{ "cells": [ { "cell_type": "markdown", "id": "29c46e38-9ad2-425e-8ba8-23b9f61c4887", "metadata": {}, "source": [ "Math 420/620 Quiz 9" ] }, { "cell_type": "markdown", "id": "81a6d260-0ac4-42da-84c8-62c03d783123", "metadata": {}, "source": [ "1. This problem studies the striking difference between the behavior of\n", "continuous-time and discrete-time dynamical systems that can\n", "occur even in simple models." ] }, { "cell_type": "markdown", "id": "3a91238b-3bd3-4de5-8b5d-293d5cb892a8", "metadata": {}, "source": [ "(i) Show that the continuous-time dynamical system\n", "$$\n", " {dy\\over dt}=(a-1) y - a y^2\n", "$$\n", "has an equilibrium at $y=(a-1)/a$ for any $a>1$." ] }, { "cell_type": "markdown", "id": "7ccdea22-367e-459d-9aa0-5e0bb90591d2", "metadata": {}, "source": [ "We let $g(y)=(a-1)y-ay^2$ and solve $g(y)=0$.\n", "\n", "Since\n", "$$\n", " g(y)= (a-1)y-ay^2=y(a-1-ay)=0\n", "$$\n", "then either $y=0$ or $a-1-ay=0$.\n", "\n", "The second possibility is satisfied when $y=(a-1)/a$. Therefore\n", "that is a fixed point.\n", "\n", "Note there is another fixed point at $y=0$." ] }, { "cell_type": "markdown", "id": "f33dc6c4-161b-4f3c-8c63-7a44fea8606a", "metadata": {}, "source": [ "(ii) Linearize the dynamical system in part (i) about the\n", "equilibrium and show the fixed point is stable for all $a>1$." ] }, { "cell_type": "markdown", "id": "2c19064c-86bf-4526-b2e6-ad3e366cb0a8", "metadata": {}, "source": [ "Let $p=(1-a)/a$ be the fixed point.\n", "The linearized right hand size is given by\n", "$$\n", " g_L(y)=g(p)+g'(p)(y-p).\n", "$$\n", "Since $g'(y)=a-1-2ay$, then $a>1$ implies\n", "$$\n", " g'(p)=a-1-2ap=a-1-2(a-1)=1-a<0.\n", "$$\n", "It follows that the fixed point is linearly stable\n", "and consequently stable." ] }, { "cell_type": "markdown", "id": "d96b6a05-7985-44df-89a5-77c7ee322473", "metadata": {}, "source": [ "(iii) Show that the analogous discrete-time dynamical system\n", "$$\n", " \\Delta z_n=(a-1) z_n - a z_n^2\n", "$$\n", "also has an equilibrium at $z=(a-1)/a$ for any $a>1$." ] }, { "cell_type": "markdown", "id": "96753e01-a613-41fb-9112-351f6c386c60", "metadata": {}, "source": [ "Note that $\\Delta z_n=g(z_n)$ where $g$ is defined as\n", "in (i). Therefore, the points such that $g(z)=0$\n", "are also the same. Thus\n", "$z=(1-a)/a$ as well as $z=0$ are equilibria points." ] }, { "cell_type": "markdown", "id": "f4445ef3-e373-4130-a012-3ca924572d17", "metadata": {}, "source": [ "(iv) Write the discrete-time dynamical system in (iii) as $z_{n+1}=f(z_n)$.\n", "What is $f(z)$?" ] }, { "cell_type": "markdown", "id": "45158c95-8efa-4436-82c9-b2d5e408bf3c", "metadata": {}, "source": [ "Since $\\Delta z_n= z_{n+1}-z_n$ it follows that\n", "$$\n", " z_{n+1}=z_n+\\Delta z_n=z_n+(a-1) z_n - a z_n^2\n", " =a z_n- az_n^2= f(z_n)\n", "$$\n", "where $f(z)=a z (1-z)$." ] }, { "cell_type": "markdown", "id": "721f961b-2c5f-47da-8cdd-62ae40a98df3", "metadata": {}, "source": [ "(v) The fixed point $z=(a-1)/a$ is stable when $|f'(z)|<1$ at\n", "that fixed point. For what values of $a>1$ is the fixed point\n", "stable." ] }, { "cell_type": "markdown", "id": "851e0404-78fa-411d-9a27-eee6ee9ff9bf", "metadata": {}, "source": [ "Differentiating yields\n", "$$\n", " f'(z)={d\\over dz}\\big(az-az^2\\big)=a-2az.