Mixing
Codimension of bifurcation
$begingroup$
I understood that in bifurcation theory we call the number of parameters as codimension of a bifurcation. For example saddle node, transcritical and pitchfork all are codimension one since bifurcation happens when one parameter varies.
In Perko there is a discussion on unfolding. If we have
$dot{x}= x^{k}$
as system, then an unfolding for that is
$dot{x}= mu_{1}+ mu_{2} x +...+ mu_{k-1} x^{k-2}+x^{k}$
so the bifurcation would be codimension k,
Could you please explain Geometricly why we use word codimension?
If we consider trajectories of the original system, what are we adding geometricly to those solution curves?
Could you please give an example in 3dimensional system?
I am sorry if questions sounds a bit unclear. I do not know how to ask it in a better way.
dynamical-systems
asked Jan 29 at 15:17
Sarah19Sarah19
264
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add a comment |
$begingroup$
I understood that in bifurcation theory we call the number of parameters as codimension of a bifurcation. For example saddle node, transcritical and pitchfork all are codimension one since bifurcation happens when one parameter varies.
In Perko there is a discussion on unfolding. If we have
$dot{x}= x^{k}$
as system, then an unfolding for that is
$dot{x}= mu_{1}+ mu_{2} x +...+ mu_{k-1} x^{k-2}+x^{k}$
so the bifurcation would be codimension k,
Could you please explain Geometricly why we use word codimension?
If we consider trajectories of the original system, what are we adding geometricly to those solution curves?
Could you please give an example in 3dimensional system?
I am sorry if questions sounds a bit unclear. I do not know how to ask it in a better way.
dynamical-systems
asked Jan 29 at 15:17
Sarah19Sarah19
264
$endgroup$
add a comment |
$begingroup$
I understood that in bifurcation theory we call the number of parameters as codimension of a bifurcation. For example saddle node, transcritical and pitchfork all are codimension one since bifurcation happens when one parameter varies.
In Perko there is a discussion on unfolding. If we have
$dot{x}= x^{k}$
as system, then an unfolding for that is
$dot{x}= mu_{1}+ mu_{2} x +...+ mu_{k-1} x^{k-2}+x^{k}$
so the bifurcation would be codimension k,
Could you please explain Geometricly why we use word codimension?
If we consider trajectories of the original system, what are we adding geometricly to those solution curves?
Could you please give an example in 3dimensional system?
I am sorry if questions sounds a bit unclear. I do not know how to ask it in a better way.
dynamical-systems
asked Jan 29 at 15:17
Sarah19Sarah19
264
$endgroup$
I understood that in bifurcation theory we call the number of parameters as codimension of a bifurcation. For example saddle node, transcritical and pitchfork all are codimension one since bifurcation happens when one parameter varies.
In Perko there is a discussion on unfolding. If we have
$dot{x}= x^{k}$
as system, then an unfolding for that is
$dot{x}= mu_{1}+ mu_{2} x +...+ mu_{k-1} x^{k-2}+x^{k}$
so the bifurcation would be codimension k,
Could you please explain Geometricly why we use word codimension?
If we consider trajectories of the original system, what are we adding geometricly to those solution curves?
Could you please give an example in 3dimensional system?
I am sorry if questions sounds a bit unclear. I do not know how to ask it in a better way.
dynamical-systems
dynamical-systems
asked Jan 29 at 15:17
Sarah19Sarah19
264
asked Jan 29 at 15:17
Sarah19Sarah19
264
asked Jan 29 at 15:17
Sarah19Sarah19
264
asked Jan 29 at 15:17
Sarah19Sarah19
264
asked Jan 29 at 15:17
Sarah19Sarah19
264
264
add a comment |
add a comment |
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