ME2016 HW2
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ME_2016/.ipynb_checkpoints/HW2-checkpoint.ipynb
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342
ME_2016/.ipynb_checkpoints/HW2-checkpoint.ipynb
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "eaa65a42-a9a7-4420-acb7-4fee6610fa5b",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import sympy as sp \n",
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"import matplotlib.pyplot as plt"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a78f18ca-7b95-4f82-b501-ef9f402ecbc7",
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"metadata": {},
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"source": [
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"Dane Sabo\n",
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"October 2nd, 2024\n",
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"\n",
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"# Problem 3.7.2 (Written)"
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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": 14,
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"id": "0af0982d-25ae-4a1d-82d3-262935ab1d81",
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"metadata": {},
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"outputs": [],
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"source": [
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"def linear_phase_portrait_info(A):\n",
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" print('\\n Input Matrix:')\n",
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" print(A)\n",
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" \n",
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" print('\\n Eigenvalues and Eigenvectors:')\n",
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" print(np.linalg.eig(A).eigenvalues)\n",
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" print(np.linalg.eig(A).eigenvectors)\n",
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" \n",
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" print('\\n Trace:')\n",
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" print(np.linalg.trace(A))\n",
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"\n",
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" print('\\n Determinant:')\n",
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" print(np.linalg.det(A))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "bafdbc61-fba6-490b-8793-6b1820d6fbd4",
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"metadata": {},
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"source": [
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"Sketch phase portraits for the following linear systems:\n",
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"## $\\dot x = 0$, $\\dot y = x + 2y$"
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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": 15,
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"id": "37b37e65-7ade-473e-82ff-6391c6c6fe1d",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\n",
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" Input Matrix:\n",
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"[[0 0]\n",
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" [1 2]]\n",
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"\n",
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" Eigenvalues and Eigenvectors:\n",
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"[2. 0.]\n",
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"[[ 0. 0.89442719]\n",
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" [ 1. -0.4472136 ]]\n",
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"\n",
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" Trace:\n",
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"2\n",
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"\n",
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" Determinant:\n",
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"0.0\n"
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]
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}
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],
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"source": [
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"A = np.array([[0,0],[1,2]])\n",
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"linear_phase_portrait_info(A)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "966ca5a8-bcf4-4860-b923-f5cca9f6e1de",
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"metadata": {},
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"source": [
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"## $\\dot x = x+2y$, $\\dot y = 0$"
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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": 18,
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"id": "6e05a8d7-7bf6-4520-a3b6-0848d067b189",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\n",
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" Input Matrix:\n",
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"[[1 2]\n",
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" [0 0]]\n",
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"\n",
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" Eigenvalues and Eigenvectors:\n",
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"[1. 0.]\n",
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"[[ 1. -0.89442719]\n",
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" [ 0. 0.4472136 ]]\n",
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"\n",
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" Trace:\n",
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"1\n",
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"\n",
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" Determinant:\n",
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"0.0\n"
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]
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}
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],
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"source": [
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"A = np.array([[1,2],[0,0]])\n",
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"linear_phase_portrait_info(A)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0179406c-daa5-47bf-a10f-126f1ffe92c4",
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"metadata": {},
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"source": [
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"## $\\dot x = 3x+4y$, $\\dot y = 4x - 3y$"
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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": 17,
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"id": "0c198580-fd00-43d8-aeb7-947c88f449c0",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\n",
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" Input Matrix:\n",
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"[[ 3 4]\n",
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" [ 4 -3]]\n",
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"\n",
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" Eigenvalues and Eigenvectors:\n",
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"[ 5. -5.]\n",
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"[[ 0.89442719 -0.4472136 ]\n",
