Bifurcation from 2-dimensional to 3-dimensional invariant tori
The authors study the bifurcation of small invariant 2-tori into invariant 3-tori near an equilibrium with two pairs of conjugate pure imaginary eigenvalues \(\pm i\omega_ 1\), \(\pm i\omega_ 2\). The nonresonance condition \(k_ 1\omega_ 1+k_ 2\omega_ 2\neq 0,\) \((k_ 1,k_ 2)\in {\mathbb{Z}}^ 2\setminus \{(0,0)\}\), is assumed. After rewriting in ``polar normal form'', truncating, and rescaling, one obtains a 2-dimensional vector field in the rescaled polar amplitudes \((\rho_ 1,\rho_ 2)\). With two parameters \((\mu_ 1,\mu_ 2)\), there is, under certain conditions, a curve in \(\mu\)-space across which Hopf bifurcation of the amplitude equation occurs. Equilibria of the amplitude equation correspond to invariant 2-tori of the full equation, and closed orbits of the amplitude equation correspond to invariant 3-tori of the full equation. The problem is therefore to determine curves in \(\mu\)- space along which the bifurcation from 2-tori to 3-tori persists when the truncated terms in the full equation are restored. The authors solve this problem using an adaptation of KAM theory. The result is as follows. The Hopf bifurcation curve becomes a 3-dimensional surface \({\mathcal S}\) in (\(\mu\),\(\omega\))-space. There is a Cantor set of points P in \({\mathcal S}\) such that to each P correspond two cuspidal regions \(C_ -\), \(C_+\), one on each side of \({\mathcal S}\), with common vertex P. Along any curve in \(C_ -\cup C_+\) that crosses \({\mathcal S}\), a hyperbolically stable invariant 2-torus bifurcates to a hyperbolically stable invariant 3- torus.
scientific article; zbMATH DE number 4151445