Semi-local analysis of the \(k:1\) and \(k:2\) resonances in quasi-periodically forced systems (Q2770230)

A \(2\)-dimensional nonlinear oscillator that can sustain, for certain values of the parameters, self-excited oscillations is considered. It is assumed that in the space of the parameters there is a nondegenerate Hopf bifurcation curve at some equilibrium. An \(m\)-dimensional quasi-periodic perturbation with fixed frequency is applied to the system. In the hypothesis that the Floquet exponents of the unperturbed system at the bifurcation are in resonance with the quasi-periodic frequency of the perturbation, the change in the global structure of the bifurcation diagram of the quasi periodically forced oscillator is studied; i.e. the author looks at what happens in the complement of the cusps that were obtained in [\textit{B. L. J. Braaksma} and \textit{H. W. Broer}, Ann. Inst. Henri Poincaré Anal. Non Linéaire 4, 115-168 (1987; Zbl 0614.34043)]. NEWLINENEWLINENEWLINEIn more detail, the following situation is considered. The vector field to be studied is of the form NEWLINE\[NEWLINEX(\sigma)=\omega \partial_x + X_0(\sigma)+ \varepsilon F(x,y,\sigma,\varepsilon)\partial_y, \quad x\in \mathbb{T}^m,\;y\in N \subseteq \mathbb{R}^2,\;\sigma \in \mathbb{R}^s,\tag{Xsig}NEWLINE\]NEWLINE where \(\varepsilon F(x,y,\sigma,\varepsilon)\partial_y\) represents a non-integrable perturbation of the unforced oscillator NEWLINE\[NEWLINEX_0(\sigma):=\left[\left( \begin{matrix} \mu & - \alpha\\ \alpha & \mu \end{matrix}\right) y + |y|^2 C(\sigma)y +R(y,\sigma)\right]\partial_y,NEWLINE\]NEWLINE and \(\omega \partial_x \) is the fixed quasi-periodic `forging'. The parameter \(\sigma\) is of the form \(\sigma= (\mu, \alpha, \sigma_3, \ldots)\) and the Hopf bifurcation curve of the unforced system is given by \(\mu=0\). The vector field (\text{Xsig}) is assumed to be in normal resonance at \(\varepsilon =0\), i.e. there is some \(k\in \mathbb{Z}^m\) and \(l\in \mathbb{Z}_{+}\) such that \(\langle k,\omega\rangle+l\alpha=0.\) By combining an averaging transformation with a Van der Pol transformation, the system is put into normal form (X)\(X=Y+\mu^3r \partial_z\), with (Y)\(Y:=\frac{1}{l}\omega \partial_x+\mu^2Z\), where Z\(Z:=(\lambda z+e^{i\theta}|z|^2z+\overline{z}^{l-1})\partial_z\). NEWLINENEWLINENEWLINEHere \(z=y_1+iy_2\), \(\lambda=\mu+i\alpha\) and \(r\) is of order \(O(1)\) as \(\mu \rightarrow 0\). The idea is to look at the vector field (\text{X}) as it was a small perturbation of the integrable vector field (\text{Y}), so that the bifurcation analysis for the former can be given in terms of persistence results of the bifurcation analysis of the latter. Upon variation of parameters, the vector field \(Z\) given in (\text{Z}) admits a rich range of possible bifurcation scenarios. These are classified in section 5.3 for the case \(l=1\) in saddle-node, Hopf, cusp, Bogdanov-Takens and codimension 3 bifurcations and in section 5.4 for the case \(l=2\) in saddle-node and pitchfork bifurcations, Hopf, degenerate pitchfork, Bogdanov-Takens, and symmetric double-zero bifurcations. Supplementary illustrative bifurcation diagrams are provided. The diagrams are `semi-local', meaning that one normalizes around \((y,\varepsilon)=(0,0)\) and let \(\lambda\) vary over a set which is large enough to capture all the interesting phenomena. The bifurcation diagram of (\text{Z}) persists in the integrable system (\text{Y}), but the interpretation changes: bifurcations of equilibria of (\text{Z}) correspond to bifurcations of quasi-periodic tori of (\text{Y}). A pecularity is that the original Hopf bifurcation curve has many more resonance points. Many bifurcations of system (\text{Y}) persist in the full system (\text{X}). The central fact is that the original Hopf bifurcation curve gets more frayed by the presence of many more low order normal resonances, whose position depends also on the strength of the perturbation.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00062].