How to calculate correlation on this plot? How to calculate the correlation between Blue and Red curve. Is the Pearson correlation coefficients works for non-linear data? 

 A: Generally speaking, Pearson correlation measures linear relationship, so naturally nonlinear relationships may not be captured by it. However, having a nonlinear looking time series doesn't mean that you can't calculate Pearson correlation. For example, the two series can be exact multiples of each other, however nonlinear they may look when plotted with respect to time.
If you plot the corresponding points of the blue curve and the red curve in to x-y plane, you'll have a much better idea of how they're related, and can calculate the Pearson correlation as well.
In addition, the two series might have been correlated by a lag. In your case, it doesn't seem to be, i.e. lag is $0$, but generally you calculate the cross-correlation function for that.
A: Given the visible relationship between the variables represented by the blue curve (y1) and the red curve (y2), you will find a positive value if you calculate the correlation.
Linear correlation does not appear to be the best measure for the relationship between the variables y1 and y2. It only measures the linear relationship between them.
Something else you may need to think about. 
Let's call x the horizontal-axis variable.There is a threshold value xo on your horizontal axis (just below 0.2), where both y1 and y2 reach their maxima. There seems to be a strong positive relationship between (y1-y2) and the absolute value of (x-xo).
