Coupled nonlinear oscillators: formation and decay of local modes
In this Chapter we study the conditions for the appearance and decay of Local Modes (LMs) and Normal Modes (NMs) in systems of equivalent weakly interacting nonlinear oscillators. A topological definition of LMs and NMs is provided, based on the study of the structure of phase space in the vicinity of the considered trajectory. According to such a definition an NM is a set of resonant trajectories n i ω i = n j ω j with its close vicinity. The sizes of such resonant regions, or in other words, the widths of the resonances are found. The definitions and the applied methods are based on the theory of canonical transformations and on the method of averaging developed in Chapter 2. With the help of the method of averaging the problems, considered in this Chapter, are reduced to a nonlinear Hamilton system with two degrees of freedom. These are studied with the help of Poincare surfaces of section (Poincare map). These are intersections of two-dimensional tori (with the studied trajectories on them) with the three-dimensional manifold, which fixes one of the coordinates of the system (for example, q1 = 0 and p1 > 0). In the 4-dimensional phase space the Poincare map is a one-dimensional manifold, the projection of which on the plane ( P2, q2, P1 = q1 = 0) is of interest to us. The plots of Poincare surfaces of section reveal clearly the changes of tori during the transition from local trajectories to the resonant trajectories of normal modes.
KeywordsNormal Mode Local Mode Nonlinear Oscillator Unitary Transformation Canonical Transformation
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