| Abstract: |
| In this talk, we prove that general three- or four-dimensional systems
are real-analytically nonintegrable near degenerate equilibria in the
Bogoyavlenskij sense under additional weak conditions when the Jacobian
matrices have a zero and pair of purely imaginary eigenvalues or two
incommensurate pairs of purely imaginary eigenvalues at the equilibria.
For this purpose, we reduce their integrability to that of the
corresponding Poincar\{e}-Dulac normal forms and further to that of simple
planar systems, and use a novel approach for proving the analytic
nonintegrability of planar systems. Our result also implies that general
three- and four-dimensional systems exhibiting fold-Hopf and double-Hopf
codimension-two bifurcations, respectively, are real-analytically
nonintegrable under the weak conditions. To demonstrate these results,
we give two examples for the R\{o}ssler system and coupled van der Pol
oscillators. |
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