I have a function
In[186]:= v[k_, x_] = u1[x] - u2[x]
Out[186]= 1 - 10 (2 - k)^x + (1 - k)/(2 k) - 2 k - k/(2 (1 - k)) +
10 (1 + k)^x + 1/2 (-(1 - k)^2 - k^2) + 1/2 ((1 - k)^2 + k^2) -
Log[2/(Sqrt[15] Sqrt[(1 - k)/k])] +
Log[(2 Sqrt[3/5])/Sqrt[k/(1 - k)]]
k is the variable of interest and x is a parameter. A solution is found when v[k,x]==0. As x increases multiple equilibria arise. What I want to do is have a plot that shows the multiple equilibria, i.e. a function k[x], to see the point where multiple equilibria arise. I tried to write the function as the following:
k1[x_] := FindRoot[v[k, x] == 0, {k, 1}]
k1[x_?NumericQ] := k /. FindRoot[v[k, x] == 0, {k, .5}]
This provides a solution, but only provides a local solution around the starting point and if I change the starting point I get a different solution. How can I plot a figure that will show all the equilibria over a certain range? Any help would be appreciated.
