= Solution
The <force> is conservative, with $\mathbf F=-\nabla U$ and <potential energy> zero at the origin:
$$
\boxed{U(t)=\frac k2[x(t)^2+y(t)^2]=\frac k2\left[x_0^2\cos^2(\omega t+\phi_1)+y_0^2\cos^2(\omega t+\phi_2)\right].}
$$
Released from rest at $(c,c)$, the trajectory has $x=y=c\cos\omega t$, giving
$$
\boxed{U(t)=kc^2\cos^2\omega t=\frac{kc^2}{2}(1+\cos2\omega t).}
$$
The right panel of the figure shows this nonnegative <oscillation>: it starts at a maximum $kc^2$, reaches zero at $t=\pi/(2\omega)$, and repeats after $\pi/\omega$. Its <frequency> is twice the coordinate <frequency>. <Conservation of mechanical energy> gives $K(t)=kc^2\sin^2\omega t$, so the missing <potential energy> is <kinetic energy>, not an energy loss.
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