This study presents a study of bifurcation in a dynamic power system model. It becomes one of the major precautions for electricity suppliers and these systems must maintain a steady state in the neighborhood of the operating points. We study in this study the dynamic stability of two node power systems theory and the stability of limit cycles emerging from a subcritical or supercritical Hopf bifurcation by computing the first Lyapunov coefficient. The MATCONT package of MATLAB was used for this study and detailed numerical simulations presented to illustrate the types of dynamic behavior. Results have proved the analyses for the model exhibit dynamical bifurcations, including Hopf bifurcations, Limit point bifurcations, Zero Hopf bifurcations and Bagdanov-taknes bifurcations.
In this section, some classical concepts of the theory of dynamical systems are reviewed. In particular, an overview of the main features of the dynamic behavior of a system in the neighborhood of a specific type of non-hyperbolic equilibriumpoint, the saddle-node equilibriumpoint, is presented. More details on the content explored in this section can be found in [11, 21, 18].
The planar, circular, restricted three-body problem predicts the existence of periodic or- bits around the Lagrangian equilibriumpoint L1. Considering the Earth-lunar-probe sys- tem, some of these orbits pass very close to the surfaces of the Earth and the Moon. These characteristics make it possible for these orbits, in spite of their instability, to be used in transfer maneuvers between Earth and lunar parking orbits. The main goal of this paper is to explore this scenario, adopting a more complex and realistic dynamical system, the four-body problem Sun-Earth-Moon-probe. We defined and investigated a set of paths, derived from the orbits around L1, which are capable of achieving transfer between low- altitude Earth (LEO) and lunar orbits, including high-inclination lunar orbits, at a low cost and with flight time between 13 and 15 days.
This paper tried to presented a new model of SDOF system by GDTM. It uses in door closer and shock absorber. The results have been compared with the Taylor’s series and some other methods and show that our model works well and more accurate than other methods. The damping of the system increases when we decrease both orders. By considering fractional order of α and the reducing of damping coefficient we found that there would be an acceleration in the equilibriumpoint. But it becomes zero when damping coefficient increases.
real equilibrium solutions do not exist. The trace and the de- terminant of J show the stability of the point, since Det J and −Tr J must both be positive for stability, from Eq. (9). The discriminant shows whether local trajectories around the point are non-spiral or spiral, from Eq. (10). The picture is rich: the “active-biosphere” equilibriumpoint C exists as a stable node (non-spiral trajectories) for nearly all parameter choices. Points A and B form a pair, in that neither exists or both exist. When both exist, point A is always stable and point B always unstable. The discriminant is always posi- tive where points exist, indicating that spiral behaviour is not observed in this model over the slices of parameter space sur- veyed in Fig. 9.
Just after being inside the Planckian black hole for a short period of time, it would be the reverse process, during which the equilibriumpoint of the conscious- ness will go sharply in the opposite direction and would be at the lower level compar- ing to the level before the experiment. But it would be at this level for a short period of time, probably a few hours too. Then it would return to the level a little bit higher than it was before the experiment. A few years should pass before the equilibriumpoint will be inside the Planckian black hole constantly.
Lemma 2.1 (Sedaghat, 2003): Assume that , , , …be a system of difference equations and is an equilibriumpoint of . If all eigenvalues of the Jacobian matrix about the fixed point lie inside the open unit disk | | , then is locally asymptotically stable. If one of them has absolute value greater than one, then is unstable.
between the non trivial stable equilibrium population P3 and the unstable equilibrium population P2 which can be considered as a measure of the diameter of the attraction basin. For the present case the population at the unstable equilibriumpoint P2 increases from 5, when the male care is most effective, to 15 when P3 and P2 practically coincide which corresponds to the annihilation of the attraction basin forecasting a catastrophic collapse. This scenario comes up when α ≈ 220 corresponding to the critical surveillance level α crit . Therefore the population does not decrease steadily from 45 to zero going to extinction. It decreases from 45 to 15 when the system collapses and tends inevitably to extinction. The reduction of the attraction basin diameter given by the approximation between the points P2 and P3 with increasingly values of α is shown clearly in the Figure 7. For the particular value of α = α crit that brings the curve Dαh(P*) tangent to the parable f(P*) – see Figure 5 – the two points coincide and
Hence, examples of price bubbles for assets in positive net supply and infinite-lived agents, pre- sented by Santos and Woodford (1997), were regarded as being very fragile, by the authors them- selves. All examples dealt with the very special case of borrowing constraints precluding short-sales, as in this case existence of equilibrium dispensed with the above requirements. There were two types of examples: one (Example 4.5) where joint uniform impatience was violated and another (Examples 4.2 and 4.4) admitting, for some deflators, infinite present values of aggregate wealth. The former had very special endowments (zero beyond the initial date) and did not seem to accommodate the case of money. The latter were not robust to adding sufficiently productive assets.
EQUILIBRIUM SIMULATION: MONTE CARLO METHOD. We make several simulations using the Monte Carlo method in order to obtain the chemical equilibrium for several first-order reactions and one second-order reaction. We study several direct, reverse and consecutive reactions. These simulations show the fluctuations and relaxation time and help to understand the solution of the corresponding differential equations of chemical kinetics. This work was done in an undergraduate physical chemistry course at UNIFIEO.
In the sequel we always denote by F (T ) the set of fixed points of the nonexpansive semi-group T , VI(H,B,M) the set of solutions to the variational inequality (1.2) and MEP(F ) the set of solutions to the following auxiliary problem for a system of mixed equilibrium problems:
From this equation can be generated isotherms of equilibrium moisture content for cassava starch pellets (Figure 3), which have sigmoidal shape, which is the typical behavior showed by different authors for other products (BROOKER et al. , 1992; PARRA-CORONADO et al, 2008;. CHAYJAN & ESNA- ASHARI, 2010). Regression analysis showed no significant differences between the experimental data and those generated by the model.
Structures in general are well described by the Linear Theory of Elasticity, or simply Theory of Elasticity, in which two hypoth- eses are adopted and lead to the linearity of the formulated problems: Geometric Linearity and Linear Elasticity. In the first, no distinction is made between the deflected and undeflected shapes when formulating equilibrium equations. In the second, also known as Physical Linearity, it is assumed that stress and deformation components obey a linear relationship through the elastic stiffness modulus.
solution concepts. But some of these concepts are restrict to a certain kind of games. The most important solution concept was defined by John Nash (Nash, 1950). It will be seen that the Nash equilibrium existence is guaranteed for a large class of games.
From the conventional equations of the gravitational field, the point-mass concept has in this investigation been elaborated in terms of a revised renormalisation procedure. In a first application a black hole configuration of the Schwarzschild type has been studied, in which there is no electric charge and no angular momentum. A gravitational collapse in respect to the nuclear binding energy is then found to occur at a critical point mass in the range of about 0.4 to 90 solar masses. This result becomes modified if the collapse is related to other re- strictions such as to the formation of “primordial black holes” growing by the accretion of radiation and matter , or to phenomena such as a strong centrifugal force.