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" [ 0.4472136 0.89442719]]\n",
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"\n",
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" Trace:\n",
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"0\n",
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"\n",
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" Determinant:\n",
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"-25.000000000000007\n"
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]
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}
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],
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"source": [
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"A = np.array([[3,4],[4,-3]])\n",
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"linear_phase_portrait_info(A)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a41e8548-555b-4701-bad7-9049cb395dec",
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"metadata": {},
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"source": [
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"## $\\dot x = 3x+y$, $\\dot y = -x + 3y$"
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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": 16,
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"id": "40cd3d5a-a712-4fab-94bb-8148d2fd61ad",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\n",
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" Input Matrix:\n",
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"[[ 3 1]\n",
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" [-1 3]]\n",
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"\n",
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" Eigenvalues and Eigenvectors:\n",
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"[3.+1.j 3.-1.j]\n",
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"[[0.70710678+0.j 0.70710678-0.j ]\n",
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" [0. +0.70710678j 0. -0.70710678j]]\n",
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"\n",
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" Trace:\n",
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"6\n",
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"\n",
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" Determinant:\n",
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"10.000000000000002\n"
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]
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}
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],
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"source": [
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"A = np.array([[3,1],[-1,3]])\n",
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"linear_phase_portrait_info(A)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "6cdd8c34-e628-4bce-b37e-74f0da7a7bfd",
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"metadata": {},
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"source": [
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"# Problem 3.7.4 (Python)\n",
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"Plot the phase portraits for the following systems:\n",
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"## $\\dot x = y$, $\\dot y = x-y+x^3$"
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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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"id": "21643903-10a1-47ca-b449-4dcb6a4819e7",
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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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"cell_type": "markdown",
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"id": "c6c01322-669f-4bd4-8727-addd978c85e2",
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"metadata": {},
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"source": [
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"## $\\dot x = -2x-y+2$, $\\dot y = xy$"
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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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"id": "cd100989-95c4-4d6e-87a2-23468c3e7d30",
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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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"cell_type": "markdown",
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"id": "91f924f6-1567-47c9-893a-93cbf5753b98",
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"metadata": {},
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"source": [
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"## $\\dot x = x^2- y^2$, $\\dot y = xy-1$"
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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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"id": "d9c0e844-0458-4025-87a6-2b7f0332e8f5",
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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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"cell_type": "markdown",
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"id": "312d3ea7-a2c1-4db3-9aef-d4882adba1f3",
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"metadata": {},
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"source": [
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"## $\\dot x = 2-x-y^2$, $\\dot y = -y(x^2 + y^2 -3x +1)$"
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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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"id": "cfb3f6a8-fcf2-4669-a566-21ceea20b5ca",
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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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"cell_type": "markdown",
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"id": "12ad95a7-06dd-45b9-8d48-4a7e0451422e",
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"metadata": {},
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"source": [
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"# Problem 3.7.5 (Written)\n",
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"Construct a nonlinear system that has four critical points: two saddle points, one stable focus, and one unstable focus."
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]
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},
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{
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"cell_type": "markdown",
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"id": "cf8225d4-cedb-4daa-a7b0-ef9b032c4c92",
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"metadata": {},
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"source": [
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"# Problem 3.7.6 (Written)\n",
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"A nonlinear capacitor-resistor electrical circuit can be modeled using the differential equations\n",
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"$$ \\dot x = y $$\n",
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"$$\\dot y = -x + x^3 - (a_0 + x)y$$\n",
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"where $a_0$ is a nonzero constant and $x(t)$ represents the current in the circuit at time t. Sketch phase portraits when $a_0 > 0$ and $a_0<0$ and give a physical interpretation of the results."
