Variational equations in parametric variables and transformation of their solutions

An analytical relationship is derived which connects the matrix of partial derivatives of the current state vector with respect to the initial conditions with the matrix of solutions to variational equations in parametric variables. These equations correspond to the equations of motion obtained using the generalized coordinate-time transformation admitting an increase of the number of unknown parameters. Particular cases of transformations leading to the variational equations in regularizing variables of Sperling-Burdet and Kustaancheimo-Stiefel are considered. All formulas necessary in these cases for determination of the matrix of isochronous derivatives are obtained.     

Limit Cycle in a Four Sectors On-Off Tracking Loop

A simplified model of a bidimensional on-off tracking loop is introduced. By using geometric analysis in the time domain, it is shown that there is a simple mode of limit cycle which is stable in the loop for finite disturbances and for a finite range of change of the loop parameters. The consideration of the unsymmetric general case covers a wide class of cases in which the loop is subjected to constant velocity inputs. A nonfiltering assumption is defined which is needed to establish the connection between the simplified loop and practical systems. The main results are applied to a conventional symmetric loop. The analytic approach developed here is applicable as weil to loops in which the number of sectors is different from four and also to loops containing nonlinearities more complex than simple on-off, like hysteresis.