Bifurcation Example Using Newton's Method












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$begingroup$


I am studying dynamical systems as part of a research project.
I have been using Newton's Method and studying the dynamic properties.
Does anyone know where I could find a relatively simple example of bifurcation using Newton's Method?










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    2












    $begingroup$


    I am studying dynamical systems as part of a research project.
    I have been using Newton's Method and studying the dynamic properties.
    Does anyone know where I could find a relatively simple example of bifurcation using Newton's Method?










    share|cite|improve this question











    $endgroup$















      2












      2








      2


      2



      $begingroup$


      I am studying dynamical systems as part of a research project.
      I have been using Newton's Method and studying the dynamic properties.
      Does anyone know where I could find a relatively simple example of bifurcation using Newton's Method?










      share|cite|improve this question











      $endgroup$




      I am studying dynamical systems as part of a research project.
      I have been using Newton's Method and studying the dynamic properties.
      Does anyone know where I could find a relatively simple example of bifurcation using Newton's Method?







      reference-request numerical-methods dynamical-systems newton-raphson






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      share|cite|improve this question













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      edited Dec 14 '18 at 21:03









      Adam

      1,1951919




      1,1951919










      asked Nov 27 '14 at 14:13









      NeilNeil

      112




      112






















          1 Answer
          1






          active

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          4












          $begingroup$

          Here is an illustration of the comment that I made 5 hours ago. The idea is to examine the dynamics of $f_c(x)=x^2+c$ for various choices of $c$. When $c<0$, there are two real roots. When $c>0$, there are no real roots. Thus, we expect a distinct change in behavior as $c$ passes through zero.



          In the animation below, $f_c(x)=x^2+c$ is the light parabola and the corresponding Newton's method iteration function
          $$n_c(x)=frac{x^2-c}{2x}$$
          is shown in blue. The diagonal black line is the graph of $y=x$ and the intersection of this line with the graph of $n_c(x)$ are the fixed points of $n_c$. The extreme bifurcation at zero happens because the fixed points of $n_c$ correspond to the roots of $f_c$, which disappear.



          enter image description here






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            thanks for your help that makes sense - any ideas how I would go about programming a similar animation into maple?
            $endgroup$
            – Neil
            Nov 28 '14 at 17:02










          • $begingroup$
            @Neil Nope - I'm a Mathematica guy. :) You might also look at the family $f(z)=z^5-z-lambda$, which has a nice sequence of period doubling bifurcations around $lambda=1$.
            $endgroup$
            – Mark McClure
            Nov 28 '14 at 17:05











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          1 Answer
          1






          active

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          active

          oldest

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          4












          $begingroup$

          Here is an illustration of the comment that I made 5 hours ago. The idea is to examine the dynamics of $f_c(x)=x^2+c$ for various choices of $c$. When $c<0$, there are two real roots. When $c>0$, there are no real roots. Thus, we expect a distinct change in behavior as $c$ passes through zero.



          In the animation below, $f_c(x)=x^2+c$ is the light parabola and the corresponding Newton's method iteration function
          $$n_c(x)=frac{x^2-c}{2x}$$
          is shown in blue. The diagonal black line is the graph of $y=x$ and the intersection of this line with the graph of $n_c(x)$ are the fixed points of $n_c$. The extreme bifurcation at zero happens because the fixed points of $n_c$ correspond to the roots of $f_c$, which disappear.



          enter image description here






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            thanks for your help that makes sense - any ideas how I would go about programming a similar animation into maple?
            $endgroup$
            – Neil
            Nov 28 '14 at 17:02










          • $begingroup$
            @Neil Nope - I'm a Mathematica guy. :) You might also look at the family $f(z)=z^5-z-lambda$, which has a nice sequence of period doubling bifurcations around $lambda=1$.
            $endgroup$
            – Mark McClure
            Nov 28 '14 at 17:05
















          4












          $begingroup$

          Here is an illustration of the comment that I made 5 hours ago. The idea is to examine the dynamics of $f_c(x)=x^2+c$ for various choices of $c$. When $c<0$, there are two real roots. When $c>0$, there are no real roots. Thus, we expect a distinct change in behavior as $c$ passes through zero.



