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The former image shows the graphics of displacement $x(t)$ (in meters) and velocity $x'(t)$ (in meters per second) of an oscillating system.
Does this system have mechanical resonance?
I think it doesn't, but I haven't been able to properly justify my claim. Any help is appreciated
The system, as currently excited, is not in resonance because it's not being driven; its sinusoidal amplitude is decaying over time. However, it is not critically damped or close to it; it is underdamped and could be driven into resonance.
We know unambiguously that the system is underdamped because it oscillates after being released from a disturbed position at rest. But we can also show this mathematically from the displacement equation you provide.
The general equation for a damped harmonic oscillator is
$$x(t)\propto e^{-\frac{b}{2m}t}\cos(\omega_d t+\phi),$$
where $x$ is the displacement as a function of the time $t$, $b$ is the damping coefficient, $m$ is the mass, $\omega$ is the damped natural frequency, and $\phi$ is the phase. For near-critically damped systems, $\omega_d\approx 0$, which isn't the case here, as $\omega_d=2$ as given in the equation. For underdamped systems, the damping ratio $\zeta\equiv\frac{b}{2m\omega_n}<1$, where $\omega_n=\sqrt{k/m}$ is the natural frequency that satisfies $$\omega_d^2=\omega_n^2-\frac{b^2}{4m^2}.$$
Thus, with $\frac{b}{2m}=1$ and $\omega_d=2$ according to your equation, $$\zeta=\sqrt{\frac{1}{\omega_d^2+\frac{b^2}{4m^2}}}=\frac{1}{\sqrt{5}}<1.$$ Conclusion: The system is underdamped and could be driven into resonance.
My answer was wrong because I was using the wrong definition of critical damping. Chemomechanic's answer is correct. User926356, I suggest you "unaccept" my answer and accept his instead.
