Detection of coarse-grained unstable states of microscopic/stochastic systems: a timestepper-based iterative protocol

We address an iterative procedure that can be used to detect coarse-grained hyperbolic unstable equilibria (saddle points) of microscopic simulators when no equations at the macroscopic level are available. The scheme is based on the concept of coarse timestepping (Kevrekidis et al. in Commun. Math. Sci. 1(4):715–762, 2003) incorporating an adaptive mechanism based on the chord method allowing the location of coarse-grained saddle points directly. Ultimately, it can be used in a consecutive manner to trace the coarse-grained open-loop saddle-node bifurcation diagrams of complex dynamical systems and large-scale systems of ordinary and/or partial differential equations. We illustrate the procedure through two indicative examples including (i) a kinetic Monte Carlo simulation (kMC) of simple surface catalytic reactions and (ii) a simple agent-based model, a financial caricature which is used to simulate the dynamics of buying and selling of a large population of interacting individuals in the presence of mimesis. Both models exhibit coarse-grained regular turning points which give rise to branches of saddle points.     

Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation

The (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation of B-type (BKP) is hereby investigated. New soliton solutions and soliton-like similarity solutions are constructed for the (2+1)-dimensional BKP equation. The similarity solutions are not travelling wave solutions when the arbitrary functions involved are chosen appropriately. Painlevé test shows that there are two solution branches, one of which has the resonance ?2. And four similarity reductions for the BKP equation are given out through nontrivial variable transformations. Moreover, abundant soliton behaviour modes of the solutions, such as soliton fusion and soliton reflection, are discussed in detail.     

Weakly coupled parametrically forced oscillator networks: existence,stability, and symmetry of solutions

In this paper, we discuss existence, stability, and symmetry of solutions for networks of parametrically forced oscillators. We consider a nonlinear oscillator model with strong 2:1 resonance via parametric excitation. For uncoupled systems, the 2:1 resonance property results in sets of solutions that we classify using a combinatorial approach. The symmetry properties for solution sets are presented as are the group operators that generate the isotropy subgroups. We then impose weak coupling and prove that solutions from the uncoupled case persist for small coupling by using an appropriate Poincaré map and the Implicit Function Theorem. Solution bifurcations are investigated as a function of coupling strength and forcing frequency using numerical continuation techniques. We find that the characteristics of the single oscillator system are transferred to the network under weak coupling. We explore interesting dynamics that emerge with larger coupling strength, including anti-synchronized chaos and unsynchronized chaos. A classification for the symmetry-breaking that occurs due to weak coupling is presented for a simple example network.     

Corner Singularities and Regularity Results for the Reissner/Mindlin Plate Model

The theory for elliptic boundary value problems for general elliptic systems is used in order to investigate systematically corner singularities and regularity for weak solutions to a broad class of boundary value problems for the Reissner/Mindlin plate model in polygonal domains. The regularity results for the deflection of the midplane and for the rotation of fibers normal to the midplane are formulated in Sobolev spaces H ^s , where s>1 is a real number. The number s depends on the geometry, the material parameters and the boundary conditions in general and is related to a decomposition of the fields in a singular and a regular part. The leading singular terms are calculated for a wide class of boundary conditions (36 combinations). The results are critically compared with those known from a stress potential approach.     

Performance comparison among some nonlinear filters for a low cost SINS/GPS integrated solution

The Kalman filter is a familiar minimum mean square estimator for linear systems. In practice, the filter is frequently employed for nonlinear problems. This paper investigates into the application of the Kalman filter’s nonlinear variants, namely the extended Kalman filter (EKF), the unscented Kalman filter (UKF) and the second order central difference filter (CDF2). A low cost strapdown inertial navigation system (SINS) integrated with the global position system (GPS) is the performance evaluation platform for the three nonlinear data synthesis techniques. Here, the discrete-time nonlinear error equations for the SINS are implemented. Test results of a field experiment are presented and performance comparison is made for the aforesaid nonlinear estimation techniques.     