\n", "$$\n", "Again let $p=(a-1)/a$. Then\n", "$$\n", " f'(p)=a-2ap=a-2(a-1)=-a+2.\n", "$$\n", "Solving $|f'(p)|=|a-2|<1$ implies\n", "$$\n", " -1\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "scatter(0:30,mkpoints(a,0.7),label=\"a=2.8, z0=0.7\")\n", "plot!([0,30],[p,p],label=\"fixed point\")" ] }, { "cell_type": "markdown", "id": "d8afd08d-6917-449d-8750-5726d64c52ca", "metadata": {}, "source": [ "The discrete dynamical system converged to the fixed point\n", "at\n", "$$p=(a-1)/a\\approx 0.6428571428571428.$$\n", "This makes sense\n", "because $a=2.8$ implies $a\\in (1,3)$ so the fixed point is stable." ] }, { "cell_type": "markdown", "id": "8833ac14-dfae-4898-aa11-af4c8ba08fc2", "metadata": {}, "source": [ "(vii) Let $a=3.2$ and $z_0=0.7$.\n", "Make a scatter plot of the points $(n,z_n)$ for $n=0,1,\\ldots,30$.\n", "Describe what happened." ] }, { "cell_type": "code", "execution_count": 5, "id": "b229e9a3-96a0-4e9c-af12-1cf37ee3da57", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.6875" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a=3.2; p=(a-1)/a" ] }, { "cell_type": "code", "execution_count": 6, "id": "7cbb73ba-dfd5-4947-8ab3-1691fa3f2f7c", "metadata": {}, "outputs": [ { "data": { "image/png": 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vWrXK19f3jWVMnTpVJBL9+OOPhBBdXd2qqud3VVdWVlpZWRFCgoKCgoJeOUm8jo6Oci2ZTKahofHyx8cQEhUVNXHixDeWBErlFRJiZ6jSWKtlUFFRwUo9AFzG3YtlvNr4kJSLL/+ZuECW9bBNmzbvsNnc3NypU6fGxsY+ePDgxo0bZmZmcrmcELJ+/XqjF7w4saq1tXVkZOTJkydf3HJGRkZISMj8+fPrnlB8lSVLlly/fn3Xrl30TFR2dna5ubnV1dXKTdnZ2RFC9u3b92JVRkZG9POT6d1ZepXHjx+/al6r+Pj4hw8fDh6Mk1tvoX0bD0G66pFk7cfXPD09WakHgMu4G4STJ4wRnVpOxAUv/kgQM79jl26Ojo7vsNnCwkKRSESfwzt16pTyopXRo0dnvUD5U+WL2tra2NhY5cNrtm3bRj/ZMTMzMyQk5Kuvvho3blzd4a5fv37s2DGVGvbu3btx48bDhw8rd2q9vLzs7OzoB2OlpKTExcXRF+MMGDDgxaqysrIMDQ0JIeHh4Tt27KDjMyoqKiLi2TTKe/bsqfsMyM2bN4eHh+N60bcyaVyk+dV1JPPWs2WK0rywzsfOsHnz5qzWBcBJTD3G4v194McwKRSKvgMGadm3IvMuPn9IzbJcQddxhibmqamp77BNiqLkcnlYWFjr1q179+4dHBzcvHnzc+fOvXGtgIAANze3Tp06WVtbd+jQQfl4EX19/WPHjlEUNWXKFKFQqLxXISAggO4we/Zs5ZOwlFxdXd3c3EL+35kzZyiKOn78uJmZWbdu3czNzX/++Wd13ktlZWWPHj1atGjh7+/v4eGRl5dHtzs6Ou7YsYN+LZVKjY2Nz549+9It4DFMr/HgwYO2ncMsfIItgkeatWgzZvKMenrcT9NTXl7OdgnAGolEUltby+w2Of0Yppqamm/nL1i+/A+hmR1l5syrLKt8lBAQ2HHrxnX0BZbvRqFQJCUl8Xi81q1bV1RU6OjoCARvOBdLUVRKSkpRUZG1tXXdPdHCwkIDAwNNTU2pVFr3JB+fz6d32oKDg+fPn9+1a9e6WysrK1MoFMpFXV1d+q6G8vLyu3fvOjg4vNWjLO/evSuTyXx8fJTHRYuKivT09EQiESFELpeXl5cbGRm99AZ/PIbpjYqLi3Nzc11cXOh/T1CHWCzGRHGchccwMUwoFP60dMmcr786e/ZsZmamgYFB+/btW7Zs+Z6b5fP5yvOLBgYG6qzC4/FatGjxYruZ2bMHi+vo6Lx0ouSRI0eqpCAhhM7IFxkYGHTo0EGdeury8PBQaTE1NVW+1tDQMDY2ftttgpKJiQmzD/IGgLfF6SCkGRsbDxw