Narrow Results

A pendulum excited by the combination of vertical and horizontal forcing at the pivot point was considered and the period-1 rotational motion was studied. Analytical approximations of period-1 rotations and their stability boundary on the excitation parameters (ω, p)-plane are derived using asymptotic analysis for the pendulum excited elliptically and along a tilted axis. It was assumed that the damping is small and the frequency of the base excitation is relatively high. The accuracy of the approximations was examined for different values of the parameters e and κ controlling the shape of excitation, and it was found that using the second and third order approximations ensures a good correspondence between analytical and numerical results in the majority of cases. Basins of attractions of the coexisting solutions were constructed numerically to evaluate the robustness of the obtained rotational solutions. It was found that the horizontal component of excitation has a larger effect on the shift in position of the saddle node bifurcations for the elliptically excited case than for the pendulum excited along a tilted axis. For the elliptically excited pendulum with pivot rotating in the same direction as the pendulum the stability boundary is shifted downwards providing a larger region of the solution existence. When the pendulum and the pivot rotate in opposite directions, the boundary is shifted upwards significantly limiting the region of the solution existence. In contrast, for the pendulum excited along the tilted axis, the direction of the rotation has a minor effect for low frequency values and the addition of the horizontal component always results in a larger region of the solution existence.

This paper presents analytical derivations to study periodic solutions for the two-body problem perturbed by the first zonal harmonic parameter. In particular, three different semianalytical approaches to solve this problem have been studied: (1) the classic perturbation theory, (2) the Lindstedt–Poincaré technique, and (3) the Krylov–Bogoliubov–Mitropolsky method. In addition, the numerical integration by Runge–Kutta algorithm is established. However, the numerical comparison tests show that by increasing the value of angular momentum the solutions provided by Lindstedt–Poincaré and Krylov–Bogoliubov–Mitropolsky methods become similar, and they provide almost identical results using a smaller value for the perturbed parameter which quantify the dynamical flattening of the main body, the Krylov–Bogoliubov–Mitropolsky provides more accurate results to design elliptical periodic solutions than Lindstedt–Poincaré technique when the perturbed parameter has a relatively large value, regardless of the value of angular momentum. This study can be applied to equatorial orbits to obtain closed-form analytical solutions.

In this paper, we consider a class of predator–prey equations with state-dependent delayed feedback. Firstly, we investigate the local stability of the positive equilibrium and the existence of the Hopf bifurcation. Then we use perturbation methods to determine the sub/supercriticality of Hopf bifurcation and hence the stability of Hopf bifurcating periodic solutions. Finally, numerical simulations supporting our theoretical results are also provided.

Computational geometry algorithms branch on the signs of predicates. Prior predicate evaluation techniques are slow on degenerate (zero sign) predicates, especially on predicates on algebraic numbers. Degeneracy is common for predicates whose arguments have common antecedents. We present three randomized algorithms for degeneracy detection. The first algorithm uses modular arithmetic to detect degenerate predicates on rational numbers. The second algorithm uses quotient rings to reduce detecting degenerate predicates on algebraic numbers to multiple rational predicates, which can be evaluated deterministically or using the first algorithm. This algorithm is impractical because it is exponential in the number of algebraic numbers, yet it is still much faster than prior work that uses root separation bounds. The third algorithm uses a perturbation to eliminate degenerate algebraic predicates that are not identical to zero and a second perturbation to detect those that are. The first and third algorithms are incorporated into an exact geometric computation library. By sampling values generated by the algorithms, the library estimates the degeneracy detection failure probability over the lifetime of every calling program. We call this approach statistical degeneracy detection (SDD). Extensive testing shows that predicate evaluation is reliable and fast.

Employing higher-order perturbation theory, we find a new class of perturbative extremal rotating black hole solutions with Born–Infeld electric charge in odd D dimensional spacetime. The seed solution is an odd-dimensional extremal Myers–Perry black hole with equal angular momenta to which a perturbative, nonlinear, electric Born–Infeld field charge q is added maintaining the extremality condition. The perturbations are performed up to third-order. We also study some physical properties of these black holes. In particular, it is shown that the values of the gyromagnetic ratio of the black holes are modified by the perturbative parameter q and the Born–Infeld parameter β.

We prove the existence of fountain-like solutions, obtained by superposition of bubbles of different blow-up orders, for a nonlinear elliptic equation with critical growth and Hardy-type potential.

The well-known vibration model of axially moving beam is considered. Both axial moving speed and axial force are assumed to vary harmonically. The Method of Multiple Time Scales (a perturbation method) is used. The natural vibrations of beam are considered for different values of beam parameters. Resonances are obtained for seven different conditions. Solvability conditions for each resonance case are found analytically. Effects of transport velocity, axial force, rigidity and damping are discussed. Stability analysis are obtained for principal parametric resonances. Stable and unstable regions are obtained regarding velocity and force effects separately and together.

This study analyzes the vibration movements of multi-support micro beams placed in an electrically smooth area using the modified couple stress theory. It has been assumed that the potential voltage that creates the electrical field strength varies harmonically. Large number of experiments in recent years have indicated that classical continuum theory is unable to predict the mechanical behavior of microstructure with small size. However, nonclassical continuum theory should be used to accurately design and analyze the microstructures. Modified couple stress theory models the micro and nanomechanical systems with higher accuracy because they employ additional material parameters to the equation considering size dependent behavior. The most general nonlinear motion equations for multi-support microbeams have been obtained by considering the material size parameter, the number of support and support positions, damping effect, axial stresses, electrical field strength, and nonlinear effects resulting from elongations. The nonlinear equations of motion are obtained according to the Hamilton method using the modified couple stress theory (MCST). The resulting equations of motion are nondimensionalized. In this way, the mathematical model has been made independent of the type and geometric structure of the material. Approximate solutions of the obtained dimensionless motion equation are obtained by the multi-scale method, which is one of the perturbation methods. As a result, an increase occurs in the first mode frequencies (${\omega }_1$) and nonlinear correction effect parameters ($\lambda ({\omega }_1)$) with the progress of the center support position gradually towards $\eta =0.5$ and the increase of the microbeam elasticity coefficient (${\alpha }_2$).

A perturbation algorithm Multiple Scales Modified Lindstedt–Poincare (MSMLP), combination of method of Multiple Scales and modified Lindstedt–Poincare is proposed for the solution of Quintic Duffing equation which combines the advantages of both the methods. Solution obtained by the MSMLP method is compared with the Multiple Scales method and accurate closed form approximate solution of the Quintic Duffing equation. The proposed method produces better results for a wide range of amplitude values of oscillations and strong nonlinearities. Numerical simulation has been performed in MATHEMATICA 7.0.

Extremal Kähler metrics were introduced by E. Calabi as best representatives of a given Kähler class of a complex compact manifold, these metrics are critical points of the L² norm of the scalar curvature function. In this paper, we report some joint works with C. Arezzo and M. Singer concerning the construction of extremal Kähler metrics on blow ups at finitely many points of Kähler manifolds which already carry an extremal Kähler metric. In particular, we give sufficient conditions on the number and locations of the blown up manifold points for the blow up to carry an extremal Kähler metric.