Linearity of y(t) = x²(t)

A system is linear if it satisfies superposition: $a x_1(t) + b x_2(t) \to a y_1(t) + b y_2(t)$.

Given: $y(t) = x^2(t)$

For individual inputs: $y_1(t) = x_1^2(t)$, $y_2(t) = x_2^2(t)$.

For the combined input $x_3(t) = a x_1(t) + b x_2(t)$:

$$\begin{aligned} y_3(t) &= \left[a x_1(t) + b x_2(t)\right]^2 \\ &= a^2x_1^2(t) + b^2x_2^2(t) + 2ab\,x_1(t)x_2(t) \end{aligned}$$

Required for linearity: $a y_1(t) + b y_2(t) = a x_1^2(t) + b x_2^2(t)$.

Clearly $y_3(t) \ne a y_1(t) + b y_2(t)$ (the coefficients become $a^2$, $b^2$ and a cross term $2ab\,x_1x_2$ appears).

Simple check with homogeneity: doubling the input, $x \to 2x$, gives $y = 4x^2$, i.e. the output becomes four times, not twice.

Answer: $y(t) = x^2(t)$ is a non-linear system (a square-law device, such as that used in square-law modulators and detectors).