Application of Bernoulli shift to investigation of stability of spatially periodic solutions in lattice dynamical systems

We consider lattice dynamical systems in the space of bi-infinite sequences of vectors equipped by the supremum norm. Using the general result about the spectrum of certain operators on powers of a Banach space applied to Bernoulli shift, we are able to determine the spectrum of certain linear operators on . This enables us to decide about the stability of spatially homogeneous or spatially periodic solutions of the considered lattice dynamical system.     

On the dynamical stability of D↓

It is suggested that, due to their mutual attractive potential, atoms of spinpolarized deuterium adsorbed on liquid helium will tend to form metastable dimers. These would then provide a very efficient channel for recombination, as experimentally observed.     

Unstable periodic orbits in the Lorenz attractor

We apply a new method for the determination of periodic orbits of general dynamical systems to the Lorenz equations. The accuracy of the expectation values obtained using this approach is shown to be much larger and have better convergence properties than the more traditional approach of time averaging over a generic orbit. Finally, we discuss the relevance of the present work to the computation of unstable periodic orbits of the driven Navier-Stokes equations, which can be simulated using the lattice Boltzmann method.     

On the modelling of dynamic behavior of periodic lattice structures

Summary. The aim of this contribution is to propose and apply a new approach to the formulation of mathematical models for the analysis of dynamic behavior of dense periodic lattice structures (space or plane trusses) of an arbitrary form. The modelling approach is carried out on two levels. First, we formulate a discrete model, represented by the system of finite difference equations with respect to the spatial coordinates. The obtained equations describe both low- and high-frequency wave propagation problems. Second, two continuum models are derived directly from the finite difference equations and represented respectively by the second- and the fourth-order PDEs with constant coefficients. These models have a physical sense provided that the considerations are restricted to the long wave propagation phenomena. The proposed approach is applied to the vibration analysis for a certain plane lattice structure. Special attention is given to the determination of the range of applicability of the continuum models.     

On the analysis of time-periodic nonlinear dynamical systems

In this study, a general technique for the analysis of time-period nonlinear dynamical systems is presented. The method is based on the fact that all quasilinear periodic systems can be replaced by similar systems whose linear parts are time invariant via the well-known Liapunov-Floquet (L-F) transformation. A general procedure for the computation of L-F transformation in terms of Chebyshev polynomials is outlined. Once the transformation has been applied, a periodic orbit in original coordinates has a fixed point representation in the transformed coordinates. The stability and bifurcation analysis of the transformed equations are studied by employing thetime-dependent normal form theory and time-dependent centre manifold reduction. For the two examples considered, the three generic codimension-one bifurcations, viz, Hopf, flip and tangent, are analysed. The methodology is semi-analytic in nature and provides a quantitative measure of stability even under critical conditions. Unlike the perturbation or averaging techniques, this method is applicable even to those systems where the periodic term in the linear part does not contain a small parameter or a generating solution does not exist due to the absence of the time-invariant term in the linear part.     

Pullback and forward attractors for dissipative lattice dynamical systems with additive noises

This work investigates the dissipative dynamical system in the infinite lattice ?. The dynamics of each node depends on itself and nearby nodes by a nonlinear function. When each node is perturbed with weighted Gaussian white noise, a unique pullback attractor and forward attractor exists whose domain of attraction are random tempered sets. Furthermore, we prove that the pullback and forward attractors are equivalent to a random equilibrium which is also tempered. Both convergence to the pullback and forward attractors are exponentially fast.     

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

A complete characterization of the stability boundary of an asymptotically stable equilibrium point in the presence of type-k saddle-node non-hyperbolic equilibrium points, with k ≥ 0, on the stability boundary is developed in this paper. Under the transversality condition, it is shown that the stability boundary is composed of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-0 saddle-node equilibrium points on the stability boundary and the stable centre and centre manifolds of the type-r saddle-node equilibrium points with r ≥ 1 on the stability boundary. This characterization is the first step to understanding the behaviour of stability regions and stability boundaries in the occurrence of saddle-node bifurcations on the stability boundary.     

