The Orbital Dynamics of Synchronous Satellites: Irregular Motions in the 2 : 1 Resonance

<table>Lyapunov exponents <svg height="15.1125" id="M367" style="vertical-align:-2.3205pt" version="1.1" viewbox="0 0 29.424999 15.1125" width="29.424999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D706"></use></g><g transform="matrix(.017,-0,0,-.017,9.446,12.162)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.327,12.162)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,23.487,12.162)"><use xlink:href="#x29"></use></g>
</svg> and <svg height="15.1125" id="M368" style="vertical-align:-2.3205pt" version="1.1" viewbox="0 0 29.424999 15.1125" width="29.424999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D706"></use></g><g transform="matrix(.017,-0,0,-.017,9.446,12.162)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,15.327,12.162)"><use xlink:href="#x32"></use></g><g transform="matrix(.017,-0,0,-.017,23.487,12.162)"><use xlink:href="#x29"></use></g>
</svg>, corresponding to the variables <svg height="15.25" id="M369" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 19.924999 15.25" width="19.924999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D44B"></use></g>
<g transform="matrix(.012,-0,0,-.012,13.525,15.187)"><use xlink:href="#x34"></use></g>
</svg> and <svg height="15.25" id="M370" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 18 15.25" width="18" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D44C"></use></g>
<g transform="matrix(.012,-0,0,-.012,11.6,15.187)"><use xlink:href="#x34"></use></g>
</svg>, respectively, for <svg height="19.924999" id="M371" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 188.83749 19.924999" width="188.83749" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D436"></use></g>
<g transform="matrix(.012,-0,0,-.012,12.05,19.863)"><use xlink:href="#x31"></use></g>
<g transform="matrix(.017,-0,0,-.017,23.1,15.775)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,37.787,15.775)"><use xlink:href="#x2212"></use></g><g transform="matrix(.017,-0,0,-.017,47.765,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,55.925,15.775)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,59.8,15.775)"><use xlink:href="#x34"></use></g><g transform="matrix(.017,-0,0,-.017,67.96,15.775)"><use xlink:href="#x36"></use></g><g transform="matrix(.017,-0,0,-.017,76.119,15.775)"><use xlink:href="#x37"></use></g><g transform="matrix(.017,-0,0,-.017,84.279,15.775)"><use xlink:href="#x37"></use></g><g transform="matrix(.017,-0,0,-.017,92.438,15.775)"><use xlink:href="#x37"></use></g><g transform="matrix(.017,-0,0,-.017,100.597,15.775)"><use xlink:href="#x38"></use></g><g transform="matrix(.017,-0,0,-.017,108.757,15.775)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,116.916,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,125.075,15.775)"><use xlink:href="#x33"></use></g><g transform="matrix(.017,-0,0,-.017,140.986,15.775)"><use xlink:href="#xD7"></use></g><g transform="matrix(.017,-0,0,-.017,160.416,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,168.575,15.775)"><use xlink:href="#x30"></use></g>
<g transform="matrix(.012,-0,0,-.012,176.737,7.613)"><use xlink:href="#x31"></use></g><g transform="matrix(.012,-0,0,-.012,182.449,7.613)"><use xlink:href="#x31"></use></g>
</svg> m<sup>2</sup>/s and <svg height="19.924999" id="M372" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 188.83749 19.924999" width="188.83749" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D436"></use></g>
<g transform="matrix(.012,-0,0,-.012,12.05,19.863)"><use xlink:href="#x31"></use></g>
<g transform="matrix(.017,-0,0,-.017,23.1,15.775)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,37.787,15.775)"><use xlink:href="#x2212"></use></g><g transform="matrix(.017,-0,0,-.017,47.765,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,55.925,15.775)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,59.8,15.775)"><use xlink:href="#x34"></use></g><g transform="matrix(.017,-0,0,-.017,67.96,15.775)"><use xlink:href="#x36"></use></g><g transform="matrix(.017,-0,0,-.017,76.119,15.775)"><use xlink:href="#x37"></use></g><g transform="matrix(.017,-0,0,-.017,84.279,15.775)"><use xlink:href="#x38"></use></g><g transform="matrix(.017,-0,0,-.017,92.438,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,100.597,15.775)"><use xlink:href="#x39"></use></g><g transform="matrix(.017,-0,0,-.017,108.757,15.775)"><use xlink:href="#x34"></use></g><g transform="matrix(.017,-0,0,-.017,116.916,15.775)"><use xlink:href="#x35"></use></g><g transform="matrix(.017,-0,0,-.017,125.075,15.775)"><use xlink:href="#x34"></use></g><g transform="matrix(.017,-0,0,-.017,140.986,15.775)"><use xlink:href="#xD7"></use></g><g transform="matrix(.017,-0,0,-.017,160.416,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,168.575,15.775)"><use xlink:href="#x30"></use></g>
<g transform="matrix(.012,-0,0,-.012,176.737,7.613)"><use xlink:href="#x31"></use></g><g transform="matrix(.012,-0,0,-.012,182.449,7.613)"><use xlink:href="#x31"></use></g>
</svg> m<sup>2</sup>/s, <svg height="17.012501" id="M373" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 48.575001 17.012501" width="48.575001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.875)"><use xlink:href="#x1D465"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.563,16.95)"><use xlink:href="#x34"></use></g>
<g transform="matrix(.017,-0,0,-.017,20.625,12.875)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,35.312,12.875)"><use xlink:href="#x30"></use></g>
<g transform="matrix(.012,-0,0,-.012,43.475,4.712)"><use xlink:href="#x2218"></use></g>
</svg> and <svg height="17.375" id="M374" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 48.924999 17.375" width="48.924999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.875)"><use xlink:href="#x1D466"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.9,16.95)"><use xlink:href="#x34"></use></g>
<g transform="matrix(.017,-0,0,-.017,20.963,12.875)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,35.649,12.875)"><use xlink:href="#x30"></use></g>
<g transform="matrix(.012,-0,0,-.012,43.825,4.712)"><use xlink:href="#x2218"></use></g>
</svg>.</table>

Mathematical Problems in Engineering

fig15

Figure 15

Figure 15: The Orbital Dynamics of Synchronous Satellites: Irregular Motions in the 2 : 1 Resonance 