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For this reason (among others), the Solar System is chaotic in the technical sense defined by mathematical chaos theory, [1] and that chaotic behavior degrades even the most precise long-term numerical or analytic models for the orbital motion in the Solar System, so they cannot be valid beyond more than a few tens of millions of years into the ...
Some dynamical systems, like the one-dimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor , since then a large set of initial conditions leads to orbits ...
Like gravity attracts planetary bodies or an ocean current directs sea creatures, researchers found that there are invisible “attractors” that chaotic systems are drawn to.These attractors ...
The term "dynamics" came in a little later with Gottfried Leibniz, and over a century after Newton, Pierre-Simon Laplace introduced the term celestial mechanics. Prior to Kepler , there was little connection between exact, quantitative prediction of planetary positions, using geometrical or numerical techniques, and contemporary discussions of ...
A prototypical example of a planetary problem is the Sun–Jupiter–Saturn system, where the mass of the Sun is about 1000 times larger than the masses of Jupiter or Saturn. [18] An approximate solution to the problem is to decompose it into n − 1 pairs of star–planet Kepler problems , treating interactions among the planets as perturbations.
The perturbation must be tiny compared to the overall size of the attractor of the system to avoid significant modification of the system's natural dynamics. [ 2 ] Several techniques have been devised for chaos control, but most are developments of two basic approaches: the Ott–Grebogi–Yorke (OGY) method and Pyragas continuous control .
This is done in "Chaotic Exchange of Solid Material Between Planetary Systems: Implications for the Lithopanspermia Hypothesis" [12] to analyze the capture of solid material that may have arrived on the Earth early in the age of the Solar System to study the validity of the lithopanspermia hypothesis.
Newtonian dynamics can produce chaotic planetary orbits, especially in a system having large planets at high orbital eccentricity. [28] The need for stable orbits rules out stars with planetary systems that contain large planets with orbits close to the host star (called "hot Jupiters"). It is believed that hot Jupiters have migrated inwards to ...