Depending on the size of your bird a few other options would be to use Photoshop. It includes a “Puppet Warp” option where you can set the bend points and then warp the item and it will shift the pixels for you and you can create multiple frames from it. The Gimp, as Scott mentioned is a free alternative to Photoshop, but I don’t know what all animation features it has.
Adobe Illustrator and Inkscape are vector art tools which may be easier to do animations in since they objects are vectors you can just move the various pen points to re-shape each frame as needed.
If you’re object is more complex you can use Spine (Esoteric software) to do animations. It’s a complete rigging system (i.e you define the skeleton and then the body and it warps the areas much like Photoshop’s Puppet Warp does, but it does it in making the whole animation. We’ve recently added mesh support to take full advantage of Spine’s features (Note this is in beta and we want more testing before we say it’s production ready).