You wrote two equations above, one is the equation of h.o. with friction, the other one has friction term with opposite sign, which leads to run-away.
The energy here is defined as
E = p_x \dot x + p_y \dot y - L = m \dot x\dot y.
and since the Lagrangian does not depend on time, it should be constant in time.
You can verify this by multiplying the solutions for [itex]\dot x, \dot y[/itex] - the exponentials will cancel out and the result does not depend on time.
However, all this seems very artificial - I would like to see some useful application of it.