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Bifurcation of periodic solutions in a rotationally symmetric oscillation system. (English) Zbl 0555.34036

We describe the bifurcation of small periodic solutions in the system ẍ\(+f(x,\lambda)x=0\), where \(x\in {\mathbb{R}}\) and f(x,\(\lambda)\) is rotationally invariant, with \(f(0,0)>0\). This problem is essentially a 4- dimensional 0(2)\(\times 0(2)\)-equivariant bifurcation problem. One easily obtains branches of circular and collinear solutions which exist for all \(\lambda\). If \(f((\rho,0),\lambda)=a(\lambda)+b(\lambda)\rho^ 2+h.o.t.\), and if b(0)\(\neq 0\), then there are no further periodic solutions. If \(b(0)=0\) and b’(0)\(\neq 0\) then there is some further bifurcation of periodic solutions with less symmetry as \(\lambda\) crosses zero. The main idea of the proof is to bring the bifurcation function in a suitable normal form.

MSC:

34C25 Periodic solutions to ordinary differential equations
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