LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKERFdG9vbHNGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRLkxpbmVhckFsZ2VicmFGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA= First case: 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 Find equilibrium point(s): 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 Plot phase plane portrait: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnREVwbG90RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNlMtRjY2Ji1GIzY7LUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW9HRiQ2LVEwJkRpZmZlcmVudGlhbEQ7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZLLyUpc3RyZXRjaHlHRksvJSpzeW1tZXRyaWNHRksvJShsYXJnZW9wR0ZLLyUubW92YWJsZWxpbWl0c0dGSy8lJ2FjY2VudEdGSy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlpGRy1GIzYmLUYsNiNRIUYnLUYjNiVGQy1GLDYlUSJ0RidGL0YyRkdGaW5GRy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZm8vJSliZXZlbGxlZEdGSy1GRDYtUTEmSW52aXNpYmxlVGltZXM7RidGR0ZJRkxGTkZQRlJGVEZWRlhGZW4tRiw2JVEieEYnRi9GMi1GNjYkLUYjNiRGXm9GR0ZHLUZENi1RIj1GJ0ZHRklGTEZORlBGUkZURlYvRllRLDAuMjc3Nzc3OGVtRicvRmZuRmlwLUY/NigtSSNtbkdGJDYkUSMyMEYnRkctRiM2Jy1GXnE2JEZjb0ZHLUZENi1RIitGJ0ZHRklGTEZORlBGUkZURlYvRllRLDAuMjIyMjIyMmVtRicvRmZuRmlxLUklbXN1cEdGJDYlLUY2NiQtRiM2Ji1GLDYlUSJ5RidGL0YyLUY2NiQtRiM2JUZeb0YvRjJGR0YvRjJGRy1GIzYlLUZecTYkUSI0RidGR0YvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRi9GMkZhb0Zkb0Znb0Zpby1GRDYtUSgmbWludXM7RidGR0ZJRkxGTkZQRlJGVEZWRmhxRmpxLUZecTYkUSI1RidGRy1GRDYtUScmc2RvdDtGJ0ZHRklGTEZORlBGUkZURlZGWEZlbkZecC1GNjYkLUYjNiVGXm8vJStleGVjdXRhYmxlR0ZLRkdGRy1GRDYtUSIsRidGR0ZJL0ZNRjFGTkZQRlJGVEZWRlgvRmZuUSwwLjMzMzMzMzNlbUYnRj5GW3BGYnJGYXBGZXAtRj82KEZdcS1GIzYoRmNxRmVxRl5wRmVyRi9GMkZhb0Zkb0Znb0Zpb0Zhc0Zkc0Znc0ZickZqc0ZedEZHRkcvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGYHQtRjY2Ji1GIzYoRmluLUYjNiZGXnAtRkQ2LVEwJkFwcGx5RnVuY3Rpb247RidGR0ZJRkxGTkZQRlJGVEZWRlhGZW5GYXBGR0ZgdC1GIzYmRmJyRmZ1RmFwRkdGaW5GR0ZHRmp0Rl11RmB0LUYsNiVGYG8vRjBGS0ZHRmVwLUZecTYkRmBzRkctRkQ2LVEjLi5GJ0ZHRklGTEZORlBGUkZURlZGaHFGZW5GW3NGYHQtRkQ2LVEifkYnRkdGSUZMRk5GUEZSRlRGVkZYRmVuLUYsNiVGYHBGXXZGR0ZlcEZedkZgdi1GXnE2JFEkNC41RidGR0ZgdC1GLDYlRmRyRl12RkdGZXBGXnZGYHZGaHZGYHQtRjY2Ji1GIzY/LUY2NiYtRiM2LEZecC1GNjYkLUYjNiRGXnZGR0ZHRmVwRl52RmB0RmJyRmV3RmVwLUZecTYkUSQwLjVGJ0ZHRkdGR0ZqdEZddUZgdC1GNjYmLUYjNi1GXnBGZXdGZXBGXnZGYHRGY3ZGYnJGZXdGZXAtRl5xNiRRJDEuNUYnRkdGR0ZHRmp0Rl11RmB0LUY2NiYtRiM2LUZecEZld0ZlcEZpd0ZgdEZjdkZickZld0ZlcEZedkZHRkdGanRGXXVGYHQtRjY2Ji1GIzYsRl5wRmV3RmVwRmB4RmB0RmJyRmV3RmVwRl52RkdGR0ZqdEZddUZgdC1GNjYmLUYjNixGXnBGZXdGZXAtRl5xNiRRJDIuNUYnRkdGYHRGYnJGZXdGZXBGXnZGR0ZHRmp0Rl11RmB0LUY2NiYtRiM2LUZecEZld0ZlcC1GXnE2JFEkMy41RidGR0ZjdkZgdEZickZld0ZlcEZedkZHRkdGanRGXXVGYHQtRjY2Ji1GIzYtRl5wRmV3RmVwRmh2RmN2RmB0RmJyRmV3RmVwRml3RkdGR0ZqdEZddUZgdC1GNjYmLUYjNi1GXnBGZXdGZXBGaHZGY3ZGYHRGYnJGZXdGZXBGYHhGR0ZHRmp0Rl11RmB0LUY2NiYtRiM2LUZecEZld0ZlcEZodkZjdkZgdEZickZld0ZlcEZfeUZHRkdGanRGXXVGYHQtRjY2Ji1GIzYtRl5wRmV3RmVwRmh2RmN2RmB0RmJyRmV3RmVwRmZ5RkdGR0ZqdEZddUZgdC1GNjYmLUYjNi1GXnBGZXdGZXBGZnlGY3ZGYHRGYnJGZXdGZXBGaHZGR0ZHRmp0Rl11RmB0LUY2NiYtRiM2LUZecEZld0ZlcEZfeUZjdkZgdEZickZld0ZlcEZodkZHRkdGanRGXXVGYHQtRjY2Ji1GIzYtRl5wRmV3RmVwRmB4RmN2RmB0RmJyRmV3RmVwRmh2RkdGR0ZqdEZddUZgdC1GNjYmLUYjNi1GXnBGZXdGZXBGaXdGY3ZGYHRGYnJGZXdGZXBGaHZGR0ZHRmp0Rl11RmluRkdGR0ZqdEZddUZgdC1GLDYlUSlkaXJmaWVsZEYnRi9GMkZlcC1GNjYmLUYjNiYtRl5xNiRRIzI1RidGR0ZgdEZgXGxGR0ZHRmp0Rl11RmB0RmN2LUYsNiVRJ2Fycm93c0YnRi9GMkZlcC1GLDYlUSVTTElNRidGL0YyRmB0LUYsNiVRK2xpbmVjb2xvdXJGJ0YvRjJGZXAtRiw2JVElYmx1ZUYnRi9GMkZgdC1GLDYlUSpudW1wb2ludHNGJ0YvRjJGZXAtRl5xNiRRJTExMDBGJ0ZHRmB0RmN2LUYsNiVRJ2xhYmVsc0YnRi9GMkZlcC1GNjYmLUYjNihGaW4tSSNtc0dGJDYjUSN4fkYnRmB0LUZdXmw2I1EjfnlGJ0ZpbkZHRkdGanRGXXVGaW5GXnRGR0ZHLUZENi1RIjtGJ0ZHRklGY3RGTkZQRlJGVEZWRlhGanBGXnRGRw== Explore the linearization about the equilibrium: First, define the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGOQ== 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 Now find the Jacobian matrix: 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 Evaluate the Jacobian matrix at the equilibrium point: DON'T FORGET TO INSERT THE VALUES FOR LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUSJ+RidGLy9GNUY5RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUEY6Ri9GPkY3and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2J1EieUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUSJ+RidGLy9GNUY5RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUEY6RjpGL0Y+Rjc=in the substitution command !! 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 Find the eigenvalues of the Jacobian to determine the nature of the linearization: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEsRWlnZW52YWx1ZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSZEZmVxbUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y9RkA= Second case: 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 Find equilibrium point(s): 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 Now pick out the three equilibrium points from the list of solutions. Plot phase plane portrait: 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 Use the above commands (with G suitably redefined) to explore the linearizations about each of the three equilibrium points. JSFH