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Bifurcation with two parameters

1

$begingroup$

Question:

Consider the system

beginalign
fracdxdt & = y \
fracdydt & = -(x^2+mu)y - (x^2+nu)x
endalign

Conduct Hopf analysis at the parameter values where Hopf bifurcations occur.

---

Attempt:

When we only have one parameter, I understand that a bifurcation occurs at the parameter value where the critical point(s) change stability. However, what is the definition of a bifurcation point when there are two or more parameter values?

In this particular example, I see that the critical points are at $(0,0)$ and $(pm sqrt-nu,0)$, where the latter only exists when $nu<0$.

Going from $nu>0$ to $nu<0$, the critical point $(0,0)$ goes from being stable to unstable, and we also gain two extra critical points - so is it right to say that we have a bifurcation at $nu=0$?

ordinary-differential-equations dynamical-systems bifurcation

share|cite|improve this question

asked Mar 25 at 21:06

glowstonetreesglowstonetrees

2,390418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Bifurcation points are like borderline between the regions of systems on parameter plane (or space) with qualitatively the same behaviour. If you are studying bifurcations of equilibria of planar systems, local bifurcations only happen when you pass through a non-hyperbolic equilibrium: i.e., its linearization has eigenvalues $lambda$ with $Relambda = 0$. So, yeah, $nu = 0$ belongs to a set of bifurcation points: it follows both from change in qualitative picture (number of equilibria is different depending on point in the neighbourhood) and from non-hyperbolicity of equilibrium.
  $endgroup$
  – Evgeny
  Mar 26 at 10:09
  

  
  
  
  

add a comment | 

1

$begingroup$

Question:

Consider the system

beginalign
fracdxdt & = y \
fracdydt & = -(x^2+mu)y - (x^2+nu)x
endalign

Conduct Hopf analysis at the parameter values where Hopf bifurcations occur.

---

Attempt:

When we only have one parameter, I understand that a bifurcation occurs at the parameter value where the critical point(s) change stability. However, what is the definition of a bifurcation point when there are two or more parameter values?

In this particular example, I see that the critical points are at $(0,0)$ and $(pm sqrt-nu,0)$, where the latter only exists when $nu<0$.

Going from $nu>0$ to $nu<0$, the critical point $(0,0)$ goes from being stable to unstable, and we also gain two extra critical points - so is it right to say that we have a bifurcation at $nu=0$?

ordinary-differential-equations dynamical-systems bifurcation

share|cite|improve this question

asked Mar 25 at 21:06

glowstonetreesglowstonetrees

2,390418

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Bifurcation points are like borderline between the regions of systems on parameter plane (or space) with qualitatively the same behaviour. If you are studying bifurcations of equilibria of planar systems, local bifurcations only happen when you pass through a non-hyperbolic equilibrium: i.e., its linearization has eigenvalues $lambda$ with $Relambda = 0$. So, yeah, $nu = 0$ belongs to a set of bifurcation points: it follows both from change in qualitative picture (number of equilibria is different depending on point in the neighbourhood) and from non-hyperbolicity of equilibrium.
  $endgroup$
  – Evgeny
  Mar 26 at 10:09
  

  
  
  
  

add a comment | 

$begingroup$

Question:

Consider the system

beginalign
fracdxdt & = y \
fracdydt & = -(x^2+mu)y - (x^2+nu)x
endalign

Conduct Hopf analysis at the parameter values where Hopf bifurcations occur.

---

Attempt:

When we only have one parameter, I understand that a bifurcation occurs at the parameter value where the critical point(s) change stability. However, what is the definition of a bifurcation point when there are two or more parameter values?

In this particular example, I see that the critical points are at $(0,0)$ and $(pm sqrt-nu,0)$, where the latter only exists when $nu<0$.

Going from $nu>0$ to $nu<0$, the critical point $(0,0)$ goes from being stable to unstable, and we also gain two extra critical points - so is it right to say that we have a bifurcation at $nu=0$?

ordinary-differential-equations dynamical-systems bifurcation

share|cite|improve this question

asked Mar 25 at 21:06

glowstonetreesglowstonetrees

2,390418

$endgroup$

share|cite|improve this question

asked Mar 25 at 21:06

glowstonetreesglowstonetrees

2,390418

share|cite|improve this question

asked Mar 25 at 21:06

glowstonetreesglowstonetrees

2,390418

asked Mar 25 at 21:06

glowstonetreesglowstonetrees

2,390418

asked Mar 25 at 21:06

glowstonetreesglowstonetrees

2,390418