  The dynamical evolution of the solar system is chaotic with a Lyapunov time
of only $\sim$5 Myr for the inner planets. Due to the chaos it is fundamentally
impossible to accurately predict the solar system's orbital evolution beyond
$\sim$50 Myr based on present astronomical observations. We have recently
developed a method to overcome the problem by using the geologic record to
constrain astronomical solutions in the past. Our resulting optimal
astronomical solution (called ZB18a) shows exceptional agreement with the
geologic record to $\sim$58 Ma (Myr ago) and a characteristic resonance
transition around 50 Ma. Here we show that ZB18a and integration of Earth's and
Mars' spin vector based on ZB18a yield reduced variations in Earth's and Mars'
orbital inclination and Earth's obliquity (axial tilt) from $\sim$58 to
$\sim$48 Ma -- the latter being consistent with paleoclimate records. The
changes in the obliquities have important implications for the climate
histories of Earth and Mars. We provide a detailed analysis of solar system
frequencies ($g$- and $s$-modes) and show that the shifts in the variation in
Earth's and Mars' orbital inclination and obliquity around 48 Ma are associated
with the resonance transition and caused by changes in the contributions to the
superposition of $s$-modes, plus $g$-$s$-mode interactions in the inner solar
system. The $g$-$s$-mode interactions and the resonance transition (consistent
with geologic data) are unequivocal manifestations of chaos. Dynamical chaos in
the solar system hence not only affects its orbital properties, but also the
long-term evolution of planetary climate through eccentricity and the link
between inclination and axial tilt.
