examples/Periodic points of the logistic map.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "using ValidatedNumerics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "logistic (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logistic(r, x) = -r*((x-0.5)^2 - 0.25) #r*x*(1-x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-0.0001, 1.0001000000000002]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = @interval(-0.0001, 1.0001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{(Interval{Float64},Symbol),1}:\n",
       " ([-6.968514798756479e-17, 1.3517743393731552e-16],:unique)\n",
       " ([0.6774193548387095, 0.6774193548387097],:unique)        "
      ]
     },
     "execution_count": 4,
     "metadata": {},
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   ],
   "source": [
    "r = @interval(3.1)\n",
    "newton(x -> logistic(r,x)-x, X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{(Interval{Float64},Symbol),1}:\n",
       " ([-6.233702290330726e-17, 1.1931689491945523e-16],:unique)\n",
       " ([0.5580141252026949, 0.5580141252026971],:unique)        \n",
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       " ([0.7645665199585938, 0.7645665199585949],:unique)        "
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     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "r = @interval(3.1)\n",
    "newton(x -> logistic(r, logistic(r,x))-x, @interval(-1, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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   "outputs": [],
   "source": [
    "function iterate_logistic(n)\n",
    "    ex = :(x)\n",
    "    for i in 1:n\n",
    "        ex = :(logistic(r,$ex))\n",
    "    end\n",
    "    ex = :($(ex) - x)\n",
    "    ex\n",
    "end\n",
    "\n",
    "for n in [1, 2, 4, 5, 8:16, 32]\n",
    "    name = symbol(\"logistic_iterated_$n\")\n",
    "    ex = iterate_logistic(n)\n",
    "    @eval $name(r,x) = $ex\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       ":(logistic(r,logistic(r,logistic(r,logistic(r,x)))) - x)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "iterate_logistic(4)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 14,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0"
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     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "logistic_iterated_16(3.1, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{(Interval{Float64},Symbol),1}:\n",
       " ([-9.947150462130687e-17, 1.1038772564320529e-16],:unique)\n",
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     "execution_count": 23,
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   ],
   "source": [
    "r = @interval(3.1)\n",
    "newton(x -> logistic_iterated_2(r, x), X)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "8-element Array{(Interval{Float64},Symbol),1}:\n",
       " ([-9.361862579270038e-17, 1.0515086331210236e-16],:unique)\n",
       " ([0.3828196830173237, 0.3828196830173243],:unique)        \n",
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       " ([0.874997263602464, 0.8749972636024642],:unique)         "
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     "execution_count": 22,
     "metadata": {},
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   "source": [
    "r = @interval(3.5)\n",
    "newton(x -> logistic_iterated_4(r, x), X)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
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       "16-element Array{(Interval{Float64},Symbol),1}:\n",
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       " ([0.8873708969850299, 0.8873708969850314],:unique)        "
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   "source": [
    "r = @interval(3.55)\n",
    "newton(x -> logistic_iterated_8(r, x), X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
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       "16-element Array{(Interval{Float64},Symbol),1}:\n",
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   "source": [
    "r = @interval(3.55)\n",
    "krawczyk(x -> logistic_iterated_8(r, x), X)"
   ]
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   "cell_type": "code",
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       " ⋮                                                         \n",
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   "source": [
    "r = @interval(4)\n",
    "newton(x -> logistic_iterated_8(r, x), X)"
   ]
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   "cell_type": "code",
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       " ([-9.998882189393042e-5, -9.998789038509129e-5],:unknown)\n",
       " ⋮                                                        \n",
       " ([1.0000999888218949, 1.0000999897534038],:unknown)      \n",
