Controlling the reactive current of an PMSM in an open loop setting
It sounds to me that you are running an open loop "Volts per Hertz" algorithm similar to what you could run with an induction machine.  The problem you are seeing, I think, is that a permanent magnet synchronous machine has very little damping (contrary to induction machines), and your rotor is oscillating pretty wildly.  When your currents are high, this is from a high phase lag on the rotor, which makes high torque, which requires high current (and there is a lag involved because of the machine inductance).  Then the rotor accelerates, overshoots the synchronous speed and phase, and the current drops, then the stator field catches up, overshoots the rotor, etc.  You don't seem to be breaking synchronism (motor would stop and your currents would skyrocket), which is something to be thankful for.  To confirm or disprove my theory, you could look at rotor speed.  I suspect you will see the oscillations around your fundamental frequency.

You make this claim:  "For my investigations the system needs to be in a [steady] state with exactly periodic currents and signals, hence no closed loop control is used."  I think you are implying that a closed loop control cannot get you exactly periodic currents and signals, but the reverse is true.  In my experience, you will not be able to get steady, uniform speed on a permanent magnet synchronous machine without using closed loop control.  I should mention that when I say "closed loop control", I mean you are closing the loop around speed.  You have to have a position sensor (or a position sensing algorithm) to do this because you have to know where the magnetic axis of the rotor is in order to maintain your stator field at a constant phase with respect to the rotor field.  If you do this, and your load is constant, then you will have steady currents and steady speed.

As I am thinking about how you would make your setup work, I think you have to have damping to eliminate the resonance.  One (impractical) way to get damping would be to rebuild your motor with damper bars on the rotor.  But if your shaft load could provide damping, maybe this would work, too.  You would need a torsional damper with viscous damping behavior as your shaft departs from synchronous phase.  This might be more attainable because you can obtain such dampers commercially (maybe they come this small?) or even perhaps build one?  You would have to compare that effort to adding a position sensor (if you don't already have one) and a closed loop control circuit.

Just to "close the loop" on another statement, you always have reactive current in a pmsm.  Because you have inductance in the stator, to get current on the quadrature (i.e., torque producing) axis of the rotor reference frame, you have to put in leading terminal voltage.  Measuring at the terminals of the motor, the voltage will lead the current in motoring.  If you could somehow measure the back-emf of the motor (you can't), and you had the system under closed loop speed control, you could get the terminal current to line up directly with the emf of the motor, but, again, from what you can measure at the terminals, you would find you have to put in leading voltage to accomplish this.  You can see some phasor diagrams at this link:  http://electricalacademia.com/synchronous-machines/synchronous-motor-equivalent-circuit-phasor-diagram/

I am sorry I cannot come up with something more useful than this, but as I understand what you are working with, this is the best I can do.