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To show that a 2D nonlinear ODE undergoes a pitchfork bifurcation
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Show that a pitchfork bifurcation occurs when $mu = 0$ for the system
$$dotx = mu x + xy + 3y^2$$
$$doty = -2y + x^2 + 2xy^2$$.
Attempt at a solution
If $dotx = 0$ then
$$x = frac-3ymu + y.$$
The Jacobian is
$$J(x,y) = beginpmatrixmu + y & 6y + x\ 2x + 2y^2 & -2 + 4xyendpmatrix$$
I know that $det J(0, 0) = -2mu$ so the $(0, 0)$ fixed point passed from unstable to stable as we pass $mu = 0$, so we probably have a subcritical pitchfork.
Related question: What would you look for in a transcritical or saddle node bifurcation? What's the general procedure for determining the bifurcation type in a 2D system?
nonlinear-system nonlinear-analysis non-linear-dynamics
asked Jul 18 at 10:58
LMZ
310110
add a comment |Â
up vote
2
down vote
favorite
Show that a pitchfork bifurcation occurs when $mu = 0$ for the system
$$dotx = mu x + xy + 3y^2$$
$$doty = -2y + x^2 + 2xy^2$$.
Attempt at a solution
If $dotx = 0$ then
$$x = frac-3ymu + y.$$
The Jacobian is
$$J(x,y) = beginpmatrixmu + y & 6y + x\ 2x + 2y^2 & -2 + 4xyendpmatrix$$
I know that $det J(0, 0) = -2mu$ so the $(0, 0)$ fixed point passed from unstable to stable as we pass $mu = 0$, so we probably have a subcritical pitchfork.
Related question: What would you look for in a transcritical or saddle node bifurcation? What's the general procedure for determining the bifurcation type in a 2D system?
nonlinear-system nonlinear-analysis non-linear-dynamics
asked Jul 18 at 10:58
LMZ
310110
1
1. After the pitchfork bifurcation the number of equilibriums changes from 1 to 3. It looks like that you can analyze how many equilibria there are depending on $mu$. 2. The general procedure is center manifold reduction. If you don't want to reinvent the wheel, Y. Kuznetsov's book provides explicit formulas that allows to recognize codimensione-1 and -2 bifurcations.
– Evgeny
Jul 27 at 15:08
Thanks so much!
– LMZ
Jul 29 at 12:39
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Show that a pitchfork bifurcation occurs when $mu = 0$ for the system
$$dotx = mu x + xy + 3y^2$$
$$doty = -2y + x^2 + 2xy^2$$.
Attempt at a solution
If $dotx = 0$ then
$$x = frac-3ymu + y.$$
The Jacobian is
$$J(x,y) = beginpmatrixmu + y & 6y + x\ 2x + 2y^2 & -2 + 4xyendpmatrix$$
I know that $det J(0, 0) = -2mu$ so the $(0, 0)$ fixed point passed from unstable to stable as we pass $mu = 0$, so we probably have a subcritical pitchfork.
Related question: What would you look for in a transcritical or saddle node bifurcation? What's the general procedure for determining the bifurcation type in a 2D system?
nonlinear-system nonlinear-analysis non-linear-dynamics
asked Jul 18 at 10:58
LMZ
310110
Show that a pitchfork bifurcation occurs when $mu = 0$ for the system
$$dotx = mu x + xy + 3y^2$$
$$doty = -2y + x^2 + 2xy^2$$.
Attempt at a solution
If $dotx = 0$ then
$$x = frac-3ymu + y.$$
The Jacobian is
$$J(x,y) = beginpmatrixmu + y & 6y + x\ 2x + 2y^2 & -2 + 4xyendpmatrix$$
I know that $det J(0, 0) = -2mu$ so the $(0, 0)$ fixed point passed from unstable to stable as we pass $mu = 0$, so we probably have a subcritical pitchfork.
Related question: What would you look for in a transcritical or saddle node bifurcation? What's the general procedure for determining the bifurcation type in a 2D system?
nonlinear-system nonlinear-analysis non-linear-dynamics
asked Jul 18 at 10:58
LMZ
310110
asked Jul 18 at 10:58
LMZ
310110
asked Jul 18 at 10:58
LMZ
310110
asked Jul 18 at 10:58
LMZ
310110
310110
1
1. After the pitchfork bifurcation the number of equilibriums changes from 1 to 3. It looks like that you can analyze how many equilibria there are depending on $mu$. 2. The general procedure is center manifold reduction. If you don't want to reinvent the wheel, Y. Kuznetsov's book provides explicit formulas that allows to recognize codimensione-1 and -2 bifurcations.
– Evgeny
Jul 27 at 15:08
Thanks so much!
– LMZ
Jul 29 at 12:39
add a comment |Â
1
1. After the pitchfork bifurcation the number of equilibriums changes from 1 to 3. It looks like that you can analyze how many equilibria there are depending on $mu$. 2. The general procedure is center manifold reduction. If you don't want to reinvent the wheel, Y. Kuznetsov's book provides explicit formulas that allows to recognize codimensione-1 and -2 bifurcations.
– Evgeny
Jul 27 at 15:08
Thanks so much!
– LMZ
Jul 29 at 12:39
1
1
1. After the pitchfork bifurcation the number of equilibriums changes from 1 to 3. It looks like that you can analyze how many equilibria there are depending on $mu$. 2. The general procedure is center manifold reduction. If you don't want to reinvent the wheel, Y. Kuznetsov's book provides explicit formulas that allows to recognize codimensione-1 and -2 bifurcations.
– Evgeny
Jul 27 at 15:08
1. After the pitchfork bifurcation the number of equilibriums changes from 1 to 3. It looks like that you can analyze how many equilibria there are depending on $mu$. 2. The general procedure is center manifold reduction. If you don't want to reinvent the wheel, Y. Kuznetsov's book provides explicit formulas that allows to recognize codimensione-1 and -2 bifurcations.
– Evgeny
Jul 27 at 15:08
Thanks so much!
– LMZ
Jul 29 at 12:39
Thanks so much!
– LMZ
Jul 29 at 12:39
add a comment |Â
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1
1. After the pitchfork bifurcation the number of equilibriums changes from 1 to 3. It looks like that you can analyze how many equilibria there are depending on $mu$. 2. The general procedure is center manifold reduction. If you don't want to reinvent the wheel, Y. Kuznetsov's book provides explicit formulas that allows to recognize codimensione-1 and -2 bifurcations.
– Evgeny
Jul 27 at 15:08
Thanks so much!
– LMZ
Jul 29 at 12:39