Bifurcations of Nontwisted Heteroclinic Loop with Resonant Eigenvalues

By using the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits to establish the local coordinate systems in the small tubular neighborhoods of the heteroclinic orbits, we study the bifurcation problems of nontwisted heteroclinic loop with resonant eigenvalues. The existence, numbers, and existence regions of 1-heteroclinic loop, 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits are obtained. Meanwhile, we give the corresponding bifurcation surfaces.

Bifurcations and Stability of Nondegenerated Homoclinic Loops for Higher Dimensional Systems

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence ofk-homoclinic loop andk-periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.

On the Stability of Rotors in Cylindrical Journal Bearings

The regions of stability for plain cylindrical journal bearings have been determined analytically here. The linear “variational” equation of motion has been employed to obtain the stability regions bounded by families of load-carrying capacity and operating eccentricity curves. The results were applied to the “quasi-static” equilibrium case for gas lubricated cylindrical journal bearings of L/D = 2. They show that there exists a “minimum” in the stability curves, a prediction supported by experimental evidence. The results of this work seem to bridge together observation on stability at very small clearances and large ones.