Bifurcation Example Using Newton's Method
$begingroup$
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?
reference-request numerical-methods dynamical-systems newton-raphson
edited Dec 14 '18 at 21:03
Adam
1,1951919
asked Nov 27 '14 at 14:13
NeilNeil
112
$endgroup$
add a comment |
$begingroup$
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?
reference-request numerical-methods dynamical-systems newton-raphson
edited Dec 14 '18 at 21:03
Adam
1,1951919
asked Nov 27 '14 at 14:13
NeilNeil
112
$endgroup$
add a comment |
$begingroup$
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?
reference-request numerical-methods dynamical-systems newton-raphson
edited Dec 14 '18 at 21:03
Adam
1,1951919
asked Nov 27 '14 at 14:13
NeilNeil
112
$endgroup$
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
reference-request numerical-methods dynamical-systems newton-raphson
edited Dec 14 '18 at 21:03
Adam
1,1951919
asked Nov 27 '14 at 14:13
NeilNeil
112
edited Dec 14 '18 at 21:03
Adam
1,1951919
asked Nov 27 '14 at 14:13
NeilNeil
112
edited Dec 14 '18 at 21:03
Adam
1,1951919
edited Dec 14 '18 at 21:03
Adam
1,1951919
edited Dec 14 '18 at 21:03
Adam
1,1951919
1,1951919
asked Nov 27 '14 at 14:13
NeilNeil
112
asked Nov 27 '14 at 14:13
NeilNeil
112
asked Nov 27 '14 at 14:13
NeilNeil
112
112
add a comment |
add a comment |
1 Answer
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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.
answered Nov 28 '14 at 0:28
Mark McClureMark McClure
23.7k34472
$endgroup$
$begingroup$
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
add a comment |
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$begingroup$
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.
answered Nov 28 '14 at 0:28
Mark McClureMark McClure
23.7k34472
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
answered Nov 28 '14 at 0:28
Mark McClureMark McClure
23.7k34472
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
answered Nov 28 '14 at 0:28
Mark McClureMark McClure
23.7k34472
$endgroup$
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.
answered Nov 28 '14 at 0:28
Mark McClureMark McClure
23.7k34472
answered Nov 28 '14 at 0:28
Mark McClureMark McClure
23.7k34472
answered Nov 28 '14 at 0:28
Mark McClureMark McClure
23.7k34472
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
add a comment |
$begingroup$
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
add a comment |
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