Oscillation Modes of Forced Nonlinear Systems of van der Pol Type

Abstract

Oscillation modes of forced nonlinear systems of van der Pol type: \\ddotx−μ(1−x²)\\dotx+x=fsin(ωt), (\\dotx=(d⁄dt)x) are studied, where μ denotes a parameter of nonlinear damping characteristics, and f is an amplitude of external periodic force with the angular frequency ω. Introducing a multiplicity M of the oscillation mode π_r,s(m,m+1) associated with r times m peaks and s times (m+1) peaks; M=rm+s(m+1), which represents a degree of periodicity of the oscillation mode, the relations of M to the angular frequency ω are obtained for varying ω, where the amplitude f is fixed as the value 1. By numerical experiments, it is shown that what state of the oscillation mode π_r,s(m,m+1) appears for variation in ω. At first, we shall show that a set of the rotation number of the oscillation mode π_r,s(m,m+1) belongs a subset of Farey series. Secondly, we shall show that the region of angular frequency to the characteristics of oscillation modes consists of three disjoint regions of different types; Type1, Type2 and Type3. Furthermore, we shall show that in the region of Type1, the multiplicity M is proportional to inverse of the angular frequency ω, and in the region of Type2, only an oscillation mode π(1) does exist. Finally, we shall show that in the region of Type3, for the case of small damping, the value of multiplicity M decreases hyperbolically for increasing inverse of the angular frequency ω, and for the case of large damping, the value of multiplicity M increases hyperbolically for increasing inverse of the angular frequency ω. The results obtained enables us to see what oscillation mode appears for varying the angular frequency.