4kO0q3tFnn33GdgkAAI0bdy+WAQAAIAjCevL11187OTkFBgbGx8e/220YSuHh4dHR0e+zBQcHh9TU1Ff9VCaTnT179n22DwDQqOHQKPPS09M3bdqUmZmpo6NTUFDw3Xffvc/WKioqlPf/vZulS5e+5okQeXl5AwYMKC0tfZ8hAAAaLwQhw7Kysn777Tc+n79y5UovLy8fHx+JREIIefDgwcmTJydNmsTn8xUKxapVq3r16uXq6ioWi//666+MjIw2bdoMGzaMz+cTQmQy2fr16588efKqk5e7d+92d3e/cOFCWlpaWFhYWFgY3V5aWhoVFZWTk+Pn56d8VFNFRQV9EWlMTEyzZs3u3LmTkJDQrl27IUOGEEK2bdtWVVX1008/EUImTJiA2wEBgGtwaJRhAoFAX1+fz+cbGxvr6uo+fvz4hx9+IIS4uLjs3r37l19+IYR8//330dHRzs7OZWVlvr6+GRkZbdu23bt375gxYwghFEX17t374sWLbdu2/emnn5KSkl4cZePGjf379y8qKmrVqtW4ceO2bt1KCKmoqPDz83vw4IGnp+fvv/9OPzWCEPLNN9/k5+cTQnbu3DlkyJB79+61bt167ty5mzdvJoTo6+vzeDxjY2NjY2M6hgEAOKUJ7hGW7FlRk5nygQbTEJiMnCMwsVQ2WFlZDRgwIDo6mn5ug/KBDAKBYOfOnf7+/jweb+3atfHx8RoaGn/88UdQUBD9cMFBgwbZ2dllZ2enpaWlpaWdOHFCIBAMHDiQngvtRb169aIPulpbW0+bNm3kyJFRUVH29vbr1q0jhISFhTk5Oc2bN09l9eDgYHoW09ra2sOHD0dGRg4YMGDRokXKp0wAAHBNEwxC/e4RCqn4w4zF4/EERuZqdra3t1+yZMmYMWN27dpla2tLCElMTLx+/bqfnx/dQSqVpqamPnz40MfHh74HX0dH51WTTyrXCggIePToUVVV1f3799u1a0c3WltbN2vW7MGDBypB2Lp1a2WHwsLCt3u3AABNURMMQoGpFTG1YruKl6iurl63bp23t/eOHTvoE3h6enqRkZHTpk1T9tHT08vKyqJPK9Lqvq5LKn326IyKigpNTU2hUKinp1e380sfmYSDn2rKzc09dfp0ztOCtl6ewcHB+HcDaMLwv/eHM3fuXDMzs2vXrhUWFi5fvpwQ0rdv3127dhFC6FN0MplMIBAEBQXFxcXRs1rfunUrMTHxpVv7+++/a2pqCCFbt27t1q0bn8/v3r37gQMHiouLCSFHjhyRy+XK/b/XMDY2lkgkr4pbblq+Zn2b7v3HHsmdfd8k4rcDHu26KJ/CAQBNTxPcI2yYjh49unfv3oSEBE1NzV27dvn7+3fo0CEiIuLu3butWrXy9PQsLy8vLi5+8OCBg4PDkiVL2rVr5+3tLZVKlYdAVdjb2/v6+hoYGOTk5MTGxhJCevToMWzYMA8PDzc3t4cPH27ZskWdRyoaGRmNHTvW1dVVW1v77Nmz9vb2DL/zxubq1auLN+4r/vIs0RASQkraDS1Nv95n8Kg7V8+yXRoA1AtOT7pdT2prayUSCT3hp0KhqKqq0tbWrqio4PP5yvlCJRIJj8ejFysrK1NSUgwNDZs1a6bcSFlZWWZmZqtWrWQymaamJj1rtlJoaOjYsWN79OiRnZ3t5uZWd1LvsrKynJwcFxcXoVBIt0ilUm1tbR6PJ5FIBAIBPblzTU1NZWWlcuZiiqJKS0sNDQ2ZPQbYGCfdHjxm0h7zCOLSsW6jxZbBF6N+cnV1rY8R4W1h0m0uw6TbjYNAIFBOe83n8+mnuqt8a9fdV9PS0nrxGKahoSG9kdd83dMP0X3VikrK9K07qFAoVCYlIYS+g+INb4wbHmdmk1ZOKo01ps5ZWVkIQoAmCecIG6XBgwe3atWK7SqaJgd7W1KYrtIoLEqnL/QFgKYHQdgojR071svLi+0qmqZpY0eYnFhKFLXKFl7GDfOawpc+JwsAmgAcGgX4jw4dOsz9bMDPf3Qt9Rlao2tumHXN6umNw/u2s10XANQXBCGAqllTPh82qP+xf4/n5Of6DewVFrYM9xECNGEIQoCXsLGxiRz9GdtVAMCHgL9zAQCA0xr9HqFAIOjXr5/KbXbQQJSUlPj7+7NdBQDA6zT6IDx16hQmj27InJ2d2S4BAOB1Gn0QOjg4NKhpZQAAoHHBOUIAAOA0BCEAAHAaghAAADgNQQgAAJyGIAQAAE5DEAIAAKchCAEAgNMQhND4VFZWFhQUsF0FADQRCEJoTO7evevbpYdju5D2gydZt/BetmotRVFsFwUAjVujn1kGuCMjI6P7oBFPh24itp6EEFIjW3Bg9tOCH35c9C3bpQFAI4Y9Qmg0Fv68PD9swbMUJIQItcWfLNv8977KykpW6wKAxu0t9ghXrVq1adMmPp8/ceLEcePGqfx069at27Ztq9vyzz//aGlp/fDDD2fPnqVb9PT0Dhw48H4FA3ddT0ikPp3/nya+BmnmnZKS0rp1a5aKAoBGT90gjImJ+fHHHw8cOFBbWztw4EAHB4ewsLC6HYKCgmxsbOjX27ZtS01N1dLSIoTcuXPH3d29X79+hBChUMho8cAtIi0tUiUhWvp1G3lVUvo3DQDg3agbhGvWrJk+fTr9bLnJkyf/+eefKkHo7OysfODOzJkzp06dqvyRm5tbSEgIQwUDd33cK/R+wt7KblOeN1UUahalubi4sFcUADR66p4jvHv3rq+vL/3az8/vzp07r+oZFxf36NGjiIgIZcv69eu7du06fvz45OTk96kVOG7GlM+dUw/qnPqNSEuIopakXDRd1//P35fweDy2SwOARkzdPcL8/HwjIyP6tZGRUX5+/qt6bt68efDgwfr6z45fDRw4cOTIkXp6evv37w8ICEhKSmrWrNlLVxSLxT4+Pnz+s2zu2rXrihUrXjWKTCYTCoUCAa565