2020/04/28 by D. D. Carpintero, J. C. Muzzio
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Classical mechanics #Eigenvalues and eigenvectors #Lyapunov exponent #Lyapunov function #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Monodromy #Monodromy matrix #Neighbourhood (mathematics) #Nonlinear system #Periodic orbits #Physics #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #astro-ph.GA #nlin.CD #physics.pop-ph
paper · pdf · doi:10.1093/mnras/staa1227
5 pages, 1, figure. Accepted for publication in MNRAS
arxiv created 2020/04/28 · openalex publication_date 2020/05/13 · arxiv updated 2020/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to show that stable, simply unstable, and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially, and fully chaotic orbits in their neighbourhood.