Existence and uniqueness of periodic orbits in the cubic autocatalator

Periodic orbits are sought in a mathematical model of a simple prototype chemical reaction involving essentially only two reacting species. Physically, these periodic orbits correspond to time-periodic oscillations in the concentrations of the two chemicals. Using the results for the existence and uniqueness of periodic orbits for Lienard systems, necessary and sufficient conditions are obtained for the existence of exactly one periodic orbit and of no periodic orbits. The results apply to a closed system where the quadratic autocatalytic reaction and decay step are present but the uncatalysed reaction is not and where there is only one physically relevant equilibrium solution     

Mean-square stability of linear stochastic dynamical systems under parametric wide-band excitations

A new simplified condition is developed for determining the exponential meansquare stability margins of linear stochastic dynamical systems. It is well-known that under parametric wide-band noise disturbances, the governing equations of motion of such a system can be approximated by linear Itô stochastic differential equations (SDE). A necessary and sufficient condition for exponential mean square stability of the resulting ltô SDE is that the real parts of all the eigenvalues of the matrix describing the system of second-order moments are negative. Equivalently, the Routh-Hurwitz procedure provides conditions for stability in the form of several inequalities. In this study, it is shown that a necessary condition for the system configuration to correspond to a point on the stability boundary is that the determinant of the matrix describing the system of secondorder moments be zero. This condition is a single algebraic expression allowing for the straightforward calculation of all candidate stability boundaries. In addition, the topological properties of the stability domain are presented and shown to be useful in identifying stability boundaries and stability domains from the developed single stability boundary condition. This simplified condition provides significant advantages in the analytical and numerical estimation of the stability border and stability region of dynamical systems. The usefulness and superiority of the new condition is demonstrated by applications to example dynamical systems, including a long-span bridge model subjected to turbulent wind.     

Efficient numerical treatment of periodic systems with application to stability problems

Two efficient numerical methods for dealing with the stability of linear periodic systems are presented. Both methods combine the use of multivariable Floquet–Liapunov theory with an efficient numerical scheme for computing the transition matrix at the end of one period. The numerical properties of these methods are illustrated by applying them to the simple parametric excitation problem of a fixed end column. The practical value of these methods is shown by applying them to some helicopter rotor blade aeroelastic and structural dynamics problems. It is concluded that these methods are numerically efficient, general and practical for dealing with the stability of large periodic systems.     

Symmetric and non-symmetric periodic orbits for the digital filter map

We exhibit instances of non-symmetric periodic orbits for the digital filter map, resolving a question posed in the literature as to whether such orbits can exist. This piecewise irrational rotation, depending on a parameter a = 2cos θ, is an isometry of [?1, 1) × [?1, 1) and reflections in the two diagonals are time-reversing symmetries for the map. Symmetric orbits are plentiful and have been much investigated. Each periodic orbit is paired with a symbolic string, from the alphabet {?, 0, +}, arising under iteration of the map because of the presence of a line of discontinuity. We prove the existence of an infinite family of non-symmetric orbits where the period N starts at 29 and increases in steps of 5; they correspond to the strings (+00)⁵(+?)² 0 ^N?19. We describe several computer algorithms to find non-symmetric periodic orbits and their symbolic strings and list non-symmetric strings both for a = 0.5, and for N ≤ 100 across the parameter range. Our evidence suggests that non-symmetric orbits, though not plentiful, are characteristic of the dynamics of the map for all parameter values.     

Pseudospectral method for assessing stability robustness for linear time-periodic delayed dynamical systems

The article presents a pseudospectral approach to assess the stability robustness of linear time-periodic delay systems, where periodic functions potentially present discontinuities and the delays may also periodically vary in time. The considered systems are subject to linear real-valued time-periodic uncertainties affecting the coefficient matrices, and the presented method is able to fully exploit structure and potential interdependencies among the uncertainties. The assessment of robustness relies on the computation of the pseudospectral radius of the monodromy operator, namely, the largest Floquet multiplier that the system can attain within a given range of perturbations. Instrumental to the adopted novel approach, a solver for the computation of Floquet multipliers is introduced, which results into the solution of a generalized eigenvalue problem which is linear w.r.t. (samples of) the original system matrices. We provide numerical simulations for popular applications modeled by time-periodic delay systems, such as the inverted pendulum subject to an act-and-wait controller, a single-degree-of-freedom milling model and a turning operation with spindle speed variation.     

First principles prediction of structural stability,elastic, lattice dynamical and thermal properties of osmium carbides

Abstract

First principles calculations are performed to investigate the structural stability, elastic, lattice dynamical and thermal properties of osmium carbides with various crystal structures. Our calculation indicates that the I₄Te type structure is energetically the most favourable for Os₄C. Based on stress–strain relationships, elastic constants are obtained, and the relevant mechanical properties are also discussed. The phonon dispersion relation and the dynamical stability are also predicted. We have found that the predicted structures are mechanically stable as well as dynamically stable except for cubic-Os₄C. Through the quasi-harmonic Debye model, the temperature and pressure effects on the bulk modulus, thermal expansion coefficient, heat capacity, Grüneisen parameter and Debye temperature are presented.     