ZbzRw/8sXbD/l1DKsRiL0+P//29vnnz5mKxmO264IOqqKhguwRgjUwmq66u1tDQULO/lpbWG8/K8dS8DcvCwmL//v1BQUGEkBMnTowfPz49Pf2lJdrY2Bw+fLhjx44v/rRr1659+vT56quvXjqEiYnJyZMnlXFrbm6uTNOXDoQg5CyKoiQSiZ6eHtuFADvEYvFrvhygaZNKpSKRSP0gVIe6QeLk5JScnEwHYXJyspOT00u77dmzx8LCIjAw8KU/NTc3l0gkrxqCx+M5OjqamJioWRIAAMD7U/cc4ciRI9euXSuTySoqKtatWzdixAi6fe7cuWlpacpumzdvHjNmjPKcjVwuP3fuHL3TeebMmaNHj+KqGQAAaFDUDcJx48a1bNnSxsbGzs4uICBAGYRbtmzJy8ujXz9+/PjevXsjR45UrkVR1MSJE7W1tY2MjD799NPly5fT+5QAAAANhLrnCGkymYwQoq2t/VZjVFZW1tbWvvGMjqmpaUpKipqHRnGOkMtwjpDjcI6Qy9g8R0h72wik4X5nAABosDDXKAAAcBqCEAAAOA1BCAAAnIYgBAAATkMQAgAAp+H2A2BMfn7+lStXJBJJmzZt3N3d2S4HAEAtCEJgxtLfVizftF3q0btaqGe44u+2tvrRWzfo6uqyXRcAwBsgCIEBB2IO/XzgYum0c4QvIIQUBH955upfn30xfe+W9WyXBgDwBjhHCAz4ceWG0r4/0ClIq24/6lxcQmVlJYtVAQCoA0EIDMjNyyVmjiqNfDNH5Ty0AAANFoIQGGBsZETKn6o0UqU5pqamrNQDAKA+BCEwYMKoofqnfq3bwnt4pqW9FWZGBoCGDxfLAAMmjhl9MW7KiU2fFHoPJVp6+qmn7Z5e3/3PXrbrAgB4MwQhMIDP5+/cuPrGjRv/HDtZXF7RfVyXfv1+5fNxvAEAGgEEITDG19fX19eX7SoAAN4O/mYHAABOQxACAACnIQgBAIDTEIQAAMBpCEIAAOA0BCEAAHAabp/ghNzc3KtXr1ZVVfn6+rq6urJdDgBAA4IgbPrmLlqyae8/EvfeCr6m/q+bu3g67li/SlNTk+26AAAaBBwabeI2Rm1dc/5RwZdnpWFzKkNmFEw8crjaderX37JdFwBAQ4EgbOJ+/3Nzeb8fCO/5B13Z7csDR/+lKIrFqgAAGg4EYRNXJq4gev99FhKPx9M3LysrY6kiAICGBUHYxGlpCkmVRKVRUVGEByQBANAQhE3cqCGDdM78UbdF49ahDr5tNDQ02CoJAKBBwVWjTdzcmV9eGx55deuIIs9BRFPL8OG/juX3txzex3ZdAAANBYKwiRMKhYf3bLt8+fLRk2fFUlmPKb179VrJ4/HYrgsAoKFAEHJCYGBgYGAg21UAADREOEcIAACchiAEAABOw6FRlkkkknv37lEU5e7urqenx3Y5AACcgyBk07JVf/60ar3c0Z/w+Pz0uCljRnz71XS2iwIA4BYEIWs2/7V90Z4LZdPOE4GIEEJqq3/ZO9VAb/3Uz8ezXRoAAIfgHCFrlvyxpuyTZc9SkBAi0Cz/+LffVm9gtSgAAM5BELKmQlZFdIz/0yTSlSkIpsMGAPiQEISs4VMKQilUGnm11bjbHQDgQ0IQsiakayeNG9F1W/hJRwID/NiqBwCAm3CxDGv+WLroRs/+TwruV3h8RHh83XtHbB8dW/9vDNt1AVDZk08AACAASURBVABwC4KQNcbGxkmXz0Rt3X749HpKQfUM6zBu9HmBAJ8IAMAHxWs4l2aYmpqmpKSYmJio01kmkwmFwnqNjXv37l2+clUoEHTsGOji4lJ/A8HboihKIpFg/gHOEovFeKAmZ0mlUpFIxOyD5LD/8RLV1dVDIj+/kJJX2qInX1Fr8MeEfoHeG1f8yufjlCoAQFODb/aXmPXtomPyFoXjDtZ2mVjdbXLhpGN7H2v8vHwl23UBAADzEIQvsTfmiKzbl3VbKnp9u/6vv9mqBwAA6g+CUFVtba1cICL8/x6A1tSR1dSyVBEAANQjBKEqgUDAr5ERlWuIaqs0+bjPHQCgCXqLi2UOHjx48uRJS0vLiRMnmpubq/x0586dmZmZykUrK6tRo0YRQuRy+V9//XX9+vUWLVpMnDhRW1ubkbpLSkp0dXUNDQ0Z2ZqK3qHdd1yJqg6MVLZon1o2YvDH9TEWAACwS909wrVr13755ZdeXl5paWmdOnWqrq5W6SAWi0v+38aNG8+ePUu3z5o1a/Xq1f7+/sePHx84cOD7V7xj1x57d1+ffiNbdO7n0b7r1bi499+milW//NA+75jJ9tEkfh/v2m7TqMEhGqkL5sxifCAAAGCdWvcRKhSK5s2br1q1qk+fPhRFeXt7z5s3b8iQIS/tLJPJbGxs/vnnn6CgoOLiYjs7u6SkJBcXF5lMZmVldebMmbZt2750RXXuI/xzY9TcqNjSwauJjhEhhBSkmW3/7Pj2NT4+Pm9+r2/p4sWLpy5c1hQIQrt19vPDzGcNCO4j5DjcR8hlrN1HmJWV9fjx4+7duxNCeDxecHDwxYsXXxWEe/futbCw6NixIyHkxo0bFhYW9N3o2tragYGBFy9efFUQvhFFUYt/W1E6+QwR6T5rMncuHLRi5oKlpw/tebdtvkZQUFBQUBDjmwUAgAZFrSDMzc3V09PT0tKiFy0sLG7cuPGqzps3b46MjKQfoZCXl1f3bKKFhUVubu6rVpRKpaNGjdLU1KQXAwICpkyZUrdDaWlprbbJ8xSkNWuTvO+RVCpV541A00BRlFQqxfwGnCWTyZjdIYBGRCqVyuVy9X8BNDU13zgHmVpBqKmpWVNTo1ysrq4WiUQv7ZmWlnb58uWdO3cqV6ytfX7XQU1NzatWJIQIhcIBAwYoj3c1a9ZMpbOenh5VU6m6mqJWg897zWah6aEoqra2Fh86Z73mKwiaPLlc/laHRtX5i1mtILS1ta2srCwqKjI1NSWEZGVl2dravrTnpk2bevXqZWNjo1wxOzuboih6BzErK6tr166vGkUoFA4cOPA15wgNDAzMdYUFBWnE3FnZqJF4qHvXzvjzkFMoitLQ0MCHzln49LlM4/8xuE21Di5ZWFgEBgbu2rWLECIWi48ePdq/f39CSGlpaWxsrLKbXC7ftm1bZOTzuw7atWunoaFx6tQpQsjjx4/j4+P79u37PuVuWfWrxZYhvDvHSG01qRSLLm+2v/Drzwvnvc82AQCAy9S9j3DJkiWDBg06f/78nTt3OnfuTF9Fcu/evd69eyuvOz127Fh1dXXv3r2VawmFwp9++mnYsGGhoaEXLlyYNWuWcmfx3fj7+SWcPvz1wqVX/vxBS6TVJ6zbwpVndXV137wmAADAy7zFY5iePn165coVS0vL9u3b04c6pVJpamqql5cX3SE7O1smk734xKL09PSEhIQWLVq0bt36NdtvaI9hggYLt09wHG6f4LL6uH0CzyOExgdByHEIQi6rjyDEBegAAMBpCEIAAOA0BCEAAHAaghAAADgNQQgAAJyGIAQAAE5DEAIAAKchCAEAgNMa6w3plXHHyuNP0RPcAAcpFAopHsPEVXK5XIZJt7lKoVAYffKFhpM7g9tsrEEoatNZy7EVZqDnJoqiZDKZjo4O24UAO6RSKT59zqqqrhbaqU7k+Z4aaxDyRDpCOxdMscZNFEXVSCSamGKNq6rEYk1MscZVtVIp4TO8C4SDSwAAwGkIQgAA4DQEIQAAcBqCEAAAOA1BCAAAnIYgBAAATkMQAgAApyEIAQCA0xCEAADAaQhCAADgNAQhAABwGoIQAAA4jRNBeP/+/W79wu3c/ew9/D4eMTY7O5vtigAAoKFo+k9vOH/h4qAJMws/WUn6+xBCsu+duBLc51LsfmdnZ7ZLAwAA9jX9PcIJM+cVfraLNPOhFyn30KcDlk2Zu5DVogAAoKFo4kFYXV1dJKkiJnZ1GynXoMTbd9kqCQAAGpQmHoQKhYLH9CMcAQCgKWniQailpaWnoSDi/P+0Pk5o6erCUkUAANCwNPEgJIT8sXShyZbhpCD92fKTm+Z7J61YMp/VogAAoKFo+leN9u3VM8ZA/4vZk54Wl/EI5eLYbMPBHS1btmS7LgAAaBCafhASQoI6drx18aRCoeDz63EPWCaT/bpizfHzV3g8Xs8ugTOmfK6lpVV/wwEAACOa/qFRpXpNwZycHPeAzkvuCC52/vFC0JLFtyj3gM55eXn1NyIAADCCE3uEH8C46XMehy2mWnWnFyu7Tskwc/181jcHtm9itzAAAHg9Du0R1qv4m7eUKUijPHtejU9gqx4AAFATgpAh/JfsWyt4+OcFAGjo8E3NDF0tIZGW/KeposhABxfLAAA0dAhCZiz6eprR7i9IVcWz5Uqx0e7Pv587g9WiAADgzXCxDDNGDBtSVVX93U/BCutWhKI08h4s+earwZ8MYrsuAAB4Ax5FUWzX8IypqWlKSoqJiYk6nWUymVAoFAgaVpDL5fLU1FQej9e8eXMNDcxxWl8oipJIJHp6emwXAuwQi8X6+vpsVwHskEqlIpGI2S/YhhUkjZ2GhoabmxvbVQAAwFvAOUIAAOA0BCEAAHAagrCRyc3NjRg90aF1gL2HX5+IkY8ePWK7IgCAxg1B2Jg8evSobbde0YZ9nky9mDX9SqxzZIc+Edfj49muCwCgEUMQNiZfzJ6fN+APhWdPwuMRQiiXjgWjdo6dNpvtugAAGjEEYWOSdPc+cQn8T5O5c25xuVwuZ6kiAIBGD0HY+PH4CoWC7SIAABqrt7iPUCKRnDlzhs/nd+vWTVtb+6V9CgsLL168KBAIAgICLCwsCCHp6enFxcX0TzU0NNq0afP+RXOWUzP73Oy7xNbjeVNZnpGWQCgUslcUAEDjpu7MMk+fPg0MDHR1dZXL5ZmZmZcuXTI1NVXps2/fvokTJwYEBNDfyzExMYSQoUOHXr582dzcnBBiYGBw+vTpVw3RBGaWqW83b94MGz6hcPD6Z1lYmGGyc+yOX7/tGRbKdmkfFGaW4TjMLMNl9TGzjLpBOHfu3EePHu3Zs4cQ0q9fv4CAgO+++65uh+zs7JYtWx47dqxjx46EEIqieDweIWTo0KGdOnWaNGnSG4dAEKojKSlp7PS5T3LzCV/D3EBnzc//6xQUxHZRHxqCkOMQhFxWH0Go7jnCmJiYwYMH06+HDBlC7+3VtX///vbt23t7e1+5ciUvL49OQVpubu65c+dycnIYqZjjvLy8rp06knMnLivx4u3LpzmYggAAzFJ3jyorK8ve3p5+bWdnl52drdIhNTVVIpEEBAS4uLhcvnz5m2++mT59OiGEz+fHxsaeO3cuMTFx5MiRq1atetUQNTU1UVFRurq69KK7uzu9c/lScrmcz+fXjVuu4fF4nL1YlKIouVzO2bcP+PS57G0/fXWSQt0grK2tVe6KCoXC6upqlQ5SqfTu3bvJycmWlpZJSUkBAQERERG2trabN28WiUSEkCdPnvj5+YWEhAwYMOBVQ9y8eZPuTAiRyWR+fn6vqqeqqkqhUOB/Bm6iKKqqqgqXCHFWdXV1VVUV21UAO+iPXv1Do5qamm88iaZuEFpZWRUUFNCv8/Pzra2tVTpYW1t7eXlZWloSQry8vMzMzO7du2dra6sMtmbNmnXv3v369euvCkJtbe0VK1aoeY6Qx+Nx8xwhEEIoiqIoSkdHh+1CgB1yuRyfPpexdo6wU6dOygs+T5061blzZ/p1VVUVfblNly5dMjMz6V208vLyoqIilbBUKBT379+3sbFhrHYAAID3pu4e1fTp07t06WJiYlJbW7tt27arV6/S7fr6+idOnOjSpUtwcLCdnd2IESN69uy5bdu2kJAQT0/P6urqPn36hIaGamtrx8TEVFRUjBgxot7eCwAAwFtTd4+wTZs2Fy5cKC4urqiouHz5csuWLen23377zcXFhRDC4/H+/fffgICA+Pj4oUOHHjhwgBAiEAhGjRpVUFCQmpoaHh5+69YtAwODenonAAAA70Dd+wg/ANxHCGrCfYQch/sIuYzN+wgBAACaJAQhvMSjR4+69x9s3crXspVv68DgM2fPsl0RAEB9waFFUJWamhrUNyL/4xVUv3aEkPzizEHTx//5bUnEoIFslwYAwDzsEYKqWQuW5H/0C+Xc7tmyiX3J6L9nLfiB1aIAAOoLghBUJSQmUa6d/tOkY1SlZVxUVMRSRQAA9QhBCKp4PB6hXnjSr0LO5+O3BQCaIHy1garAdv68eyf+01T+VFchNTY2ZqkiAIB6hCAEVb8smmd1bL7GnaPPlrPvmm6OWPPzYlaLAgCoL7hqFFTZ2dndPHfsy7kLLy1brCA8p2Z2q3es9fb2ZrsuAIB6gSCEl7C0tNy1eS3bVQAAfAg4NAoAAJyGIAQAAE5DEAIAAKchCAEAgNMQhAAAwGkIQgAA4DQEIQAAcBqCEAAAOA1BCAAAnIYgBAAATkMQAgAApyEIAQCA0xCEAADAaQhCAADgNDyGCdiUnp5+6vSZknJxQNs2Xbp0YbscAOAi7BECa2YvWNyu/8jPz0pm3zH8+PutbTuHFhYWsl0UAHAO9giBHfv2H1h/Lrl0yinC4xFCijsML7t77JPPJpw9HM12aQDALdgjBHYs27C1tMe3dArS5B4972fml5aWslgVAHAQghDYkZuTQ0wdVFtNHfLy8tgoBwC4C0EI7LC2sSFFj1Vbix5bWVmxUQ4AcBeCENgxY/woo9j/EYpStmjciXVvZmlkZMRiVQDAQbhYBtgxaOCA67fubF4ZXOI9WC4yMM4451iTtTd6J9t1AQDn8Kg6f5Kzy9TUNCUlxcTERJ3OMplMKBQKBAjyxi0jI+PkqdMl5eJ2vj6dO3dWcy2KoiQSiZ6eXr3WBg2WWCzW19dnuwpgh1QqFYlEGhoaDG4TQQJscnR0HDsmku0qAIDTcI4QAAA4DUEIAACchiAEAABOQxACAACnIQgBAIDTEIQAAMBpCEIAAOA0BCEAAHAaghAAADgNQQgAAJyGIAQAAE5DEAIAAKchCAEAgNMQhAAAwGlvF4SlpaVVVVWv6UBRVHFxsUofurG2tvZdCgQAAKhP6gbhkydP/Pz8WrZsaWlpuWrVqpf2+eeffxwdHZs3b25mZrZ69Wq68caNGy4uLq1bt7a1tT106BAzVQMAADBE3SCcOXNm+/bt8/Lyrl69Om/evEePHql0SExMHDFixObNm0tKSoqKivr370+3jx49etq0adnZ2Tt27Bg1apREImGyfAAAgPejVhCWl5fHxMRMnz6dENKyZcsePXps375dpc/KlStHjhzZvXt3QoimpqadnR0hJDExMT09ffz48YSQkJAQe3t77BQCAECDolYQZmZmEkKcnZ3pxZYtW6anp6v0uXv3LiHEy8vL1NS0Z8+e9CppaWmOjo4ikUi5Ylpa2qtGUSgUGRkZaf+vpKTk7d8OAADA2xGo06msrExHR4fH49GLurq6L6ZUfn5+TEzMpUuXrK2tJ0yYMGbMmOPHj5eXl+vo6Cj76OnplZaWvmoUiUQycOBAPv9ZNnft2nXlypWv6iyTyYRCoUCgVv3QxFAUJZVK2a4CWCORSJRfR8A1Uqm0pqZGQ0NDzf5aWlpvTAq1gsTCwkIsFisUCjqlSktLLS0tX+zTpUsX+ojo9OnTfXx8ampqzM3Ny8rKlH1KSkrc3d1fNYq+vv7NmzdNTEzUKUlDQwNByFkURfF4PD09PbYLAXZQFIVPn7P4fL5IJFI/CNXapjqd7O3t9fX1ExIS6MX4+HhPT0+VPl5eXpWVlfTryspKgUDA5/M9PDzS09OLi4sJIRRF3bhx48UVAerbrVu35n//47gvv476a+vr7/8BAA5SKwhFItHo0aNnz56dnJy8ZcuWGzdufPrpp4SQO3fudOzYke7z+eef79y58/Tp0+np6d9++214eLiGhoajo2NISMiMGTPS0tIWL16sqakZGhpaj+8G4AWTZs4NGTN7cZbjRmHoF4czWvh2fPDgAdtFAUADou7tE0uWLPH29h44cODOnTuPHDlCH8AUCARmZmZ0Bx8fn40bNy5cuHDQoEHe3t5r1qyh27ds2UJRVL9+/RISEmJjY5ndnwV4vQMxh3beyC6c+A/xG0RadpX1mPskYtOAT8eyXRcANCA8iqLYruEZU1PTlJQUNc8R4mIZLqMoSiKRqHOWKGzQ8BNes4jtfw7IW0RFXN76a/PmzeutQKhfYrFYX1+f7SqAHVKplJ1zhACNVH5BATGyUWmUG9nm5+ezUg8ANEAIQmjKXJycSK7qGUGNvAeOjo5slAMADRGCEJqy2ZPHmhxbSKqf33SoceufVlYG1tbWLFYFAA0KzrFBU+bv7//7VxNmL+5a1TJMpm1m8ORKKyNyYPsmtusCgAYEF8tA46P+xTK08vLyuLi4wsJCLy8vDw+Peq0NPgBcLMNl9XGxDIIEmj4DAwPcwAoAr4JzhAAAwGkIQgAA4DQEIQAAcBqCEAAAOA1BCAAAnIYgBAAATkMQAgAApyEIAQCA0xCEAADAaQhCAADgNAQhAABwGoIQAAA4DUEIAACchiAEAABOQxACAACnIQgBAIDTEIQAAMBpCEIAAOA0BCEAAHAaghAAADgNQQgAAJyGIAQAAE5DEAIAAKchCAEAgNMQhAAAwGkIQgAA4DQEIQAz9u0/0OOTET5deoz6fFpaWhrb5QCAugRsFwDQ6FEU1fuT4VcrDEo7fkUMLG49vnG079C1P8z7ZGB/tksDgDdDEAK8r+j9By5LjMvDl9OLlEdYoXO7yXO69+vdUyQSsVsbALwRDo0CvK/tB46W+434T5O2YY1zx/j4eJYqAoC3gCAEeF/iCgnRMVRprNU2qqioYKUeAHgrCEKA9+Xv7a6Rfk2lUZQR5+HhwUY5APB2EIQA72v6pPFm55eRnPvPlilKdHZV+xZ2dnZ2rNYFAGrBxTIA78vS0vJk9PZh46fk12gqDK2ozKRBvUKXLf2T7boAQC0IQgAGeHp6Jl0+U1BQkJeX16JFC1wsCtCIIAgBGGNubm5ubs52FQDwdnCOEAAAOA1BCAAAnIYgBAAATkMQAgAApyEIAQCA0xCEAADAaQhCAADgtLe4j3DDhg2bNm3i8/kTJ04cOXKkyk/T0tLmzJmjXJw4cWJwcDAhZPny5ZcvX6YbdXV1o6Ki3rtmAAAAxqgbhIcPH54/f/7evXvlcnl4eLidnR2dc0olJSUXLlzYtm0bvejm5ka/iIuLs7a27tevHyFEKBQyVzkAAAAD1A3C1atXz5gxIygoiBAyefLkNWvWqAQhIURbWzskJOTFdd3c3F7aDgAAwDp1zxHevn3b39+ffu3v75+UlPRin8LCwr59+4aHh//1118URSnbN2/e3LNnz0mTJqWlpb1/xQAAAAxSd4+woKDA0PDZo0eNjY3z8/NVOpiZmS1ZssTDw+Px48ffffddenr6woULCSF9+/bV1dXV1dU9ePCgn5/f7du3bW1tXzqEWCz28fHh859lc7du3VasWPGqemQymVAoFAgwVyoXURQllUrZrgJYI5FIeDwe21UAO6RSaU1NjYaGhpr9tbS03pgU6gaJgYGB8qunoqLCyMhIpYODg8PkyZPp16ampuPGjaODcPjw4XRjaGjo7du3d+3aNXPmzJcOoaure+DAAeWWjY2N9fT0XlWPhoYGgpCzKIri8Xiv+fWApo2iKHz6nMXn80UikfpBqA51g8TJySklJaVjx46EkJSUFEdHx9d0tra2FovF9LdV3XYrK6vy8vJXrcXn8x0dHU1MTNQsCQAA4P2pe45w+PDhf/75Z1VVlUwmW79+/aeffkq3L1iwICMjgxBy+/ZtOuQqKiqWLFnSrVs3Ho+nUCiuXLlC97x06dLRo0dfvMQGAACAReoG4cSJE+3t7e3s7Ozt7T09PUeNGkW3r1y5MisrixBy7Ngxa2trW1tbCwuLqqqqdevWEUIUCsWwYcP09fWtrKwGDBjw448/dunSpZ7eCQAAwDvg1b28841KS0t5PJ7yqhkV1dXVxcXFJiYmmpqaddvFYrFMJrOwsHj9xk1NTVNSUtQ8NIqLZbiMoiiJRIKzRJwlFov19fXZrgLYIZVKWTtHSHvxGpm6NDU1raysXmzX19fHby0AADRMmGsUAAA4DUEIAACchiAEAABOQxACAACnIQgBGhm5XF5YWMh2FQBNB4IQoNHIy8vrN2SUjUeAZ6/hli285iz4vqqqiu2iABo93IcH0DjIZLIOof2ehCxQhGwkhBBF7YpTy+6OHPfP7q1slwbQuGGPEKBxiNq6Pc/9Y4VHj2fLfIEs9Ku4tIKUlBRW6wJo9BCEAI3DuWuJlS6dVRrFzbskJCSwUg9Ak4EgBGgctESapFqm0iiokYpEIlbqAWgyEIQAjcMnvbsbJkX/p0lRq33/WFBQEEsVATQRCEKAxqFvnz5thfn6h+YRcQEhhOQ+MNkYPm3McDMzM7ZLA2jccNUoQOPA4/FOHtq7IeqvtX99VlxU7OTk9P3vczphdxDgvb3dY5jqFR7DBGrCY5g4Do9h4rL6eAwTDo0CAACnIQgBAIDTEIQAAMBpCEIAAOA0BCEAAHAaghAAADgNQQgAAJyGIAQAAE5DEAIAAKchCAEAgNMwRRkAqKIo6tGjR5mZmc7Ozg4ODmyXA1C/EIQA8B/3798Pj/y8QGBWY+IozLvvYqixJ2qtra0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In this case the dynamical\n", "system does not converge to the fixed point.\n", "However, as it converges it appears to oscillate\n", "between two points $p_1\\approx 0.8$ and $p_2\\approx 0.51$.\n", "\n", "This is a cycle a length two." ] }, { "cell_type": "markdown", "id": "f0b50e41-9499-4800-be50-ce380fdb539c", "metadata": {}, "source": [ "The following is an optional discussion about\n", "and description of the limit cycle.\n", "I decided to more\n", "accurately compute the points $p_1$ and $p_2$ that the\n", "limit converges two. The values of $p_1$ and\n", "$p_2$ can be obtained by running the dynamical system\n", "for a long time and examining the points $z_n$ and $z_{n+1}$ for\n", "large $n$.\n", "Instead of that, I used Newton's method to solve $f(f(z))=z$.\n", "\n", "A cycle of length two satisfies the equation $f(f(z))=z$.\n", "Substituting yields\n", "$$\n", " f(f(z))=f(az(1-z))=a(az(1-z))(1-az(1-z))=z.\n", "$$\n", "Therefore\n", "$$\\eqalign{\n", " a^2 z(1-z)(1-az(1-z))&=a^2 z (1-z-az(1-z)^2)\\cr\n", " &= a^2 z(1-z -az (z^2-2z+1))\\cr\n", " &= a^2 z(1-(1+a)z+2az^2-az^3)=z.