small noise, slow forcing limit using time dependent large deviation theory, that is, it was shown asymptotically the escape times are given by the adiabatic approximation. This result is only for the leading term, whether the analogue result holds for the Kramers rate is unknown.

Mexican Hat Toy Model {#sum_chap_mexican_hat_toy_model}
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\[chap\_mexican\_hat\_toy\_model\]

The main object of consideration of this paper, which is called the Mexican Hat Toy Model, is now introduced. Let $a>0$, $b>0$ and $V_0:\mathbb{R}^2\longrightarrow \mathbb{R}$ be a real function from the plane to the line. The unperturbed potential is defined as $$\begin{aligned}
V_0(x,y)=\frac{1}{4}r^4-\frac{1}{2}r^2-ax^2+by^2
\quad \text{where} \quad r=\sqrt{x^2+y^2}\end{aligned}$$ Let $F_x, F_y \in  \mathbb{R}$ be the forcing. The potential with forcing $V_F$ is defined as $$\begin{aligned}
V_F(x,y)
&=\frac{1}{4}r^4-\frac{1}{2}r^2-ax^2+by^2+F_xx+F_yy\\
&=\frac{1}{4}r^4-\frac{1}{2}r^2-ax^2+by^2+\mathbf{F}\cdot\mathbf{x}\\
&=V_0+\mathbf{F}\cdot\mathbf{x}\end{aligned}$$ written more compactly in vector notation. The forcing will clearly have a magnitude and direction given by $$\begin{aligned}
F=\sqrt{F_x^2+F_y^2}
\quad \text{and} \quad 
\phi=\tan^{-1}\left(\frac{F_y}{F_x}\right)\end{aligned}$$ We will study the critical points which are solutions to the simultaneous equations $$\begin{aligned}
\frac{\partial V_F}{\partial x}=0
\quad \text{and} \quad 
\frac{\partial V_F}{\partial y}=0
\label{solv_equation_hat}\end{aligned}$$ It is easy to show that when there is no forcing $F=0$, there are five critical points, that is two wells, two saddles and one hill. Stochastic resonance is studied when the forcing 