PROBLEM 1: Investigate the discrete time Logistic equation

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

(a) First try values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== such that 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Explain the results.

(b) Then try several values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RidGNQ==between 1 and 3. Explain the results.

(c) Try LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEkMy4zRidGOQ== and explain the result.

NiI=

(d) Try LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEkMy41RidGOQ== and explain what you see.

(e) if you try 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you will see a 4-cycle, a 8- cycle, a16-cycle and a 32-cycle respectively.

(f) For 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for example LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEkMy45RidGOQ== you get chaotic behavior.

(g) One important property of chaotic behavior is sensitive dependence on initial conditions. In other words if two seeds are very close

initially, their orbits will quickly separate and diverge from each other. Verify that for the logistics equation in the chaotic regime.

SOLUTION:

(a) First let us graph a few members of the family LyZJInhHNiI2IywmSSJrR0YlIiIiRilGKSomSSJyR0YlRiktJkYkNiNGKDYjLCZGKUYpRi0hIiJGKQ==when r goes from 0 to 1.

QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi

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Let us obtain some time series for the logistic when r takes the values 0.4 , 0.9.

PkkiZkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCooJCIiJSEiIiIiIkYnRi8sJkYvRi9GJ0YuRi9GJEYkRiQ=

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigkIiIlISIiIiIiRiRGLSwmRi1GLUYkRixGLUYlRiVGJQ==

print();

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigkIiIlISIiIiIiRiRGLSwmRi1GLUYkRixGLUYlRiVGJQ==

LUkqcG9pbnRwbG90RzYiNiQ3Iy1JJHNlcUclKnByb3RlY3RlZEc2JDckSSJrR0YkLS1JI0BARzYkRilJKF9zeXNsaWJHRiQ2JEkiZkdGJEYsNiMkIiImISIiL0YsOyIiISIjNS9JJnN0eWxlR0YkSSVsaW5lR0Yk

NiQtSSdDVVJWRVNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IzctNyQkIiIhIiIhJCIiJiEiIjckJCIiIiIiISQiJCsiISIkNyQkIiIjIiIhJCInKytPISIoNyQkIiIkIiIhJCIrKys7KVEiISM2NyQkIiIlIiIhJCIrdC9jdmEhIzc3JCQiIiYiIiEkIislW0ojeUAhIzc3JCQiIiciIiEkIittciVScCkhIzg3JCQiIigiIiEkIit4YWJ1TSEjODckJCIiKSIiISQiKyFIUiQqUSIhIzg3JCQiIioiIiEkIitiXWVjYiEjOTckJCIjNSIiISQiKz4wXkFBISM5LUkmU1RZTEVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0klTElORUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ig==

PkkiZ0c2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCooJCIiKiEiIiIiIkYnRi8sJkYvRi9GJ0YuRi9GJEYkRiQ=

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigkIiIqISIiIiIiRiRGLSwmRi1GLUYkRixGLUYlRiVGJQ==

print();

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigkIiIqISIiIiIiRiRGLSwmRi1GLUYkRixGLUYlRiVGJQ==

LUkqcG9pbnRwbG90RzYiNiQ3Iy1JJHNlcUclKnByb3RlY3RlZEc2JDckSSJrR0YkLS1JI0BARzYkRilJKF9zeXNsaWJHRiQ2JEkiZ0dGJEYsNiMkIiImISIiL0YsOyIiISIjNS9JJnN0eWxlR0YkSSVsaW5lR0Yk

NiQtSSdDVVJWRVNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IzctNyQkIiIhIiIhJCIiJiEiIjckJCIiIiIiISQiJEQjISIkNyQkIiIjIiIhJCIodiRwOiEiKDckJCIiJCIiISQiKyE0dDI+IiEjNTckJCIiJSIiISQiKzg4IjNXKiEjNjckJCIiJiIiISQiKz0qcFhwKCEjNjckJCIiJyIiISQiK3FfRCNSJyEjNjckJCIiKCIiISQiKydSIkcmUSYhIzY3JCQiIikiIiEkIithPnUmZSUhIzY3JCQiIioiIiEkIisnXDF6JFIhIzY3JCQiIzUiIiEkIit5PWIvTSEjNi1JJlNUWUxFRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJUxJTkVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI=

It seems that the orbits tend to zero. This is not surprising given that the fixed point x = 0 is attracting. Let us verify that.

PkkiRkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCooSSJyR0YkIiIiRidGLSwmRi1GLUYnISIiRi1GJEYkRiQ=

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKihJInJHRiUiIiJGJEYrLCZGK0YrRiQhIiJGK0YlRiVGJQ==

print();

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKihJInJHRiUiIiJGJEYrLCZGK0YrRiQhIiJGK0YlRiVGJQ==

PkkjRlBHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLy1JIkZHRiQ2I0kieEdGJEYvRi8=

NiQiIiEqJiwmSSJyRzYiIiIiRighIiJGKEYmRik=

print();

NiQiIiEqJiwmSSJyRzYiIiIiRighIiJGKEYmRik=

Since 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 there is only one nonegative fixed point, i.e, x =0. Let us evaluate the derivative at the fixed point.

LS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiRkdGKDYjJkkjRlBHRig2IyIiIg==

SSJyRzYi

print();

SSJyRzYi

Therefore the fixed point is attracting.

(b) Again we will draw a few members of the family 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for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIjxGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EvdGhpY2ttYXRoc3BhY2VGJy8lJ3JzcGFjZUdGSi8lKG1pbnNpemVHRi4vJShtYXhzaXplR1EpaW5maW5pdHlGJy1JI21pR0YkNiVRInJGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGMi1GLDYkUSMzLkYnRi8=

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