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       " ([1.0000999972054745, 1.000099998136983],:unknown)       \n",
       " ([1.0000999981369831, 1.0000999990684916],:unknown)      \n",
       " ([1.0000999990684918, 1.0001000000000002],:unknown)      "
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     "metadata": {},
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   "source": [
    "r = @interval(4)\n",
    "newton(x -> logistic_iterated_16(r, x), X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
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    {
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       "65069-element Array{(Interval{Float64},Symbol),1}:\n",
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       " ([2.730081891125118e-5, 2.7300818911420588e-5],:unique) \n",
       " ⋮                                                       \n",
       " ([0.9999734437821953, 0.9999734437821955],:unique)      \n",
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       " ([0.9999879218367506, 0.9999879218367508],:unique)      "
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     "execution_count": 26,
     "metadata": {},
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   "source": [
    "roots = ans;\n",
    "filter(x -> x[2]==:unique, roots)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
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       "0.0"
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     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "logistic_iterated_16(3.1, 0)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
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   "outputs": [
    {
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       " ([0.32634737357758975, 0.32634737357758997],:unique)      \n",
       " ⋮                                                         \n",
       " ([0.7201970757788171, 0.7201970757788173],:unique)        \n",
       " ([0.7499999999999999, 0.7500000000000001],:unique)        \n",
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       " ([0.9977359612865422, 0.9977359612865424],:unique)        "
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     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "newton(x->logistic_iterated_5(r, x), X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "LoadError",
     "evalue": "BoundsError()\nwhile loading In[30], in expression starting on line 1",
     "output_type": "error",
     "traceback": [
      "BoundsError()\nwhile loading In[30], in expression starting on line 1",
      "",
      " in getindex at array.jl:246"
     ]
    }
   ],
   "source": [
    "roots[roots[2]==:unique]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1024-element Array{(Interval{Float64},Symbol),1}:\n",
       " ([-1.0220696221544258e-16, 8.283033843053199e-17],:unique)\n",
       " ([9.394002137712196e-6, 9.394002137833982e-6],:unique)    \n",
       " ([9.430769118680364e-6, 9.430769118773784e-6],:unique)    \n",
       " ([3.757565556196494e-5, 3.7575655562059124e-5],:unique)   \n",
       " ([3.772272071721069e-5, 3.772272071733585e-5],:unique)    \n",
       " ([8.454390131859121e-5, 8.454390131868317e-5],:unique)    \n",
       " ([8.487478753608691e-5, 8.487478753617993e-5],:unique)    \n",
       " ([0.00015029697452838232, 0.0001502969745285893],:unique) \n",
       " ([0.0001508851908543323, 0.0001508851908544388],:unique)  \n",
       " ([0.00023483240445339927, 0.00023483240445349297],:unique)\n",
       " ([0.0002357514405563716, 0.00023575144055646552],:unique) \n",
       " ([0.0003381470145895322, 0.00033814701458962634],:unique) \n",
       " ([0.00033947033522630486, 0.0003394703352264008],:unique) \n",
       " ⋮                                                         \n",
       " ([0.999714745464859, 0.9997147454648591],:unique)         \n",
       " ([0.9997158574619074, 0.9997158574619076],:unique)        \n",
       " ([0.9998090384816738, 0.999809038481674],:unique)         \n",
       " ([0.999809782923107, 0.9998097829231072],:unique)         \n",
       " ([0.9998844771639072, 0.9998844771639074],:unique)        \n",
       " ([0.9998849275276295, 0.9998849275276297],:unique)        \n",
       " ([0.9999410586657799, 0.9999410586657801],:unique)        \n",
       " ([0.9999412884518407, 0.9999412884518408],:unique)        \n",
       " ([0.9999787808528637, 0.9999787808528638],:unique)        \n",
       " ([0.9999788635779219, 0.9999788635779221],:unique)        \n",
       " ([0.9999976423021615, 0.9999976423021617],:unique)        \n",
       " ([0.99999765149395, 0.9999976514939501],:unique)          "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = @interval(4)\n",
