![]() Allan's values of i = 0.42 and e 0 (Allan, 1969) are therefore called into question, and taking e 0 is shown to be absolutely necessary if we want to understand the phenomena at work in the Mimas–Tethys system. A probability of capture into this secondary resonance as a function of the eccentricity of Tethys on encounter is derived, using Malhotra's method (Malhotra, 1990). It means that the system could have experienced several captures in the primary resonance, instead of a single one, and that i could have been, with this assumption, much lower than 0.4. A capture into the 1/1 secondary resonance (between the libration period of the primary ii resonance and the period of about 200yr) is found possible. ![]() The role of each one of the three terms is examined in the appearance of chaos. We also show, thanks to this model, that these terms may have brought about a stochastic layer of noticeable width at the time of capture in the ii resonance, with the consequence that the possible values of the inclination i of Mimas before capture range from 0.4 to 0.6 (these uncertainties arise from the present uncertainties on e). Although e is now very small, it is shown that this quantity could have been much greater in the past. We present a perturbedpendulum model in which these terms bring about a perturbation to the ideal ii resonance pendulum, which is in a direct ratio to the eccentricity e of Tethys. Three terms are found to generate this period. We have investigated the role of the 200yr period discovered by Vienne and Duriez (1992) on the tidal evolution of the Mimas–Tethys system through the 2:4 ii present resonance. These Hamiltonian systems were analyzed and controlled based on the practical applications, but there is no chaotic systems which is constructed based on the generalized Hamiltonian system. The chaos control of Hamiltonian systems is mainly in stability analysis and control algorithm, for example, Tao used a piecewise linear output feedback control to realize the anti-control of a generalized Hamiltonian system, and the chaotic dynamics have been analyzed Khan and Shahzad proposed the control of chaos in the Hamiltonian system of Mimas-Tethys. ![]() studied a quasi-integrable Hamiltonian system with two degree-of-freedom (DOF) and this system can show chaos by changing the deterministic intensity of bounded noise. The analysis of chaotic dynamics of Hamiltonian systems is mainly reflected in the integrable and non-integrable systems in the classical mechanics, such as Li and Chen proposed a new periodic driven nonlinear non-integrable Hamiltonian system and the chaotic phenomena were observed when the system was driven by the periodic signal with specific frequencies Farina and Pozzoli looked the beam-plasma system as a reference Hamiltonian system with many degrees of freedom and found that the development of self-consistent large amplitude oscillations occurs in correspondence with the onset of chaos Gan et al. The diffusion of the trajectory in the phase space is not only diminished, but turns the motion into strictly periodic. The latter allows to construct a control term an order of magnitude smaller than the potential to which it is added. This allows to utilize the Hamiltonian formalism to its extent, first to examine the diffusion of a dense set of initial conditions in the (now two-dimensional) phase space, and second, to employ a control method of suppressing chaos. The equation of motion is transformed to a two-dimensional model in which $f$ is the independent variable. ![]() an mLE map in a mixed space of $\omega^2$ and the initial condition. The onset of chaos, usually attributed to overlapping of the major spin-orbit resonances, for a sufficiently high value of the satellite's oblateness, $\omega^2$, is easily visible with a so called generalized bifurcation diagram, i.e. The twist is in good agreement with bifurcation diagrams constructed against $f_0$, which reveal a complicated mixture of chaotic and quasiperiodic trajectories. one orbital period, but the distribution and strength of chaos is unchanged, meaning that the character of motion does not change, but simply migrates through the phase space. It is showed that the phase flow in the full three-dimensional space is twisted with a period of $2\pi$, i.e. Maximal Lyapunov exponent (mLE) is computed in a two-dimensional space of the angular initial conditions for various initial conditions $f_0$. The model is represented in a three-dimensional phase space. A model of planar oscillations of an oblate satellite is investigated in terms of the dependence of its dynamics on the true anomaly $f$.
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