Multiple crack propagation along the interface of a non-homogeneous composite subjected to anti-plane shear loading

The present work concerns with the multiple crack propagation along the interface of two bonded dissimilar strips. The crack faces are subjected to anti-plane shear traction. Galilean transformation is employed to reduce the problem to a quasi-static one. Then, using Fourier transforms and asymptotic analysis, the quasi-static problem is reduced to a pair of singular integral equations. That are solved numerically, using Gauss-Chebyshev integration formulae. The values of the dynamic stress intensity factors are obtained and compared with the previous similar works. Further, a parametric study is introduced to investigate the effect of crack growth rate, geometric and elastic characteristics of the composite on the values of dynamic stress intensity factors.     

Implicit and explicit integration in the solution of the absolute nodal coordinate differential/algebraic equations

This investigation is concerned with the use of an implicit integration method with adjustable numerical damping properties in the simulation of flexible multibody systems. The flexible bodies in the system are modeled using the finite element absolute nodal coordinate formulation (ANCF), which can be used in the simulation of large deformations and rotations of flexible bodies. This formulation, when used with the general continuum mechanics theory, leads to displacement modes, such as Poisson modes, that couple the cross section deformations, and bending and extension of structural elements such as beams. While these modes can be significant in the case of large deformations, and they have no significant effect on the CPU time for very flexible bodies; in the case of thin and stiff structures, the ANCF coupled deformation modes can be associated with very high frequencies that can be a source of numerical problems when explicit integration methods are used. The implicit integration method used in this investigation is the Hilber–Hughes–Taylor method applied in the context of Index 3 differential-algebraic equations (HHT-I3). The results obtained using this integration method are compared with the results obtained using an explicit Adams-predictor-corrector method, which has no adjustable numerical damping. Numerical examples that include bodies with different degrees of flexibility are solved in order to examine the performance of the HHT-I3 implicit integration method when the finite element absolute nodal coordinate formulation is used. The results obtained in this study show that for very flexible structures there is no significant difference in accuracy and CPU time between the solutions obtained using the implicit and explicit integrators. As the stiffness increases, the effect of some ANCF coupled deformation modes becomes more significant, leading to a stiff system of equations. The resulting high frequencies are filtered out when the HHT-I3 integrator is used due to its numerical damping properties. The results of this study also show that the CPU time associated with the HHT-I3 integrator does not change significantly when the stiffness of the bodies increases, while in the case of the explicit Adams method the CPU time increases exponentially. The fundamental differences between the solution procedures used with the implicit and explicit integrations are also discussed in this paper.     

Chaos bursting synchronization of mismatched Hindmarsh–Rose systems via a single adaptive feedback controller

Chaotic bursting synchronization of mismatched Hindmarsh–Rose neuron systems is investigated. Based on the Lyapunov stability theory, an adaptive feedback control scheme for the synchronization of the neuron systems is proposed when partially parameters of the response system are unknown and different with those of the drive system. Furthermore, in the proposed scheme, only a single adaptive feedback controller is needed, which is efficient and easy to implement. Finally, numerical simulations are provided to show the effectiveness of the developed methods.     

Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

Bifurcations and route to chaos of the Mathieu–Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.     

Dynamic model based nonlinear tracking control of a planar parallel manipulator

Based on the dynamic model, a novel nonlinear tracking controller is developed to overcome the nonlinear dynamics and friction of a planar parallel manipulator. The dynamic model is formulated in the active joint space, and the active joint friction is described with the Coulomb + viscous friction model. A nonlinear tracking controller is designed to eliminate the tracking error by using the power function. The nonlinear tracking controller is proven to guarantee asymptotic convergence to zero of both the tracking error and error rate with the Barbalat’s lemma. The trajectory tracking experiment of the proposed controller is implemented on an actual five-bar planar parallel manipulator both at the low-speed and high-speed motion. Moreover, the control performances of the proposed controller are compared with the results of the augmented PD (APD) controller.     

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.     

The Strong Ellipticity Condition Under Changes in the Current and Reference Configuration

In this note we study the condition of strong ellipticity under changes in the current and reference configuration for the finite hyperelastostatic case. The outcome is that strong ellipticity is preserved provided one adjusts the vectors used in the definition of this condition accordingly.     

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.     

