Consider the family of ODE's with parameter r > 0: if(x,r)=rx sina, r>0 - (A) Show...

   

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Consider the family of ODE's with parameter r > 0: if(x,r)=rx sina, r>0 - (A) Show graphically that for r 1, there is exactly one fixed point. Further by drawing phase portraits (indicating as usual the stability of the fixed points), show graphically that for r = there are exactly three fixed points, and that for r = 3, there are seven fixed points. (B) Find a value of r for which there are (2k + 1) positive fixed points and (2k + 1) negative fixed points. Use this to show graphically that there must be infinitely many bifurcation values as r0. Let 0 < <3 <2 <1 1 be the values of r where bifurcation occurs. So r is the largest value at which bifurcation occurs, 12 the next largest value, etc... (C) What is the value of r? Find graphically the type of bifurcation that occurs at r and at 12. (You do not need to find the exact value of 12.) (D) Sketch the bifurcation diagram for r > 0 and x [-4, 4], indicating (as usual) the stability of the fixed points. Consider the family of ODE's with parameter r > 0: if(x,r)=rx sina, r>0 - (A) Show graphically that for r 1, there is exactly one fixed point. Further by drawing phase portraits (indicating as usual the stability of the fixed points), show graphically that for r = there are exactly three fixed points, and that for r = 3, there are seven fixed points. (B) Find a value of r for which there are (2k + 1) positive fixed points and (2k + 1) negative fixed points. Use this to show graphically that there must be infinitely many bifurcation values as r0. Let 0 < <3 <2 <1 1 be the values of r where bifurcation occurs. So r is the largest value at which bifurcation occurs, 12 the next largest value, etc... (C) What is the value of r? Find graphically the type of bifurcation that occurs at r and at 12. (You do not need to find the exact value of 12.) (D) Sketch the bifurcation diagram for r > 0 and x [-4, 4], indicating (as usual) the stability of the fixed points.