Suppose F R R is continuous and suppose there are closed intervals Io, I such that...

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Suppose F R R is continuous and suppose there are closed intervals Io, I₁ such that F(Io) CI₁ F(1₁) C Io. a. Show that F has at least one periodic orbit {x0, x1} of period q = 2 such that xo E Io and x₁ € I₁. b. Show that if Io, I₁ are disjoint, then such an orbit has minimal period q = 2. c. Give an example to show that if Ion I₁0, then an orbit of minimal period 2 might not exist. Suppose F R R is continuous and suppose there are closed intervals Io, I₁ such that F(Io) CI₁ F(1₁) C Io. a. Show that F has at least one periodic orbit {x0, x1} of period q = 2 such that xo E Io and x₁ € I₁. b. Show that if Io, I₁ are disjoint, then such an orbit has minimal period q = 2. c. Give an example to show that if Ion I₁0, then an orbit of minimal period 2 might not exist.

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SOLUTION a To show that F has at least one periodic orbit xo x1 of period q 2 such that xo Io and x1 ... View the full answer