It is rather straightforward to add a new planet, say Jupiter. Jupiter has mass $$ M_J=1.9\times 10^{27}\mathrm{kg}, $$ and distance to the Sun of \( 5.2 \) AU. The additional gravitational force the Earth feels from Jupiter in the \( x \)-direction is $$ F_{x}^{EJ}=-\frac{GM_JM_E}{r_{EJ}^3}(x_E-x_J), $$ where \( E \) stands for Earth, \( J \) for Jupiter, \( r_{EJ} \) is distance between Earth and Jupiter $$ r_{EJ} = \sqrt{(x_E-x_J)^2+(y_E-y_J)^2}, $$ and \( x_E \) and \( y_E \) are the \( x \) and \( y \) coordinates of Earth, respectively, and \( x_J \) and \( y_J \) are the \( x \) and \( y \) coordinates of Jupiter, respectively. The \( x \)-component of the velocity of Earth changes thus to $$ \frac{dv_x^E}{dt}=-\frac{GM_{\odot}}{r^3}x_E-\frac{GM_J}{r_{EJ}^3}(x_E-x_J). $$