Periodic orbits and their bifurcations in a 3-D system

We study some simple periodic orbits and their bifurcations in the Hamiltonian . We give the forms of the orbits, the characteristics of the main families, and some existence diagrams and stability diagrams. The existence diagram of the family 1a contains regions that are stable (S), simply unstable (U), doubly unstable (DU) and complex unstable (). In the regionsS andU there are lines of equal rotation numberm/n. Along these lines we have bifurcations of families of periodic orbits of multiplicityn. When these lines reach the boundary of the complex unstable region, they are tangent to it. Inside the region there are linesm/n, along which the orbits 1a, describedn-times, are doubly unstable; however, along these lines there are no bifurcations ofn-ple periodic orbits. The families bifurcating from 1a exist only in certain regions of the parameter space (, ). The limiting lines of these regions join at particular points representing collisions of bifurcations. These collisions of bifurcations produce a nonuniqueness of the various families of periodic orbits. The complicated structure of the various bifurcations can be understood by constructing appropriate stability diagrams.     

Inversion of potential field data

The problem of inversion of potential field data is a challenging one because of the difficulty in obtaining a unique solution. This paper identifies various types of nonuniqueness and argues that it is neither possible nor necessary to remove all categories of nonuniqueness. Some types of nonuniqueness are due to human limitations and choice and these would always persist. Listing all the solutions, imposing additional constraints on the acceptable solutions, a priori idealization, use of a priori or supplementary information, characterizing what is common to all the solutions, obtaining extremal solutions, seeking a distribution of all possible solutions, etc. are various responses in the face of nonuniqueness. It is shown that merely the form of nonuniqueness is changed by all these techniques. Some algorithms, which should be used to obtain the global minimum of the objective function are discussed. The conceptual commonality underlying seemingly different approaches and the possibility of nonunique interpretations of the same numerical results due to different axiomatic contexts are both elucidated.