# Accession Number:

## AD0763598

# Title:

## Infinitely Many Periodic Trajectories of the Generalized Lienard Differential System,

# Descriptive Note:

# Corporate Author:

## ILLINOIS UNIV URBANA COORDINATED SCIENCE LAB

# Personal Author(s):

# Report Date:

## 1973-05-01

# Pagination or Media Count:

## 90.0

# Abstract:

Several recent theorems provide sufficient conditions for the existence or for the existence and stability of infinitely many periodic trajectories of the the generalized Lienard system dxdt muFx - y, dydt gx, mu,x,y real. First by applying a diffeomorphism of the plane two of these theorems which require that gx identically equal to x are extended to the case where g is spring-like, i.e. g satisfying the hypothesis xgx 0 for x not 0. For a third theorem which requires g to be spring-like, the conditions on F are relaxed by using a diffeomorphism of the plane. Next the generalized Lienard system is considered under the assumption that there exists an interval x1, x2 such that gx is negative to the left of x1 and gx is positive to the right of x2. Sufficient conditions are established which, when is sufficiently small, guarantee that infinitely many periodic trajectories exist and that those trajectories are alternately stable and unstable. A useful asymptotic expansion for a function involved in these sufficient conditions is obtained. The results are applied to the system with Fx periodic of mean zero and gx asymptotic to the identity function. Author

# Descriptors:

# Subject Categories:

- Numerical Mathematics
- Mechanics