Degenerate Hopf bifurcation from multiple eigenvalues and stability
If \(\omega_ 0>0\), K is a bounded linear operator from the real Banach space A into the real Banach space B, and \(f: A\times {\mathbb{R}}\to B\) has the value zero at (0,\(\alpha)\) for all \(\alpha\in {\mathbb{R}}\), we study the existence and (linear) stability of small amplitude, \((2\pi /(\omega +\omega_ 0))\)-periodic solutions of the dynamical system \[ (*)\quad K\frac{da}{dt}=f(a,\alpha) \] with \(\alpha\) and \(\omega\) both close to zero. It is assumed that there are positive integers \(m_ 1,m_ 2,...,m_{\rho}\) and \(1=n_ 1<n_ 2<...<n_{\rho}\) such that, for \(j=1,2,...,\rho\), as \(\alpha\) increases through zero, \(m_ j\) K- eigenvalues of \(f_ a(0,\alpha)\) intersect the imaginary axis at \(in_ j\omega_ 0\) in such a way that their first k-2 derivatives at \(\alpha =0\) are zero, where \(2\leq k<\infty\). Our main result gives conditions under which the finite dimensional equation, that determines the small amplitude, \((2\pi /(\omega +\omega_ 0))\)-periodic solutions of (*) with \(\alpha\) and \(\omega\) close to zero, also determines the stability of these solutions.
periodic solutions of the dynamical system
scientific article; zbMATH DE number 4017709
scientific article