Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations

The necessary form of a linearizable system of two second-order ordinary differential equations y1″=f1(x,y1,y2,y1′,y2′), y2″=f2(x,y1,y2,y1′,y2′) is obtained. Some other necessary conditions were also found. The main problem studied in the paper is to obtain criteria for a system to be equivalent to a linear system with constant coefficients under fiber preserving transformations. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. Examples demonstrating the procedure of using the linearization theorems are presented.

A NOTE ON LINEARIZABILITY AND THE GLOBAL TIME AXIOM

The assumption of the existence of global time, which significantly simplifies the analysis of distributed systems, is generally safe since most of the conclusions obtained under the global time axiom can be transferred to the frame where no such assumption is made. In this note, we show that the compositionality of the well-known correctness condition for concurrent objects called linearizability does not satisfy this simplification rule: we present a simple non-linearizable system composed of two objects which are individually linearizable.