According the Wikipedia, Doubochinski's pendulum does not vary the ac frequency. Instead, the starting conditions dictate the steady-state amplitude.
If you merely want to keep the pendulum swinging uniformly using an electromagnet, the simplest way is to drive the electromagnet with a negative-resistance circuit. This will automatically sense the swinging of the pendulum and adjust the current appropriately to keep the pendulum swinging uniformly.
A fine and elegant early example of this is a tunnel diode motor (also a tunnel diode pendulum) that appeared in C. L. Strong's "The Amateur Scientist" column in "Scientific American" magazine for October 1965 on page 112. Besides the swinging magnet, it uses just a coil of wire for the electromagnet, a tunnel diode for the negative resistance, two resistors to set the appropriate bias voltage for the tunnel diode, and a 1.5 volt battery. The tunnel diode and swinging magnet apply a combination of dc and ac to the coil. (Mr. Strong says he also successfully made a pendulum with the tunnel diode powered with a battery improvised with blotting paper, moistened with saliva, and sandwiched between a nickel and a penny.)
These days, tunnel diodes are scarce, but the tunnel diode can be replaced with a simple negative resistance circuit made of a couple of FETs or a couple of bipolar transistors with a few resistors. Google "lambda circuit" for various examples. I have had good results using either the tunnel diode or a lambda circuit. You can merely change the bias voltage to change the amplitude of the swings.
If you are set on demonstrating Doubochinski's pendulum, you can use most any signal generator, as suggested by another responder. As long as you are using a typical pendulum, hanging from a low-friction pivot or on a thread, and swinging in air, there will be very little energy loss per swing, and a signal generator should provide plenty of power. If you don't already have a signal generator, you could use an el cheapo model such as a "1HZ-500KHz DDS Function Signal Generator" that you can get from a number of on-line vendors. It is settable in 1-Hz steps, giving good frequency resolution at all but the lowest frequencies. You said you don't want to use a 555, but if you don't object to an IC, you could use one of the resistance-controlled clock chips, such as from Linear Technology. With any of these schemes, if you want to drive a huge pendulum and you find you do need more power, you could merely add a buffer amplifier chip having a low-impedance output, or you could use an audio amplifier from a sound system.