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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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"id": "737e53cb-5f98-42b4-9ca5-696cc03cc724",
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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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"cell_type": "markdown",
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"id": "66d21987-bc58-4f31-af7b-94bcaf967921",
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"metadata": {},
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"source": [
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"# Problem 4.5.6 (Python)\n",
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"A predator-prey system may be modeled using the differential equations\n",
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"$$ \\dot x = x (1-y-\\epsilon x)$$\n",
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"$$ \\dot y = y (-1 + x - \\epsilon y)$$\n",
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"where $x(t)$ is the population of prey and $y(t)$ is the predator population size at time t, respectively. Classify the critical points for $\\epsilon>0$ and plot phase portraits for the different types of qualitative behavior. Interpret the results in physical terms"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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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": "3.12.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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342
ME_2016/HW2.ipynb
Normal file
342
ME_2016/HW2.ipynb
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@ -0,0 +1,342 @@
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{
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"cells": [
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{
|
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"cell_type": "code",
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"execution_count": 1,
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||||
"id": "eaa65a42-a9a7-4420-acb7-4fee6610fa5b",
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"metadata": {},
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"outputs": [],
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"source": [
|
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"import numpy as np\n",
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"import sympy as sp \n",
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"import matplotlib.pyplot as plt"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a78f18ca-7b95-4f82-b501-ef9f402ecbc7",
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"metadata": {},
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"source": [
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"Dane Sabo\n",
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"October 2nd, 2024\n",
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"\n",
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"# Problem 3.7.2 (Written)"
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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": 14,
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"id": "0af0982d-25ae-4a1d-82d3-262935ab1d81",
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"metadata": {},
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"outputs": [],
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"source": [
|
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"def linear_phase_portrait_info(A):\n",
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" print('\\n Input Matrix:')\n",
|
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" print(A)\n",
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" \n",
|
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" print('\\n Eigenvalues and Eigenvectors:')\n",
|
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" print(np.linalg.eig(A).eigenvalues)\n",
|
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" print(np.linalg.eig(A).eigenvectors)\n",
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" \n",
|
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" print('\\n Trace:')\n",
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" print(np.linalg.trace(A))\n",
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"\n",
|
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" print('\\n Determinant:')\n",
|
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" print(np.linalg.det(A))"
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]
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},
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{
|
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"cell_type": "markdown",
|
||||
"id": "bafdbc61-fba6-490b-8793-6b1820d6fbd4",
|
||||
"metadata": {},
|
||||
"source": [
|
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"Sketch phase portraits for the following linear systems:\n",
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"## $\\dot x = 0$, $\\dot y = x + 2y$"
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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": 15,
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"id": "37b37e65-7ade-473e-82ff-6391c6c6fe1d",
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"metadata": {},
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"outputs": [
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{
|
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\n",
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" Input Matrix:\n",
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"[[0 0]\n",
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" [1 2]]\n",
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"\n",
|
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" Eigenvalues and Eigenvectors:\n",
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"[2. 0.]\n",
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"[[ 0. 0.89442719]\n",
|
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" [ 1. -0.4472136 ]]\n",
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"\n",
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" Trace:\n",
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"2\n",
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"\n",
|
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" Determinant:\n",
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"0.0\n"
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]
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}
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],
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"source": [
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"A = np.array([[0,0],[1,2]])\n",
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"linear_phase_portrait_info(A)"
