Periodic solutions of a perturbed Kepler problem in the plane: from existence to stability (Q288750)

The authors consider the perturbed Kepler problem in the plane, defined by NEWLINE\[NEWLINE\ddot x={-x\over\| x\|^3}+ \varepsilon\nabla_xU(t,x),NEWLINE\]NEWLINE where \(x\in\mathbb{R}^2-\{0\}\), \(\varepsilon\) is a small parameter, and \(U\) is a smooth function periodic in the variable \(t\) with period \(2\pi\). The aim is to investigate stability properties of \(2\pi\)-periodic solutions arising as perturbations of the integrable \(\varepsilon=0\) case. These periodic solutions are assumed to be without collisions; their restriction to an interval of length \(2\pi\) defines a loop in \(\mathbb{R}^2-\{0\}\) with winding number \(N\neq 0\). If \(\Sigma_N\) is the set of initial conditions \((x_0,y_0)\) in \((\mathbb{R}^2-\{0\})\times\mathbb{R}^2\) producing \(2\pi\)-periodic solutions with winding number \(N\neq 0\), then \(\Sigma_N\) is invariant under the flow associated with the Kepler problem NEWLINE\[NEWLINE\dot x= y,\quad\dot y={-x\over\| x\|^3}NEWLINE\]NEWLINE and we can average \(U(x,t)\) with respect to the flow over \(\Sigma_N\) to obtain NEWLINE\[NEWLINE\Gamma_N(x_0, y_0)= {1\over 2\pi} \int^{2\pi}_0 U(t,x(t,x_0,y_0))\,dt.NEWLINE\]NEWLINE One can then express \(\Gamma_N\) in Poincaré coordinates \((\lambda,\eta,\xi)\) and re-write \(\Gamma_N\) as \(\gamma_N\) in these coordinates. If \((\lambda^*,\eta^*,\xi^*)\) is a non-degenerate critial point of \(\gamma_N\) and \(\varepsilon>0\) is small, then the bifurcating solution \(x_\varepsilon(t)\) is elliptic (all Floquet multipliers \(\mu_i\) satisfy \(|\mu_i|= 1\), \(\mu_i\neq\pm 1\)) if NEWLINE\[NEWLINE\partial^2_{\lambda\lambda}\gamma_N(\lambda^*,\eta^*,\xi^*)>0\quad \text{and}\quad\det D^2\gamma_N(\lambda^*,\eta^*,\xi^*)>0.NEWLINE\]NEWLINE The Floquet multipliers are eigenvalues of the monodromy matrix of the linearized system.NEWLINENEWLINEThis is the authors' main result. It is not sufficient to guarantee Lyapunov stability of the periodic solution or even the stability of the linearized system.NEWLINENEWLINEThe authors' proof relies heavily on a local description of the symplectic group with two degrees of freedom.