Is this a nonlinear time series? Could someone please help me to find out whether a time series is linear? 
And if it's nonlinear, what degree of nonlinearity?
I searched for an appropriate function in Matlab, but it seems there's no function which takes a vector as an argument and returns a boolean as a result or something like that.
I'm not really sure how to determine it visually.
The link for data: https://datamarket.com/data/set/22px/annual-sheep-population-1000s-in-england-wales-1867-1939#!ds=22px&display=line
 A: i took your data (73 values) and used AUTOBOX ( a piece of software that I have helped develop ) and found a result that is akin to non-linearity. AUTOBOX found that the parameters of the model had changed at period 30 thus segmenting the data 1-29 and 30-73. Here is the summary : a plot of the original data  with acf of  . The breakpoint (chow test ) is here  An analysis of the last 44 values led to this equation  (1,1,0) with three pulses , The statistics of the model are here  and here  . The acf of the residuals somewhat suggests randomness  with the residual plot here  . The actual/fit/forecast is here with forecasts here  . The whole idea is that sometimes parameters and or error variance change over time. One shouldn't simply model all the data as if it were homogeneous as the composite/global parameters may not be uniformly representative due to an exogenous (i.e.unknown) factor. You could say that two sets of parameters .. one for the first 29 and one for the last 47 is a kind of nonlinearity as was (partly) suggested here. This is one of the possible "elephants in the room" that nobody talks about but I do.
A: Showing that an observed time series is non-linear is notoriously difficult. Briefly, you would have to find a measure indicating nonlinearity (e.g., a positive Lyapunov exponent or a non-integer dimension) and then show with surrogates that the value cannot be explained by the linear properties (frequency and amplitude distribution) of the time series.
Your time series is too short for this. Your data points are few in comparison to what is needed to expect a useful result from non-linearity measures. Then you have at most two periods of your main oscillation in your data, which is far too few for any useful analysis. Finally, you have an obvious trend in your time series making the time series non-stationary (at least for the purposes of analysing it – if this is due to a stationary oscillatory behaviour, you would have to have data from at least one oscillation to make any use of this). You can correct for this by subtracting a linear trend, but you have to expect that this is not the only effect of the non-stationarity.
