250 lines
7.0 KiB
Plaintext
250 lines
7.0 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"%matplotlib notebook\n",
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"\n",
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"import numpy as np\n",
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"from scipy.integrate import odeint\n",
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"import matplotlib.pyplot as plt\n",
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"from matplotlib import animation\n",
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"plt.rcParams[\"animation.html\"] = \"jshtml\""
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"m1 = 1\n",
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"m2 = 1\n",
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"c1 = 0.1\n",
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"c2 = 0.1\n",
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"c3 = 0.1\n",
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"a1 = 0.12\n",
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"a2 = 0.05\n",
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"a3 = 0.08\n",
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"k1 = 0.05\n",
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"k2 = 0.21\n",
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"k3 = 0.11\n",
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"C1 = 0.15\n",
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"C2 = 0.30\n",
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"omega = 0.1\n",
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"\n",
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"f1 = lambda t: C1*np.cos(omega*t)\n",
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"f2 = lambda t: C2*np.cos(omega*t)\n",
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"\n",
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"def deriv(X, t):\n",
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" \"\"\"Return the derivatives dx/dt and d2x/dt2.\"\"\"\n",
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" x1, x2, xd1, xd2 = X\n",
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" F1 = f1(t)\n",
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" F2 = f2(t)\n",
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" xdd1 = F1 - xd1*(c1 + c2) + xd2*c2 - x1*(k1 + k2) + x2*k2 - a1*x1**3 + a2*(x2 - x1)**3\n",
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" xdd2 = F2 - xd2*(c2 + c3) + xd1*c1 - x2*(k2 + k3) + x1*k2 - a3*x2**3 + a2*(x2 - x1)**3\n",
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" return xd1, xd2, xdd1/m1, xdd2/m2\n",
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"\n",
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"\n",
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"def solve_duffing(tmax, dt_per_period, t_trans, x0, v0):\n",
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" \"\"\"Solve the Duffing equation with the standard odeint solver.\n",
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" \n",
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" Find the numerical solution to the Duffing equation using a suitable\n",
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" time grid: tmax is the maximum time (s) to integrate to; t_trans is\n",
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" the initial time period of transient behaviour until the solution\n",
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" settles down (if it does) to some kind of periodic motion (these data\n",
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" points are dropped) and dt_per_period is the number of time samples\n",
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" (of duration dt) to include per period of the driving motion (frequency\n",
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" omega).\n",
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" \n",
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" Returns the time grid, t (after t_trans), position, x, and velocity,\n",
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" xdot, dt, and step, the number of array points per period of the driving\n",
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" motion.\n",
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" \n",
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" \"\"\"\n",
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" # Time point spacings and the time grid\n",
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"\n",
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" period = 2*np.pi/omega\n",
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" dt = 2*np.pi/omega / dt_per_period\n",
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" step = int(period / dt)\n",
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" t = np.arange(0, tmax, dt)\n",
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" # Initial conditions: x, xdot\n",
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" X0 = x0 + v0\n",
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" X = odeint(deriv, X0, t)\n",
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" idx = int(t_trans / dt)\n",
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" return t[idx:], X[idx:], dt, step"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Set up the motion for a oscillator with initial positions and velocities\n",
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"x0, v0 = (-1, -2.5), (1.5, -0.1)\n",
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"tmax, t_trans = 200, 0\n",
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"dt_per_period = 100"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Solve the equation of motion.\n",
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"t, X, dt, pstep = solve_duffing(tmax, dt_per_period, t_trans, x0, v0)\n",
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"print(\"# samples: %d\" % len(t))\n",
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"print(\"dt: %.4f\" % dt)\n",
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"print(\"Steps per period: %d\" % pstep)\n",
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"x1, x2, xd1, xd2 = X.T"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"fig1, ((ax11, ax12), (ax21, ax22)) = plt.subplots(nrows=2, ncols=2)\n",
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"\n",
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"ax11.plot(t, x1)\n",
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"ax11.set_xlabel(r'$x_1$')\n",
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"ax12.plot(t, x2)\n",
