+
\frac{1}{2}\delta\left(u-T\right)\end{aligned}$$ where $\delta(\cdot)$ is the Dirac delta function. This approximation is used because in the SDEs which we will simulate, the times when transition into the left well is greatest is at half the period $u=\frac{T}{2}$ and the times when transition into the right well is greatest is at $u=0$ and $u=T$. Due to the fact that $m_-(u)$ and $m_+(u)$ are probabilities a factor of $\frac{1}{2}$ is used in $m_+(u)$. Progressing we have $$\begin{aligned}
p_{tot}(t)&=
\frac{1}{2}
\int_{0}^{T}p_-(t+u,u)m_-(u)+p_+(t+u,u)m_+(u)\,du\\
&\approx
\frac{1}{2}
\int_{0}^{T}
p_-(t+u,u)\delta\left(u-T/2\right)
+p_+(t+u,u)
\left(
\frac{1}{2}\delta\left(u\right)
+
\frac{1}{2}\delta\left(u-T\right)
\right)
\,du\\[0.5em]
&=
\frac{1}{2}
\left\{
p_-(t+T/2,T/2)
+\frac{1}{2}
p_+(t,0)
+\frac{1}{2}
p_+(t+T,T)
\right\}\\[0.5em]
&=
\frac{1}{2}
\left\{
p_-(t+T/2,T/2)
+\frac{1}{2}
p_+(t,0)
+\frac{1}{2}
p_+(t+0,0)
\right\}\\[0.5em]
&=
\frac{1}{2}
\left\{
p_-(t+T/2,T/2)
+
p_+(t,0)
\right\}\\[0.5em]
&=p_+(t,0)\end{aligned}$$ This is because for the simulations which we are going to do, the Kramers rate satisfy $R_{-1+1}(t)=R_{+1-1}(t+T/2)$ (see later in Section \[chap\_mexican\_hat\_toy\_model\] and \[sum\_chap\_results\] for the geometry of the Mexican Hat Toy Model which justifies this). Thus the following approximation $$\begin{aligned}
p_{tot}\approx p_+(t,0)\end{aligned}$$ is only valid for the simulations we do, and not for a general potential. We call this way of approximating $m_-(u)$ and $m_+(u)$ the perfect phase approximation.

Six Measures of Stochastic Resonance {#chap_analysis_theory}
------------------------------------

\[sum\_chap\_analysis\_theory\]

We introduce six possible criteria of measuring 