Continuation of Periodic Solutions in Conservative Systems Leveraging the First Integral of Motion

ABSTRACT Dynamical systems are often analysed with respect to a system parameter, for which numerical continuation methods are typically used. In most cases, dissipative systems are considered. Here, the continuation of periodic solutions requires the existence of an explicit parameter that can be varied during the continuation process. On the other hand, a dynamical system can be conservative, which is characterised by the existence of a first integral of motion. In contrast to dissipative systems, a family of periodic solutions typically exists even without the variation of an explicit parameter. Consequently, established continuation algorithms cannot be applied directly to compute such families of periodic solutions. Existing techniques in the literature that deal with this problem are often limited regarding their capability to control the start and end of a continuation and to isolate a specific solution. Based on the idea of an existing approach, this contribution extends the established continuation framework to conservative systems by incorporating the value of the first integral itself as continuation parameter. The proposed method is tested by computing periodic solutions of the mathematical pendulum.