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




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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














up vote
2
down vote

favorite
1












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




share|cite|improve this question















  • 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












up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





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




share|cite|improve this question











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





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 18 at 10:58









LMZ

310110




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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












  • 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















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