The influences of both offset and mass moment of inertia of a tip mass on the dynamics of a centrifugally stiffened visco-elastic beam

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end of the beam and is located along its extended axis. This system can be thought of as an extremely simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem is solved by using Frobenius method of solution in power series. The characteristic equation is then solved numerically. The simulation results are tabulated for a variety of the nondimensional rotational speeds, tip mass, tip mass offset, mass moment of inertia and internal damping parameters. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained. Some numerical results are given in graphical form. The numerical results obtained, indicate clearly that the tip mass offset and mass moment of inertia are important parameters on the eigencharacteristics of rotating beams so that they have to be included in the modeling process.     

Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems

The problem of reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly a reliable impulsive controller is designed by the impulsive control theory. Then, some sufficient conditions for reliable impulsive lag synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

Stabilizing control of Hopf bifurcation in the Hodgkin–Huxley model via washout filter with linear control term

It is a significant issue to control bifurcation because many neuronal diseases have close relevance to bifurcation of neuron system. Some studies have been done on bifurcation control in the Hodgkin–Huxley (HH) model, but there is no clear mathematical criterion for bifurcation stabilization. In this paper, according to Routh–Hurwitz stability criterion, we employ linear control term of washout filter-aided dynamic feedback controller to stabilize bifurcation of the HH model. As a result, we can deduce linear control gain based on the criterion, and simulation shows the method is effective for making the HH model stable. The controller designs described here are achieved by electrical stimulus, so it may have potential applications in the diagnosis and therapy of dynamical diseases.     

Deformations and stresses in annular disks made of functionally graded materials subjected to internal and/or external pressure

In this paper, an analytical solution is developed to determine deformations and stresses in circular disks made of functionally graded materials subjected to internal and/or external pressure. Taking mechanical properties of the materials of circular disks to be linear variations, the governing equation is derived from basic equations of axisymmetric, plane stress problems in elasticity. By transforming the governing equation into a hypergeometric equation, an accurate analytical solution of deformations and stresses in circular disks is obtained. The comparison with the numerical solution indicates that both approaches give very agreeable results, indicating correctness of the proposed analytical solution. The obtained analytical solution is employed to determine the radial displacement and stresses in circular disks subjected to external pressure, internal pressure, and internal and external pressure, respectively. How the radius ratio of circular disks affects deformations and stresses is also investigated.     

Asymptotic Analysis of Shell-like Inclusions with High Rigidity

We study the problem of an elastic shell-like inclusion with high rigidity in a three-dimensional domain by means of the asymptotic expansion method. The analysis is carried out in a general framework of curvilinear coordinates. After defining a small real adimensional parameter ε, we characterize the limit problems when the rigidity of the inclusion has order of magnitude \frac1e\frac{1}{\varepsilon } and \frac1e³\frac{1}{\varepsilon^{3}} with respect to the rigidities of the surrounding bodies. Moreover, we prove the strong convergence of the solution of the initial three-dimensional problem towards the solution of the simplified limit problem.     

Continuously Dislocated Elastic Bodies with a Neo-Hookean Like Energy Subjected to Anti-plane Shear

In this paper, we consider a materially uniform but inhomogeneous body and we are interested in three particular cases of inhomogeneities corresponding to three distinct distributions of dislocations. The field of defects enters the equilibrium equations through the components of the tensor field describing the relaxation procedure. We examine what form should these components take in order for the material to admit states of anti-plane shear. The results obtained in this paper hold for a class of materials that obey a specific form for the stored energy function. In the special case of no dislocations, this class falls under the well known class of Neo-Hookean materials.      

The effects of variable properties on MHD unsteady natural convection heat and mass transfer over a vertical wavy surface

A mathematical model will be analyzed in order to study the effects of variables viscosity and thermal conductivity on unsteady heat and mass transfer over a vertical wavy surface in the presence of magnetic field numerically by using a simple coordinate transformation to transform the complex wavy surface into a flat plate. The fluid viscosity is assumed to vary as a exponential function of temperature and thermal conductivity is assumed to vary linearly with temperature. An implicit marching Chebyshev collocation scheme is employed for the analysis. Numerical solutions are obtained for different values of variable viscosity, variable thermal conductivity and MHD variation parameter. Numerical results show that, variable viscosity, variable thermal conductivity and MHD variation parameter have significant influences on the velocity, temperature and concentration profiles as well as for the local skin friction, Nusselt number and Sherwood number.