A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles.
In this interesting paper, the authors study the phenomenon of stability breakdown for nonautonomous ordinary differential equations whose time dependence is determined by a strictly ergodic flow. More precisely, an approach for a nonautonomous version of the classical transcritical bifurcation is provided. The authors work in a skew-product setting and, therefore, to consider systems of the form \[ y'=\omega(y),\quad x'=| \varepsilon| f(y,x,\varepsilon),\tag{*} \] on \(Y\times{\mathbb R}\). Here, \(Y\) is a compact subset of \({\mathbb R}^m\) invariant under the flow \(\{\tau_t\}\) induced by \(y'=\omega(y)\), \(f:{\mathbb R}^m\times{\mathbb R}\times I\to{\mathbb R}\), \(I\) denotes an open interval containing \(0\), is assumed to possess continuous partial derivatives w.r.t.~\(x\) up to order \(3\), and one has \(f(y,0,\varepsilon)\equiv 0\). Moreover, the hypotheses that the flow \((Y,{\mathbb R})\) is strictly ergodic holds throughout the paper. In order to investigate a transcritical bifurcation for \((\ast)\), i.e., its nonautonomous generalization, the authors write \[ f(y,x,\varepsilon)=a(y,\varepsilon)x+b(y,\varepsilon)x^2+p(y,\varepsilon,x) \] and assume \(\bar{a}(0)=0\), \(\bar{b}(0)<0\) for the mean values \(\bar{a}(\varepsilon),\bar{b}(\varepsilon)\) of \(a,b\), respectively. In a first result, they show that for sufficiently small \(\varepsilon>0\), \(| x_0| \) and arbitrary initial values \(y_0\), the \(\omega\)-limit set of \((y_0,x_0)\) in \(Y\times{\mathbb R}\) contains at least one and at most two minimal subsets, and there are almost automorphic extensions of \((Y,\{\tau_t\})\) [cf. \textit{W.~Shen} and \textit{Y.~Yi}, Mem. Am. Math. Soc. 647 (1998; Zbl 0913.58051)]. Under further assumptions on \(a\), there exists a local attractor \(A_{\varepsilon}\subset Y\times{\mathbb R}\) such that \(A_{\varepsilon}\cap(Y\times\{0\})=\emptyset\), and also a statement on the Hausdorff distance between \(A_{\varepsilon}\) and \(Y\times\{0\}\) holds. These results are sharpened in the quasiperiodic case of a \(k\)-dimensional torus \(Y\) with angular components \((y_1,\ldots,y_k)\), \(\tau_t(y_1,\ldots,y_k)=(y_1+\omega_1t,\ldots,y_k+\omega_kt)\,mod\,1\) and rationally independent \(\omega=(\omega_1,\ldots, \omega_k)\). Then, writing \[ a(y,\varepsilon)=a_0(y)+ \varepsilon a_1(y)+\varepsilon^2 a_2(y,\varepsilon) \] and under the conditions \(a_0(0)\equiv 0\), \(\bar{a}_1<0\) for the mean value of \(a_1\), the local attractor \(A_{\varepsilon}\) tends to \(Y\times\{0\}\) in the Hausdorff distance as \(\varepsilon\searrow 0\). In effect, one can also apply these results inside the parameter space ``bubbles'', occurring in a bifurcation theory for ODEs with quasiperiodic coefficients [cf. \textit{H. W. Broer}, \textit{G. B. Huitema}, \textit{F.~Takens} and \textit{B. L. J. Braaksma}, Mem. Am. Math. Soc. 421 (1990; Zbl 0717.58043)]. These ``bubbles'' lead to a very limited information on the bifurcating objects and are caused by Diophantine conditions for \(\omega\), whereas the present paper assumes only that \(\omega\) is rationally independent or satisfies a Liouville-type condition. Using an integral manifold theorem, these results are applied to a nonlinearly forced ``inverted pendulum'' given by a second-order nonautonomous ODE.
scientific article; zbMATH DE number 1890733