Time-dependent Pattern Speeds in Barred Galaxies
  Based on a high quality $N$-body simulation of a double bar galaxy model, we
investigate the evolution of the bar properties, including their size, strength
and instantaneous pattern speed derived by using three distinct methods: the
Fourier, Jacobi integral, and moment of inertia methods. The interaction of the
two bars, which rotate at distinct speeds, primarily affects the size, strength
and pattern speed of the inner bar. When the two bars are perpendicular to each
other, the size and the pattern speed of the inner bar decrease and its
strength increases. The emergence of a strong Fourier $m=1$ mode increases the
oscillation amplitude of the size, strength and pattern speed of the inner bar.
On the other hand, the characteristics of the outer bar are substantially
influenced by its adjacent spiral structure. When the spiral structure
disappears, the size of the outer bar increases and its strength and pattern
speed decrease. Consequently, the ratio of the pattern speed of the outer bar
with respect to the inner bar is not constant and increases with time. Overall,
the double bar and disk system displays substantial high frequency semi-chaotic
fluctuations of the pattern strengths and speeds both in space and time,
superposed on the slow secular evolution, which invalidates the assumption that
the actions of individual stars should be well conserved in barred galaxies,
such as the Milky Way.
