1 14a. 1 14a is always a repellor, according to the derivative criterion.

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1 In[3]:= f Functionx, x^ a; Solvex fx, x Out[4]= x 4a, x 4a So the fixed points are x 4a and x 4a. f'x x f' 4a 4a f' 4a 4a Since f ' 4a 4a is always greater than (due to the fact that 4a 0 when a0), we have that the second fixed point x 4a is always a repellor, according to the derivative criterion. Now lets study the first fixed point. The derivative critetion states that it will be an attractor for those values of a such that 4a and the convergence will be monotonic when 4a 0. Reduce Abs 4a, a, Reals Reduce 0 4a, a, Reals 4 a a 0 So, according to the derivative criterion, the first fixed point x 4a is an attractor when a 3 and the convergence is always oscillatory (since the derivative 4a is nega- 4

2 SolucionEjercicioExamen.nb 4 tive when a0). Now lets study the case a 3 4 by drawing some high iterates of the function. In[5]:= a 3 4; 4a Out[6]= When a 3 4, we have that the fixed point is x, so we plot the iterates of f in an interval including this value. In[7]:= PlotNestf, x, 0, Nestf, x,, Nestf, x,, Nestf, x, 3, x,,, PlotRange, Out[7]= We see that is an attractor (since the orbits of points at both sides of - converge to -) with oscilatory convergence Now we set a. The fixed points are: a ; 4a 4a

3 SolucionEjercicioExamen.nb 3 Plotx, fx, Nestf, x,, Nestf, x, 3, Nestf, x, 4, Nestf, x, 0, x,,, PlotRange,, AspectRatio Automatic The fixed points are those where the graph of f meets the diagonal. We see that the attraction basin of the first fixed point x is x, x. It is important to observe that the figure is simetric a.; In order to see the behavior of the orbit in the long run, it is better not to plot the first iterations, so we start with the 50th one.

4 4 SolucionEjercicioExamen.nb Clearf, x; f Functionx, x^.; StartingValue ; FirstIt 50; LastIt 00; xmin ; xmax ; i 0; y NStartingValue; Whilei FirstIt, y fy; i i; DataTable y, y, y, fy; Whilei LastIt, y fy; AppendToDataTable, y, y; AppendToDataTable, y, fy; i i; AppendToDataTable, fy, fy; Cobweb ListPlotDataTable, Joined True, PlotRange xmin, xmax, xmin, xmax, AspectRatio, PlotStyle GrayLevel.3, DisplayFunction Identity; Graf Plotfx, x, x, xmin, xmax, PlotRange xmin, xmax, AspectRatio, DisplayFunction Identity; ShowCobweb, Graf We see that the orbit stabilizes oscillating between two points (a double point). We can also plot the orbit (although it is not asked):

5 SolucionEjercicioExamen.nb 5 ListPlotNestListf,, 30, Joined True, DataRange 0, Lets plot the bifurcation diagram for.3 a and initial value 0.6. Clearf, a, data, U; firstit 500; First value of t to be plotted iterates 500; Number of iterations to be calculated after firstit p0 0.6; amin.3; amax ; asteps 000; f Functionx, x^a; data ; Do a amin amaxamin asteps Initial value Minimum value for the parameter Maximum vale for the parameter Number of points evaluated between amin and amax d; U UnionDropNestListf, p0, firstit iterates, firstit; data Joindata, Tablea, Ui, i,, LengthU, d, 0, asteps ; ListPlotdata, PlotRange amin, amax,,, PlotStyle PointSize.00, AxesOrigin amin, We see clearly that there is a 5-fold window between.6 and.65, so we make a zoom.

6 6 SolucionEjercicioExamen.nb Clearf, a, data, U; firstit 500; First value of t to be plotted iterates 500; Number of iterations to be calculated after firstit p0 0.6; amin.6; amax.65; asteps 000; f Functionx, x^a; data ; Do a amin amaxamin asteps Initial value Minimum value for the parameter Maximum vale for the parameter Number of points evaluated between amin and amax d; U UnionDropNestListf, p0, firstit iterates, firstit; data Joindata, Tablea, Ui, i,, LengthU, d, 0, asteps ; ListPlotdata, PlotRange amin, amax,,, PlotStyle PointSize.00, AxesOrigin amin, We choose for example the value a.65 and we verify that this value actually provides a 5-fold orbit.

7 SolucionEjercicioExamen.nb 7 a.65; f Functionx, x^a; ListPlotNestListf, 0.6, 00, PlotRange, We can calculate the 5 limit points by printing many terms of the orbit (for example by evaluating NestList[f,0.6,00]). Since we dont need to see the first terms, we can calculate for example the 00th term and print its orbit. Nestf, 0.6, 00 NestListf,, ,.749, ,.6497,.055, 9376,.749, ,.6497,.055, 9376,.749, ,.6497,.055, 9376 We see the five values repeating. We can also try to find this attracting 5-fold point by calculating the fixed points of f 5, but in this case this points are difficult to identify since f 5 has 0 fixed points and some of them are very close to each other. SolveNestf, x, 5 x, x, Reals x.6497, x.6304, x.749, x.575, x , x , x 9376, x , x , x.0095, x.055, x.8693