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using ValidatedNumerics
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logistic(r, x) = -r*((x-0.5)^2 - 0.25) #r*x*(1-x)
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X = @interval(-0.0001, 1.0001)
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r = @interval(3.1)
newton(x -> logistic(r,x)-x, X)
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r = @interval(3.1)
newton(x -> logistic(r, logistic(r,x))-x, @interval(-1, 2))
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function iterate_logistic(n)
ex = :(x)
for i in 1:n
ex = :(logistic(r,$ex))
end
ex = :($(ex) - x)
ex
end
for n in [1, 2, 4, 5, 8:16, 32]
name = symbol("logistic_iterated_$n")
ex = iterate_logistic(n)
@eval $name(r,x) = $ex
end
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iterate_logistic(4)
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logistic_iterated_16(3.1, 0)
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r = @interval(3.1)
newton(x -> logistic_iterated_2(r, x), X)
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r = @interval(3.5)
newton(x -> logistic_iterated_4(r, x), X)
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r = @interval(3.55)
newton(x -> logistic_iterated_8(r, x), X)
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r = @interval(3.55)
krawczyk(x -> logistic_iterated_8(r, x), X)
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r = @interval(4)
newton(x -> logistic_iterated_8(r, x), X)
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r = @interval(4)
newton(x -> logistic_iterated_16(r, x), X)
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roots = ans;
filter(x -> x[2]==:unique, roots)
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logistic_iterated_16(3.1, 0)
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newton(x->logistic_iterated_5(r, x), X)
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roots[roots[2]==:unique]
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r = @interval(4)
newton(x -> logistic_iterated_10(r, x), X)
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length(filter(x->x[2]==:unique, ans))
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r = @interval(4)
newton(x -> logistic_iterated_11(r, x), X, tolerance=0, maxlevel=40)
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length(filter(x->x[2]==:unique, ans))
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newton(x->logistic_iterated_11(r, x), @interval("[0.9999994111193403, 0.9999994112365287]"))
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newton(x->logistic_iterated_11(r, x), @biginterval("[0.9999994111193403, 0.9999994112365287]"))
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diam(ans[1][1])
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When $r=4$, it is possible to calculate the periodic points exactly; see e.g. Sanders+Benet, Scipy 2014 proceedings. First we calculate the periodic points of the tent map using BigFloats:
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function tent_map_periodic_points(n)
evens = big(1.0)/(big(2)^n-1) * collect(0:2:2^n-1)
odds = big(1.0)/(big(2)^n+1) * collect(2:2:2^n)
points = zeros(BigFloat, 2^n)
points[1:2:2^n] = evens
points[2:2:2^n] = odds
points
end
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tent_map_periodic_points(4)
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These points $y_n$ are mapped onto periodic points of the logistic map by $x := h(y) := \sin^2(\frac{\pi}{2} y)$
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h(y) = sin(π*y/2)^2
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logistic_exact_period_4_points = map(h, tent_map_periodic_points(4))
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logistic_r4_period_4_points = find_roots(x -> logistic_iterated_4(4., x), -1e-10, 1+1e-10)
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These intervals do contain the exact roots:
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all([logistic_exact_period_4_points[i] in logistic_r4_period_4_points[i].interval for i in 1:16])
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Note that this calculation does not work using standard floating point for the calculation of the exact points -- rounding errors result in floating point numbers that are outside these guaranteed bounds.