    "newton(x -> logistic_iterated_10(r, x), X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1024"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "length(filter(x->x[2]==:unique, ans))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2048-element Array{(Interval{Float64},Symbol),1}:\n",
       " ([-3.7739729411576187e-17, 7.443905684961557e-17],:unique)\n",
       " ([2.350798951126237e-6, 2.3507989511639162e-6],:unique)   \n",
       " ([2.3553948387492456e-6, 2.3553948388366314e-6],:unique)  \n",
       " ([9.403173699520289e-6, 9.403173699560357e-6],:unique)    \n",
       " ([9.421557163670454e-6, 9.421557163730942e-6],:unique)    \n",
       " ([2.1157057930306742e-5, 2.115705793034727e-5],:unique)   \n",
       " ([2.11984204002431e-5, 2.119842040030646e-5],:unique)     \n",
       " ([3.761234111944355e-5, 3.76123411194866e-5],:unique)     \n",
       " ([3.768587359185487e-5, 3.768587359191397e-5],:unique)    \n",
       " ([5.8768868534615475e-5, 5.8768868534701886e-5],:unique)  \n",
       " ([5.888376140063535e-5, 5.888376140069161e-5],:unique)    \n",
       " ([8.46264412368879e-5, 8.462644123696015e-5],:unique)     \n",
       " ([8.479188410901219e-5, 8.47918841090595e-5],:unique)     \n",
       " ⋮                                                         \n",
       " ([0.9999287509423916, 0.9999287509423918],:unique)        \n",
       " ([0.9999288899616146, 0.9999288899616147],:unique)        \n",
       " ([0.9999523039753886, 0.9999523039753887],:unique)        \n",
       " ([0.9999523970393035, 0.9999523970393037],:unique)        \n",
       " ([0.9999711466680797, 0.99997114666808],:unique)          \n",
       " ([0.9999712029667102, 0.9999712029667103],:unique)        \n",
       " ([0.9999852788429373, 0.9999852788429374],:unique)        \n",
       " ([0.9999853075669987, 0.9999853075669988],:unique)        \n",
       " ([0.9999947003668137, 0.9999947003668139],:unique)        \n",
       " ([0.9999947107075408, 0.9999947107075409],:unique)        \n",
       " ([0.9999994111509435, 0.9999994111509436],:unique)        \n",
       " ([0.9999994122999167, 0.9999994122999168],:unique)        "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = @interval(4)\n",
    "newton(x -> logistic_iterated_11(r, x), X, tolerance=0, maxlevel=40)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2047"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "length(filter(x->x[2]==:unique, ans))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{(Interval{Float64},Symbol),1}:\n",
       " ([0.9999994111509435, 0.9999994111509436],:unique)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton(x->logistic_iterated_11(r, x), @interval(\"[0.9999994111193403, 0.9999994112365287]\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{(Interval{BigFloat},Symbol),1}:\n",
       " ([9.999994111509435535683237673405515492356983378423455810550216538164422108648822e-01, 9.999994111509435535683237673405515492356983378423455810550216538164422108648995e-01]₂₅₆,:unique)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton(x->logistic_iterated_11(r, x), @biginterval(\"[0.9999994111193403, 0.9999994112365287]\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.727233711018888925077270372560079914223200072887256277004740694033718360632485e-77 with 256 bits of precision"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diam(ans[1][1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exact periodic points for $r=4$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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 `BigFloat`s:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tent_map_periodic_points (generic function with 1 method)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function tent_map_periodic_points(n)\n",
    "    \n",
    "    evens = big(1.0)/(big(2)^n-1) * collect(0:2:2^n-1)\n",
    "    odds = big(1.0)/(big(2)^n+1) * collect(2:2:2^n)\n",
    "    \n",
    "    points = zeros(BigFloat, 2^n)\n",
    "    points[1:2:2^n] = evens\n",
    "    points[2:2:2^n] = odds\n",
    "    \n",
    "    points\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16-element Array{BigFloat,1}:\n",
       " 0e+00                                                                               \n",
       " 1.17647058823529411764705882352941176470588235294117647058823529411764705882353e-01 \n",
       " 1.333333333333333333333333333333333333333333333333333333333333333333333333333343e-01\n",
       " 2.35294117647058823529411764705882352941176470588235294117647058823529411764706e-01 \n",
       " 2.666666666666666666666666666666666666666666666666666666666666666666666666666687e-01\n",
       " 3.529411764705882352941176470588235294117647058823529411764705882352941176470601e-01\n",
       " 4.000000000000000000000000000000000000000000000000000000000000000000000000000052e-01\n",
       " 4.70588235294117647058823529411764705882352941176470588235294117647058823529412e-01 \n",
