Stability and chaos in nonstandard Hamiltonian planetary dynamics

It is believed today that the planets in our Solar System evolve chaotically due to secular chaos. Their rotational dynamics are therefore subject to perturbations which lead over a long period of time to instability and chaos. This problem has been studied extensively over the last decades with the aid of computer simulations which have shown that the Solar System is chaotically unstable on a timescale akin to its age. Still, no analytical theory has acceptably and satisfactory explained the origin of chaos so far. In the present study, motivated by the relevance of nonstandard (exponential and power-law) Hamiltonians in dissipative and nonconservative dynamical systems, the nonstandard Hamiltonian approach to the spin-orbit coupling has been studied by introducing two nonstandard discrete maps. These maps are dominated by the eccentricity, the asphericity parameter, and the parameters of the nonstandard Hamiltonians. It was found that chaotic behavior is a ubiquitous feature in nonstandard Hamiltonian systems, and highly irregular fractals structures arise. We considered the case of Neptune and Mercury, where we demonstrated that nonstandard Hamiltonian approach to the spin-orbit coupling is richer in patterns than the conventional approach. Several features of chaotic behavior of Neptune and Mercury have been presented based on the analyses of the Poincaré sections, the Lyapunov exponents, and the bifurcation diagrams which are reflections of the character of the trajectories in the underlying phase space. In the exponential discrete map, it was observed that both planets are non-chaotic, whereas in the power-law discrete map, Mercury may be stable in a chaotic solar system at long-term, whereas Neptune may be subject to weak instabilities. Depending on the values of the control parameters, both maps display substantially stable regimes, perpetual bifurcations, infinite number of fixed points, and chaos containing plentiful attractors. These open motivating questions addressed in the present study.