# Accession Number:

## AD0683742

# Title:

## PERIODIC FAMILIES OF SYSTEMS POSSESSING A FIRST INTEGRAL,

# Descriptive Note:

# Corporate Author:

## CALIFORNIA UNIV LOS ANGELES NUMERICAL ANALYSIS RESEARCH

# Personal Author(s):

# Report Date:

## 1968-01-01

# Pagination or Media Count:

## 28.0

# Abstract:

Dynamical systems are considered which have a first integral. Nonlinear oscillation theory shows that these systems can frequently give rise to two-parameter families of periodic solutions. The standard approach to determine stability is to examine the characteristic multipliers of the linear variational equation of the family of periodic solutions under question. Now discounting the two unity multipliers that will automatically exist due to the parameters of the periodic family, the remaining multipliers can supply a stability criterion. If they are all in absolute value less than one, the family is stable and if at least one has absolute value greater than one, the family is unstable. Quite obviously if the remaining multipliers all have norm one the characteristic exponents have zero real parts the question of stability becomes more delicate since now the criterion becomes sensitive to the higher order terms. If our system were Hamiltonian, the characteristic exponents of our linear variational equation appear in negative pairs and so in this case one either concludes instability or one cannot tell. The report considers this critical case for a two degree Hamiltonian system and a stability criterion is developed for such a situation. Author

# Descriptors:

# Subject Categories:

- Celestial Mechanics
- Mechanics