connections allows us to describe their position and organization in the parameter plane. The analytical results are successfully applied in the study of such degeneration in Chua??s equation.