       " 5.333333333333333333333333333333333333333333333333333333333333333333333333333374e-01\n",
       " 5.882352941176470588235294117647058823529411764705882352941176470588235294117683e-01\n",
       " 6.666666666666666666666666666666666666666666666666666666666666666666666666666695e-01\n",
       " 7.058823529411764705882352941176470588235294117647058823529411764705882352941202e-01\n",
       " 8.000000000000000000000000000000000000000000000000000000000000000000000000000104e-01\n",
       " 8.235294117647058823529411764705882352941176470588235294117647058823529411764721e-01\n",
       " 9.333333333333333333333333333333333333333333333333333333333333333333333333333425e-01\n",
       " 9.41176470588235294117647058823529411764705882352941176470588235294117647058824e-01 "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tent_map_periodic_points(4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These points $y_n$ are mapped onto periodic points of the logistic map by $x := h(y) := \\sin^2(\\frac{\\pi}{2} y)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "h (generic function with 1 method)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(y) = sin(π*y/2)^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16-element Array{BigFloat,1}:\n",
       " 0e+00                                                                               \n",
       " 3.376388529782209771344205408921830686870611102744153587582499406973197671517772e-02\n",
       " 4.322727117869955224893621400734141102959477031126272746950010596744255894868408e-02\n",
       " 1.304955413896704420377328450636759471200411901459511384214268446520135907271199e-01\n",
       " 1.654346968205708930868633346566097632002083905201021615912733238560166991446783e-01\n",
       " 2.771308221117308663017712253102565722861611132052862534998310635742086249501397e-01\n",
       " 3.454915028125262879488532914085904705699227050485592844661378443236848842952812e-01\n",
       " 4.538658202683490023801744464227467598184913581220883334042287350394312129893768e-01\n",
       " 5.522642316338267356999170774012490595403279347372965569966377383562423253465847e-01\n",
       " 6.368314950360414317695389677184067158124263347132527319387145577056778723541536e-01\n",
       " 7.500000000000000000000000000000000000000000000000000000000000000000000000000086e-01\n",
       " 8.013173181896281945892940774934203108094822908742071451762467471481287631862079e-01\n",
       " 9.045084971874737120511467085914095294300772949514407155338621556763151157047318e-01\n",
       " 9.251085678648070760670719614746760292353301643789564236850368282088540328017113e-01\n",
       " 9.890738003669028189642833739347997662298689044313385539425888318202984165600556e-01\n",
       " 9.914865498419508891409744224275993580493614375328164379986902296019539262761255e-01"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logistic_exact_period_4_points = map(h, tent_map_periodic_points(4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16-element Array{Root{Float64},1}:\n",
       " Root([-7.034439914540779e-17, 7.771423880327514e-17], :unique)\n",
       " Root([0.03376388529782207, 0.03376388529782214], :unique)     \n",
       " Root([0.043227271178699504, 0.04322727117869961], :unique)    \n",
       " Root([0.13049554138967037, 0.13049554138967054], :unique)     \n",
       " Root([0.16543469682057083, 0.165434696820571], :unique)       \n",
       " Root([0.2771308221117308, 0.27713082211173096], :unique)      \n",
       " Root([0.3454915028125262, 0.3454915028125264], :unique)       \n",
       " Root([0.45386582026834876, 0.4538658202683492], :unique)      \n",
       " Root([0.5522642316338267, 0.5522642316338269], :unique)       \n",
       " Root([0.6368314950360413, 0.6368314950360415], :unique)       \n",
       " Root([0.7499999999999999, 0.7500000000000002], :unique)       \n",
       " Root([0.8013173181896281, 0.8013173181896283], :unique)       \n",
       " Root([0.9045084971874736, 0.9045084971874738], :unique)       \n",
       " Root([0.925108567864807, 0.9251085678648071], :unique)        \n",
       " Root([0.9890738003669027, 0.9890738003669028], :unique)       \n",
       " Root([0.9914865498419508, 0.991486549841951], :unique)        "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logistic_r4_period_4_points = find_roots(x -> logistic_iterated_4(4., x), -1e-10, 1+1e-10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These intervals do contain the exact roots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all([logistic_exact_period_4_points[i] in logistic_r4_period_4_points[i].interval for i in 1:16])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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."
   ]
  }
 ],
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