Direct control method for improving stability and reliability of nonlinear stochastic dynamical systems

Optimal control for improving the stability and reliability of nonlinear stochastic dynamical systems is of great significance for enhancing system performances. However, it has not been adequately investigated because the evaluation indicators for stability (e.g. maximal Lyapunov exponent) and for reliability (e.g. mean first-passage time) cannot be explicitly expressed as the functions of system states. Here, a unified procedure is established to derive optimal control strategies for improving system stability and reliability, in which a physical intuition-inspired separation technique is adopted to split feedback control forces into conservative components and dissipative components, the stochastic averaging is then utilized to express the evaluation indicators of performances of controlled system, the optimal control strategies are finally derived by minimizing the performance indexes constituted by the sigmoid function of maximal Lyapunov exponent (for stability-based control)/the reciprocal of mean first-passage time (for reliability-based control), and the mean value of quadratic form of control force. The unified procedure converts the original functional extreme problem of optimal control into an extremum value problem of multivariable function which can be solved by optimization algorithms. A numerical example is worked out to illustrate the efficacy of the optimal control strategies for enhancing system performance.     

Electrical conductivity of hexagonal periodic lattice structures

Periodic lattice structures are three-dimensional arrays of unit cells having carefully engineered geometric properties. Solid Freeform Fabrication (SFF) processes have made it possible to tailor structural, thermal, or electrical properties by varying the shape and density of the unit-cell geometry. In this paper, the electrical conductivity of a hexagonal lattice structure is analytically derived using an effective unit-cell approach. The relationship between ligament length, ligament radius, relative density and electrical conductivity has been derived. The analysis indicates that the electrical conductivity increases with relative density and is linearly dependent on relative density at low lattice densities. Conductivity measurements of Ti-6Al-4V hexagonal lattices made via the Electron Beam Melting (EBM) process over a range of relative densities from 4% to 16% were taken in order to experimentally validate the analytical models.     

Time shift between unstable periodic orbits of coupled chaotic oscillators

The shift Δt between unstable periodic orbits of coupled oscillators occurring in the chaotic synchronization regime has been studied. It is shown that this time shift is the same for all equiphase orbits with various topological parameters and depends on the coupling parameter ε. This dependence obeys the universal power law Δt ∼ εⁿ with an exponent of n = −1.     

New variational principles for locating periodic orbits of differential equations

We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier-Stokes equations, simulated using the lattice Boltzmann equation.     

Stabilization of periodic orbits near a subcritical Hopf bifurcation in delay-coupled networks

We study networks of delay-coupled oscillators with the aim to extend time-delayed feedback control to networks. We show that unstable periodic orbits of a network can be stabilized by a noninvasive, delayed coupling. We state criteria for stabilizing the orbits by delay-coupling in networks and apply these to the case where the local dynamics is close to a subcritical Hopf bifurcation, which is representative of systems with torsion-free unstable periodic orbits. Using the multiple scale method and the master stability function approach, the network system is reduced to the normal form, and the characteristic equations for Floquet exponents are derived in an analytical form, which reveals the coupling parameters for successful stabilization. Finally, we illustrate the results by numerical simulations of the Lorenz system close to a subcritical Hopf bifurcation. The unstable periodic orbits in this system have no torsion, and hence cannot be stabilized by the conventional time delayed-feedback technique.     

Uniaxial local buckling strength of periodic lattice composites

To reveal the local buckling strength of periodic lattice composites, an important factor in optimal material design, analytical method based on the classical beam-column theory was applied. Buckling modes were decided according to the condition that the curvature of the strut columns is the smallest. Characteristic equations were built according to the equilibrium equations. The buckling strengths and constraint factors of various grids under uniaxial compression and tension were achieved. The strut network supports stronger rotation restrictions than pin-jointed nodes but weaker than the built-in ends. With more stacks of struts and connectivity at nodes, the restriction must be stronger and the buckling load is greater. Commonly, the constraint factors of isogrids and mixed triangle grids are greater than Kagome grids. The regular honeycomb and square grids possesses smaller buckling loads.