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"ax12.set_xlabel(r'$x_2$')\n",
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"ax21.plot(t, xd1)\n",
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"ax21.set_xlabel(r'$v_1$')\n",
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"ax22.plot(t, xd2)\n",
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"ax22.set_xlabel(r'$v_2$')\n",
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"plt.tight_layout()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"scrolled": false
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},
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"outputs": [],
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"source": [
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"%%capture\n",
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"fig2, ((ax11, ax12, ax13), (ax21, ax22, ax23)) = plt.subplots(nrows=2, ncols=3)\n",
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"\n",
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"#fig2.suptitle(\"Duffing\")\n",
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"\n",
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"# Positions\n",
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"ax11.set_xlabel(r'$x_1$')\n",
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"ax11.set_ylabel(r'$x_2$')\n",
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"ln11, = ax11.plot([], [])\n",
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"ax11.set_xlim([1.2*np.min(x1), 1.2*np.max(x1)])\n",
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"ax11.set_ylim([1.2*np.min(x2), 1.2*np.max(x2)])\n",
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"\n",
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"# Velocities\n",
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"ax21.set_xlabel(r'$v_1$')\n",
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"ax21.set_ylabel(r'$v_2$')\n",
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"ln21, = ax21.plot([], [])\n",
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"ax21.set_xlim([1.2*np.min(xd1), 1.2*np.max(xd1)])\n",
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"ax21.set_ylim([1.2*np.min(xd2), 1.2*np.max(xd2)])\n",
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"\n",
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"# x1(t)\n",
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"ax12.set_xlabel(r'$t$')\n",
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"ax12.set_ylabel(r'$x_1$')\n",
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"ln12, = ax12.plot([], [])\n",
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"ax12.set_xlim([t[0], t[-1]])\n",
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"ax12.set_ylim([1.2*np.min(x1), 1.2*np.max(x1)])\n",
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"\n",
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"# x2(t)\n",
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"ax22.set_xlabel(r'$t$')\n",
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"ax22.set_ylabel(r'$x_2$')\n",
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"ln22, = ax22.plot([], [])\n",
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"ax22.set_xlim([t[0], t[-1]])\n",
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"ax22.set_ylim([1.2*np.min(x2), 1.2*np.max(x2)])\n",
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"\n",
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"# Phase spaces\n",
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"ax13.set_xlabel(r'$\\mathbf{x}$')\n",
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"ax13.set_ylabel(r'$\\mathbf{v}$')\n",
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"ln13a, = ax13.plot([], [])\n",
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"ln13b, = ax13.plot([], [])\n",
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"ax13.set_xlim([1.2*min(np.min(x1), np.min(x2)), 1.2*max(np.max(x1), np.max(x2))])\n",
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"ax13.set_ylim([1.2*min(np.min(xd1), np.min(xd2)), 1.2*max(np.max(xd1), np.max(xd2))])\n",
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"\n",
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"# F(t)\n",
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"F1 = f1(t)\n",
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"F2 = f2(t)\n",
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"ax23.set_xlabel(r'$t$')\n",
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"ax23.set_ylabel(r'$\\mathbf{F}(t)$')\n",
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"ln23a, = ax23.plot([], [])\n",
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"ln23b, = ax23.plot([], [])\n",
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"ax23.set_xlim([t[0], t[-1]])\n",
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"ax23.set_ylim([1.2*min(np.min(F1), np.min(F2)), 1.2*max(np.max(F1), np.max(F2))])\n",
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"\n",
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"plt.tight_layout()\n",
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"\n",
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"def animate(i):\n",
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" ln11.set_data(x1[:i], x2[:i])\n",
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" ln21.set_data(xd1[:i], xd2[:i])\n",
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" ln12.set_data(t[:i], x1[:i])\n",
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" ln22.set_data(t[:i], x2[:i])\n",
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" ln13a.set_data(x1[:i], xd1[:i])\n",
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" ln13b.set_data(x2[:i], xd2[:i])\n",
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" ln23a.set_data(t[:i], F1[:i])\n",
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" ln23b.set_data(t[:i], F2[:i])\n",
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" \n",
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"\n",
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"anim = animation.FuncAnimation(fig2, animate, frames=len(t), interval=100, repeat_delay=1000)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"anim"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"anim.save('animations/coupled-duffing.mp4', writer='ffmpeg')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"anaconda-cloud": {},
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.5.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 1
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}
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