  Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t)), t\in [0,S]$ be a Gaussian
vector process and let $g(\boldsymbol{x}),\boldsymbol{x}\in\mathbb{R}^d$ be a
continuous homogeneous function. In this paper we are concerned with the exact
tail asymptotics of the chaos process $g(\boldsymbol{X}(t))+ h(t),t\in [0,S]$
with trend function $h$. Both scenarios $\boldsymbol{X}(t)$ is
locally-stationary and $\boldsymbol{X}(t)$ is non-stationary are considered.
Important examples include the product of Gaussian processes and chi-processes.