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]
|
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},
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{
|
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"cell_type": "markdown",
|
||||
"id": "966ca5a8-bcf4-4860-b923-f5cca9f6e1de",
|
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"metadata": {},
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"source": [
|
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"## $\\dot x = x+2y$, $\\dot y = 0$"
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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": 18,
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"id": "6e05a8d7-7bf6-4520-a3b6-0848d067b189",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\n",
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" Input Matrix:\n",
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"[[1 2]\n",
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" [0 0]]\n",
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"\n",
|
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" Eigenvalues and Eigenvectors:\n",
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"[1. 0.]\n",
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"[[ 1. -0.89442719]\n",
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" [ 0. 0.4472136 ]]\n",
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"\n",
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" Trace:\n",
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"1\n",
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"\n",
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" Determinant:\n",
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"0.0\n"
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]
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}
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],
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"source": [
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"A = np.array([[1,2],[0,0]])\n",
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"linear_phase_portrait_info(A)"
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]
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},
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{
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"cell_type": "markdown",
|
||||
"id": "0179406c-daa5-47bf-a10f-126f1ffe92c4",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## $\\dot x = 3x+4y$, $\\dot y = 4x - 3y$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"id": "0c198580-fd00-43d8-aeb7-947c88f449c0",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"\n",
|
||||
" Input Matrix:\n",
|
||||
"[[ 3 4]\n",
|
||||
" [ 4 -3]]\n",
|
||||
"\n",
|
||||
" Eigenvalues and Eigenvectors:\n",
|
||||
"[ 5. -5.]\n",
|
||||
"[[ 0.89442719 -0.4472136 ]\n",
|
||||
" [ 0.4472136 0.89442719]]\n",
|
||||
"\n",
|
||||
" Trace:\n",
|
||||
"0\n",
|
||||
"\n",
|
||||
" Determinant:\n",
|
||||
"-25.000000000000007\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"A = np.array([[3,4],[4,-3]])\n",
|
||||
"linear_phase_portrait_info(A)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a41e8548-555b-4701-bad7-9049cb395dec",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## $\\dot x = 3x+y$, $\\dot y = -x + 3y$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 16,
|
||||
"id": "40cd3d5a-a712-4fab-94bb-8148d2fd61ad",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"\n",
|
||||
" Input Matrix:\n",
|
||||
"[[ 3 1]\n",
|
||||
" [-1 3]]\n",
|
||||
"\n",
|
||||
" Eigenvalues and Eigenvectors:\n",
|
||||
"[3.+1.j 3.-1.j]\n",
|
||||
"[[0.70710678+0.j 0.70710678-0.j ]\n",
|
||||
" [0. +0.70710678j 0. -0.70710678j]]\n",
|
||||
"\n",
|
||||
" Trace:\n",
|
||||
"6\n",
|
||||
"\n",
|
||||
" Determinant:\n",
|
||||
"10.000000000000002\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"A = np.array([[3,1],[-1,3]])\n",
|
||||
"linear_phase_portrait_info(A)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "6cdd8c34-e628-4bce-b37e-74f0da7a7bfd",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Problem 3.7.4 (Python)\n",
|
||||
"Plot the phase portraits for the following systems:\n",
|
||||
"## $\\dot x = y$, $\\dot y = x-y+x^3$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "21643903-10a1-47ca-b449-4dcb6a4819e7",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c6c01322-669f-4bd4-8727-addd978c85e2",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## $\\dot x = -2x-y+2$, $\\dot y = xy$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "cd100989-95c4-4d6e-87a2-23468c3e7d30",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "91f924f6-1567-47c9-893a-93cbf5753b98",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## $\\dot x = x^2- y^2$, $\\dot y = xy-1$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "d9c0e844-0458-4025-87a6-2b7f0332e8f5",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "312d3ea7-a2c1-4db3-9aef-d4882adba1f3",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## $\\dot x = 2-x-y^2$, $\\dot y = -y(x^2 + y^2 -3x +1)$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "cfb3f6a8-fcf2-4669-a566-21ceea20b5ca",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "12ad95a7-06dd-45b9-8d48-4a7e0451422e",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Problem 3.7.5 (Written)\n",
|
||||
"Construct a nonlinear system that has four critical points: two saddle points, one stable focus, and one unstable focus."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "cf8225d4-cedb-4daa-a7b0-ef9b032c4c92",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Problem 3.7.6 (Written)\n",
|
||||
"A nonlinear capacitor-resistor electrical circuit can be modeled using the differential equations\n",
|
||||
"$$ \\dot x = y $$\n",
|
||||
"$$\\dot y = -x + x^3 - (a_0 + x)y$$\n",
|
||||
"where $a_0$ is a nonzero constant and $x(t)$ represents the current in the circuit at time t. Sketch phase portraits when $a_0 > 0$ and $a_0<0$ and give a physical interpretation of the results."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "737e53cb-5f98-42b4-9ca5-696cc03cc724",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "66d21987-bc58-4f31-af7b-94bcaf967921",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Problem 4.5.6 (Python)\n",
|
||||
"A predator-prey system may be modeled using the differential equations\n",
|
||||
"$$ \\dot x = x (1-y-\\epsilon x)$$\n",
|
||||
"$$ \\dot y = y (-1 + x - \\epsilon y)$$\n",
|
||||
"where $x(t)$ is the population of prey and $y(t)$ is the predator population size at time t, respectively. Classify the critical points for $\\epsilon>0$ and plot phase portraits for the different types of qualitative behavior. Interpret the results in physical terms"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.12.3"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
Loading…
x
Reference in New Issue
Block a user