Climate Vibrations and Chaotic Behavior of Earth’s Inclination in the Laskar’s Model of a Moonless Earth

Abstract

The model of the moonless Earth, introduced by J. Laskar, has the form of a non-autonomous Hamiltonian system of differential equations for two variables: the cosine of the angle of inclination and the longitude of the axis of rotation of the Earth. The system describes the rotational dynamics of the Earth under the influence of the sun and planets. Earth perturbations from other planets of the solar system are considered periodic and are taken into account using the first four terms of the Fourier expansion of the corresponding part of the Hamilton function with known amplitudes and frequencies. The initial inclination of the Earth is considered as a parameter of the problem. The system was numerically integrated over a time period of 18 million years for various values of the initial inclination from 0 to 180 degrees. Three chaotic gaps of the initial inclination were found. During the bifurcation study, singular points were found and special segments of the non-autonomous system were obtained. A bifurcation diagram of the system is constructed by the initial inclination parameter. Poincare cartographic maps are constructed. The system is written in variations on the initial conditions for the Laskar system, and with its help the dependences of the problem parameter of the senior Lyapunov exponent and the averaged MEGNO indicator are calculated. The results confirm the presence of three chaotic and one regular region of variation of the bifurcation parameter of the problem.