\n", "}$$\n", "Assuming $z\\ne 0$ then\n", "$$\n", " 1-(1+a)z+2az^2-az^3 = 1/a^2.\n", "$$\n", "Consequently\n", "$$\n", " h(z)=0\\quad\\hbox{where}\\quad h(z)=1-1/a^2-(1+a)z+2az^2-az^3.\n", "$$\n", "Now look for roots $p_1$ and $p_1$ of $h$ near $0.8$ and $0.51$ \n", "respectively using Newton's method\n", "$$\n", " w_{k+1}=w_k-h(w_k)/h'(w_k).\n", "$$\n", "Note that differentiating yields\n", "$$\n", " h'(z)=-1-a+4az-3az^2.\n", "$$" ] }, { "cell_type": "code", "execution_count": 7, "id": "2f361654-ff4f-4097-b0b9-ff4c5b5a1467", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "dh (generic function with 1 method)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h(z)=1-1/a^2-z*(1+a-z*(2*a-a*z))\n", "dh(z)=-1-a+z*(4*a-3*a*z)" ] }, { "cell_type": "code", "execution_count": 8, "id": "af84d3e6-62ed-45e9-aa39-a7b56bd76570", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "w1=0.7994591346153838\n", "w2=0.7994554906323398\n", "w3=0.7994554904673702\n", "w4=0.7994554904673702\n", "w5=0.7994554904673702\n" ] }, { "data": { "text/plain": [ "0.7994554904673702" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "wk=0.8\n", "for k=1:5\n", " wk=wk-h(wk)/dh(wk)\n", " println(\"w$k=\",wk)\n", "end\n", "p1=wk" ] }, { "cell_type": "code", "execution_count": 9, "id": "0a265580-7432-4e3d-9f3d-0e94edf30c24", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "w1=0.5129625355113645\n", "w2=0.5130444476239788\n", "w3=0.5130445095325955\n", "w4=0.5130445095326303\n", "w5=0.5130445095326303\n" ] }, { "data": { "text/plain": [ "0.5130445095326303" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "wk=0.51\n", "for k=1:5\n", " wk=wk-h(wk)/dh(wk)\n", " println(\"w$k=\",wk)\n", "end\n", "p2=wk" ] }, { "cell_type": "markdown", "id": "c2890c03-dabd-4c24-8d13-b2ee99342e25", "metadata": {}, "source": [ "It follows that when $a=3.2$ the dynamical systems\n", "converges to a cycle between the two points\n", "$$\n", " p_1\\approx 0.7994554904673702\n", " \\quad\\hbox{and}\\quad\n", " p_2\\approx 0.5130445095326303.\n", "$$" ] }, { "cell_type": "code", "execution_count": 10, "id": "58d3c2ab-55cf-46c3-b896-94f1be7dced2", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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] }, { "cell_type": "code", "execution_count": 11, "id": "554eaa0b-0138-4355-9739-d653f75d9795", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.75" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a=4.0; p=(a-1)/a" ] }, { "cell_type": "code", "execution_count": 12, "id": "33409cc6-d5bf-4c6c-80f3-f48dc21f6035", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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"d4931969-bc0a-4d31-b9ac-77b1959c6040", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.10", "language": "julia", "name": "julia-1.10" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.10.10" } }, "nbformat": 4, "nbformat_minor": 5 }