Center Equilibrium Point Research Articles

Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

The local invariant (stable, unstable, and center) manifolds of a saddle $$\times $$ center $$\times $$ center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

Einstein static universe from GUP

We consider the existence and stability of the Einstein static universe under the Generalized Uncertainty Principle (GUP) effects. We show that this solution in the presence of perfect fluid with a minimal length is cyclically stable around a center equilibrium point. By taking linear homogeneous perturbations, we find that the scale factor of Einstein static universe for closed deformed isotropic and homogeneous FLRW universe depends on the GUP coupling parameter $\alpha$. Thus, in the model by GUP effects, our universe can stay at the Einstein static state past-eternally, which means that the big bang singularity might be resolved successfully by an emergent scenario.

Stability of the Einstein static universe in f(R,\xa0T) gravity

The Einstein static (ES) universe has played a major role in various emergent scenarios recently proposed in order to cure the problem of the initial singularity of the standard model of cosmology. In the model we address, we study the existence and stability of an ES universe in the context of f(R, T) modified theories of gravity. Considering specific forms of the f(R, T) function, we seek for the existence of solutions representing ES state. Using dynamical system techniques along with numerical analysis, we find two classes of solutions: the first one is always unstable of the saddle type, while the second is always stable so that its dynamical behavior corresponds to a center equilibrium point. The importance of the second class of solutions is due to the significant role they play in constructing non-singular emergent models in which the universe could have experienced past-eternally a series of infinite oscillations about such an initial static state after which it enters, through a suitable physical mechanism, to an inflationary era. Considering specific forms for the functionality of f(R, T), we show that this theory is capable of providing cosmological solutions which admit emergent universe (EU) scenarios. We also investigate homogeneous scalar perturbations for the mentioned models. The stability regions of the solutions are parametrized by a linear equation of state (EoS) parameter and other free parameters that will be introduced for the models. Our results suggest that modifications in f(R, T) gravity would lead to stable solutions which are unstable in f(R) gravity model.

Stability of the Einstein static universe in Einstein-Cartan theory

The existence and stability of the Einstein static solution have been built in the Einstein-Cartan gravity.We show that this solution in the presence of perfect fluid with spin density satisfying the Weyssenhoff restriction is cyclically stablearound a center equilibrium point. Thus, study of this solution is interesting because it supports non-singularemergent cosmological models in which the early universe oscillates indeterminatelyabout an initial Einstein static solution and is thus past eternal.

Equilibrium points of the restricted three-body problem with equal prolate and radiating primaries, and their stability

Paper presents a complete discussion of the existence, location and stability of the equilibrium points of the coplanar restricted three-body problem with equal prolate and radiating primaries. Depending on the values of the radiation and negative oblateness parameters, two or four additional collinear equilibrium points exist, in addition to the three Eulerian points of the classical case, making up a total of up to seven collinear points. Four of these additional points, as well as the classical central equilibrium point located at the origin, are stable for certain ranges of the parameters. Also, depending on the values of the parameters, up to six additional non-collinear equilibrium points exist, in addition to the triangular Lagrangian points of the classical case. Two of these additional points are located symmetrically above and below the origin and are stable, while the other four are located symmetrically in the four quadrants and are unstable.

Stability of the Einstein static universe in the presence of vacuum energy

The Einstein static universe has played a central role in a number of emergent scenarios recently put forward to deal with the singular origin of the standard cosmological model. Here we study the existence and stability of the Einstein static solution in the presence of vacuum energy corresponding to conformally invariant fields. We show that the presence of vacuum energy stabilizes this solution by changing it to a center equilibrium point, which is cyclically stable. This allows nonsingular emergent cosmological models to be constructed in which initially the Universe oscillates indefinitely about an initial Einstein static solution and is thus past eternal.

Inverse volume corrections to emergent tachyonic inflation in loop quantum cosmology

The emergent model in the context of loop quantumcosmology with a tachyon scalar field is studied. We find that thereis a center equilibrium point in the semiclassical region and asaddle point in the classical region. If the potential of thetachyon field satisfies some conditions, the universe can stay atthe center equilibrium point past-eternally and then oscillateinfinitely around this point with the tachyon climbing up itspotential. Once the potential reaches a critical value, these twoequilibrium points coincide with each other and the oscillationphase is broken by an emergent inflation. In order to obtain asuccessful emergent tachyon inflation, a constraint onϕ̇2 of tachyon is required.

An emergent universe from a loop

Closed, singularity-free, inflationary cosmological models have recently been studied in the context of general relativity. Despite their appeal, these so called emergent models suffer from a number of limitations. These include the fact that they rely on an initial Einstein static state to describe the past-eternal phase of the universe. Given the instability of such a state within the context of general relativity, this amounts to a very severe fine tuning. Also in order to be able to study the dynamics of the universe within the context of general relativity, they set the initial conditions for the universe in the classical phase. Here we study the existence and stability of such models in the context of Loop Quantum Cosmology and show that both these limitations can be partially remedied, once semiclassical effects are taken into account. An important consequence of these effects is to give rise to a static solution (not present in GR), which dynamically is a center equilibrium point and located in the more natural semiclassical regime. This allows the construction of emergent models in which the universe oscillates indefinitely about such an initial static state. We construct an explicit emergent model of this type, in which amore » nonsingular past-eternal oscillating universe enters a phase where the symmetry of the oscillations is broken, leading to an emergent inflationary epoch, while satisfying all observational and semiclassical constraints. We also discuss emergent models in which the universe possesses both early- and late-time accelerating phases.« less

Density-vague population change

Explicit density-dependence differs from density-vagueness in the sort of regulation imposed on populations. Explicit density-dependence implies regulation toward a central equilibrium point, whereas density-vagueness implies population change away from the two extremes of extinction and immense densities. Density-vagueness does not much concern regulation within the broad range of medial densities; it is ‘liberal’.

A general analysis of genetic models with frequency-dependent mating. II. Sexual selection for heterozygotes.

Heterozygotes are assumed to mate with a frequency that is any general function, f(v), of their population frequency, v. Models are analysed in which the selection that determines the function f(v) acts either on one sex alone or on both sexes equally. The central equilibrium point v* = 1/2 always exists; it is stable if f(1/2) greater than 1/2. If the central equilibrium is unstable, other asymmetric equilibria may be stable; the fixation states may also be stable. This general analysis is applied to a number of specific models of sexual selection. The models give qualitatively different results. The outcome of selection in population cage experiments could be used to test the alternative models.