On the approximations and bifurcations of a dynamic system: PMM vol. 35, no. 5, 1971, pp. 780–796

Abstract

By means of an example of a classical problem in flight dynamics we examine the influence of approximation on the structure of the partitioning of the phase space and of the parameter space of a dynamic system. For a qualitative investigation of dynamic systems we can use the transition from the original model to a simplified or piece-wise integrable one, by approximating the characteristics in the equations of motion. Here arises the important question of the admissible deviations of the approximating functions from the real characteristics for the preservation of the necessary closeness between the original and the approximating system. The concept of necessary closeness is not unique and is determined by the aims of the investigation. For example, it can be understood as the requirement of retaining for the approximating system the same phase space and parameter space partitioning structure as for the original system [1]. In a general formulation the problem reduces to the question of preserving or losing bifurcations during the transition to the approximating system. The difficulties arisising here are connected with the fact that not all the bifurcations may be kept track of by regular methods, and furthermore, for “fused” approximating systems (piecewise-analytic ones) there may arise new types of bifurcations for which there is no complete classification as yet. Therefore, a comparative analysis of actual dynamic systems under different approximations is of interest. Below we carry out such an analysis on the basis of an example of a classical problem in flight dynamics [2–9]. The choice of this problem was dictated by the fact that in the original system a wide collection of bifurcations is possible (all types of bifurcations of the first degree of structural instability are realized) and by the fact that we have succeeded in establishing strictly the parameter space partitioning structure both for the original system (which had not been done to this time) as well as for the approximating systems. Here differences arise in the partitioning structure of the parameter space and of the phase space, permitting us to evaluate the influence of the approximations on the partitioning structure and to uncover, in particular, the important role played by the “saddle index” [10]. The retention of a quantitative closeness of the characteristics did not prove to be obligatory for the preservation of the qualitative partitioning structure of the system's phase and parameter spaces. The use of the saddle index in the qualitative investigation of “fu/.sed” systems is based on the possibility of carrying over the assertions concerning the stability conditions for a separatrix loop and the conditions for the birth of limit cycles from it, to nonanatytic systems, preserving the loop in whose composition the analytic saddle occurs. Theorems 44 – 49 in [10], with appropriate changes of formulations, remain valid for the systems mentioned because the method by which they were established carry over to these systems.