Basic dynamical mechanisms that produce an amplification of the accretion rate of gas clouds into the central regions of barred galaxies, and their subsequent effects on the evolution of barred galaxies, are discussed. Weakly dissipative orbits, representing gas clouds, are computed in a barred galaxy model with a central mass concentration such as a black hole, or a secondary small inner bar. We find amplified accretion across resonances that is especially rapid inside the Inner Lindblad Resonance, large excursions outside the galactic plane, and the existence of nontrivial attractors like strange (chaotic) attractors or limit cycles. The underlying physical mechanisms are, in general, due to the presence of broad horizontal and vertical resonances through which weakly dissipative particles can rapidly traverse. The two principal physical implications are first that the growth of a significant central mass concentration in a barred galaxy induces broad radial resonance regions which can act to enhance the fueling rate of AGNs and starbursts. Second, the broad vertical resonances allow stars to diffuse into the bulge from the disk, indicating the possibility that a significant component of all bulges has formed in this way and should possess a metal-rich rotationally supported component. Secular evolution along the Hubble sequence due to increasing bulge-to-disk ratio with time is a natural consequence of this intrinsic resonant heating.