In the previous studies, we always used the Poincaré theorem to prove the stability of order-one periodic solution.
The main idea is to set up a rectangular coordinate to debate the theorems for ascertain the stability of order-one periodic solution.
Fortunately, the analytical formula of the Poincaré map defined by the impulsive points in the phase set has been obtained, which allows us to employ it to study the existence and stability of order-1 limit cycles of model (2.2).
Investigations of the existence and stability of order-1 limit cycles of system (2.2) for the whole parameter space are quite challenging, and are similar to the study of the existence and stability of limit cycles of continuous semi-dynamical systems.
In order to employ the ideas of the Poincaré map or its successor function to address the existence and stability of order-k limit cycles, we must know the exact conditions under which the solution of model (2.2) initiating from ((x_{0}^, y_{0}^)in{mathcal{N}}) is free from impulsive effects, i.e. the more exact phase set ({mathcal{N}}) should be provided.
Cleaners like the stability of orders coming in from Rinse, the CEO said, and the fact that they don't have to do "flyering and local advertising" to drum up new business.
O2 affects the stability of ordered Br adlayer on Pt(111), but not on Ag(100) and Au(100).
In contrast to previous studies on the stability of ordered phases in Au Cu alloys, the ordered L12 phase in the Ag Au Cu alloy remained at temperatures several hundred degrees higher than predicted from the equilibrium bulk phase diagram.
Investigations on the free and entropic energies indicate that these surface-induced phase transitions are of first-order and the stabilities of order phases are intimately correlated to the strength of surface preference.
From Theorems 3.1 and 3.2, we obtain sufficient conditions for the existence and stability of positive order-1 or order-2 periodic solution.
By using the Poincaré map and the geometric method, some sufficient conditions for the existence and stability of positive order-1 or order-2 periodic solution are obtained.
