On the Asymptotic Solution of Non‐Hamiltonian Systems Exhibiting Sustained Resonance

With the exception of some special examples, much of the literature on the formal construction of asymptotic solutions of systems exhibiting sustained resonance concerns Hamiltonian problems, for which the reduced problem is of order two when a single resonance is present. In the Hamiltonian case, the resonance manifold is a curve that is explicitly defined by the governing equations and is independent of the actual sustained resonance solution. When the basic standard form system is non‐Hamiltonian, with M slow and N fast variables, the corresponding reduced problem is of order M + 1; in general it involves all of the slow variables, P 1 ,…, P M , plus the resonant phase Q . In this paper, the solution of a general non‐Hamiltonian system in standard form is formally constructed for the case of a single sustained resonance. First, a well‐known example is reviewed, for which the projection of the solutions on the resonance manifold can be derived a priori, independent of the evolution of Q . Then, the general case is solved, using a generalization of the multiple scale method of Kuzmak‐Luke, where knowledge of the asymptotic solution for Q (as well as higher‐order terms) is needed to define the projection of the solution on the resonance manifold. The results simplify significantly when initial conditions are chosen exactly on the resonance manifold; the modifications necessary for arbitrary initial conditions are also given. Two examples are discussed in detail to illustrate the procedure. The asymptotic results are confirmed for several test cases by comparison with numerical integrations of the exact equations.