          In the animation below, $f_c(x)=x^2+c$ is the light parabola and the corresponding Newton's method iteration function
          $$n_c(x)=frac{x^2-c}{2x}$$
          is shown in blue. The diagonal black line is the graph of $y=x$ and the intersection of this line with the graph of $n_c(x)$ are the fixed points of $n_c$. The extreme bifurcation at zero happens because the fixed points of $n_c$ correspond to the roots of $f_c$, which disappear.



          enter image description here






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            thanks for your help that makes sense - any ideas how I would go about programming a similar animation into maple?
            $endgroup$
            – Neil
            Nov 28 '14 at 17:02










          • $begingroup$
            @Neil Nope - I'm a Mathematica guy. :) You might also look at the family $f(z)=z^5-z-lambda$, which has a nice sequence of period doubling bifurcations around $lambda=1$.
            $endgroup$
            – Mark McClure
            Nov 28 '14 at 17:05














          4












          4








          4





          $begingroup$

          Here is an illustration of the comment that I made 5 hours ago. The idea is to examine the dynamics of $f_c(x)=x^2+c$ for various choices of $c$. When $c<0$, there are two real roots. When $c>0$, there are no real roots. Thus, we expect a distinct change in behavior as $c$ passes through zero.



          In the animation below, $f_c(x)=x^2+c$ is the light parabola and the corresponding Newton's method iteration function
          $$n_c(x)=frac{x^2-c}{2x}$$
          is shown in blue. The diagonal black line is the graph of $y=x$ and the intersection of this line with the graph of $n_c(x)$ are the fixed points of $n_c$. The extreme bifurcation at zero happens because the fixed points of $n_c$ correspond to the roots of $f_c$, which disappear.



          enter image description here






          share|cite|improve this answer









          $endgroup$



          Here is an illustration of the comment that I made 5 hours ago. The idea is to examine the dynamics of $f_c(x)=x^2+c$ for various choices of $c$. When $c<0$, there are two real roots. When $c>0$, there are no real roots. Thus, we expect a distinct change in behavior as $c$ passes through zero.



          In the animation below, $f_c(x)=x^2+c$ is the light parabola and the corresponding Newton's method iteration function
          $$n_c(x)=frac{x^2-c}{2x}$$
          is shown in blue. The diagonal black line is the graph of $y=x$ and the intersection of this line with the graph of $n_c(x)$ are the fixed points of $n_c$. The extreme bifurcation at zero happens because the fixed points of $n_c$ correspond to the roots of $f_c$, which disappear.



          enter image description here







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 28 '14 at 0:28









          Mark McClureMark McClure

          23.7k34472




          23.7k34472












          • $begingroup$
            thanks for your help that makes sense - any ideas how I would go about programming a similar animation into maple?
            $endgroup$
            – Neil
            Nov 28 '14 at 17:02










          • $begingroup$
            @Neil Nope - I'm a Mathematica guy. :) You might also look at the family $f(z)=z^5-z-lambda$, which has a nice sequence of period doubling bifurcations around $lambda=1$.
            $endgroup$
            – Mark McClure
            Nov 28 '14 at 17:05


















          • $begingroup$
            thanks for your help that makes sense - any ideas how I would go about programming a similar animation into maple?
            $endgroup$
            – Neil
            Nov 28 '14 at 17:02










          • $begingroup$
            @Neil Nope - I'm a Mathematica guy. :) You might also look at the family $f(z)=z^5-z-lambda$, which has a nice sequence of period doubling bifurcations around $lambda=1$.
            $endgroup$
            – Mark McClure
            Nov 28 '14 at 17:05
















          $begingroup$
          thanks for your help that makes sense - any ideas how I would go about programming a similar animation into maple?
          $endgroup$
          – Neil
          Nov 28 '14 at 17:02




          $begingroup$
          thanks for your help that makes sense - any ideas how I would go about programming a similar animation into maple?
          $endgroup$
          – Neil
          Nov 28 '14 at 17:02












          $begingroup$
          @Neil Nope - I'm a Mathematica guy. :) You might also look at the family $f(z)=z^5-z-lambda$, which has a nice sequence of period doubling bifurcations around $lambda=1$.
          $endgroup$
          – Mark McClure
          Nov 28 '14 at 17:05




          $begingroup$
          @Neil Nope - I'm a Mathematica guy. :) You might also look at the family $f(z)=z^5-z-lambda$, which has a nice sequence of period doubling bifurcations around $lambda=1$.
          $endgroup$
          – Mark McClure
          Nov 28 '14 at 17:05


















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