Bifurcation in a linear system with 2 equations and 1parameter
I have the following system $$frac{dx}{dt} = ax+y$$ and $$frac{dy}{dt} = -x+ay$$ By setting the derivative equals to $0$, we get the equilibrium point $(0,0)$. After drawing the phase portrait for various parameter $a$, we conclude that $(0,0)$ is stable when $a<0$ (see 1), a center when $a=0$ (see 2), and unstable when $a>0$ (see 3). Also, the eigenvalues are $lambda = a pm i$.
Do we have a bifurcation here? I was guessing that maybe we have Hopf bifurcation when $a=0$ because $(0,0)$ lose its stability when $a=0$. If it's true, I was also asked to draw the bifurcation diagram. I am not sure what it means or how to draw it.
differential-equations dynamical-systems mathematical-modeling stability-in-odes bifurcation
asked 2 days ago
aaaaaa
407
add a comment |
I have the following system $$frac{dx}{dt} = ax+y$$ and $$frac{dy}{dt} = -x+ay$$ By setting the derivative equals to $0$, we get the equilibrium point $(0,0)$. After drawing the phase portrait for various parameter $a$, we conclude that $(0,0)$ is stable when $a<0$ (see 1), a center when $a=0$ (see 2), and unstable when $a>0$ (see 3). Also, the eigenvalues are $lambda = a pm i$.
Do we have a bifurcation here? I was guessing that maybe we have Hopf bifurcation when $a=0$ because $(0,0)$ lose its stability when $a=0$. If it's true, I was also asked to draw the bifurcation diagram. I am not sure what it means or how to draw it.
differential-equations dynamical-systems mathematical-modeling stability-in-odes bifurcation
asked 2 days ago
aaaaaa
407
3
We do have a bifurcation here since equilibrium's stability has changed. However, it's not a Hopf bufurcation: since the system is linear for all values of $a$, there is no limit cycle (isolated closed trajectory). Also, typically at critical value for Hopf bifurcation we have a degenerate focus, not a center as here. I think you can take a look at simplest bifurcation diagrams in Strogatz's book.
– Evgeny
2 days ago
@Evgeny was the book you refer to is Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering by Steven H. Strogatz? and just to clarify, so Hopf bifurcation could only occur when the system is nonlinear?
– aaaaaa
yesterday
2
Yeah, that was exactly the book I was referring to. And yes, Hopf bifurcation (changing stability of equilibrium by emerging/colliding with limit cycle) can happen only when system is nonlinear: linear systems don't have limit cycles.
– Evgeny
yesterday
add a comment |
I have the following system $$frac{dx}{dt} = ax+y$$ and $$frac{dy}{dt} = -x+ay$$ By setting the derivative equals to $0$, we get the equilibrium point $(0,0)$. After drawing the phase portrait for various parameter $a$, we conclude that $(0,0)$ is stable when $a<0$ (see 1), a center when $a=0$ (see 2), and unstable when $a>0$ (see 3). Also, the eigenvalues are $lambda = a pm i$.
Do we have a bifurcation here? I was guessing that maybe we have Hopf bifurcation when $a=0$ because $(0,0)$ lose its stability when $a=0$. If it's true, I was also asked to draw the bifurcation diagram. I am not sure what it means or how to draw it.
differential-equations dynamical-systems mathematical-modeling stability-in-odes bifurcation
asked 2 days ago
aaaaaa
407
I have the following system $$frac{dx}{dt} = ax+y$$ and $$frac{dy}{dt} = -x+ay$$ By setting the derivative equals to $0$, we get the equilibrium point $(0,0)$. After drawing the phase portrait for various parameter $a$, we conclude that $(0,0)$ is stable when $a<0$ (see 1), a center when $a=0$ (see 2), and unstable when $a>0$ (see 3). Also, the eigenvalues are $lambda = a pm i$.
Do we have a bifurcation here? I was guessing that maybe we have Hopf bifurcation when $a=0$ because $(0,0)$ lose its stability when $a=0$. If it's true, I was also asked to draw the bifurcation diagram. I am not sure what it means or how to draw it.
differential-equations dynamical-systems mathematical-modeling stability-in-odes bifurcation
differential-equations dynamical-systems mathematical-modeling stability-in-odes bifurcation
asked 2 days ago
aaaaaa
407
asked 2 days ago
aaaaaa
407
edited 16 hours ago
asked 2 days ago
aaaaaa
407
asked 2 days ago
aaaaaa
407
asked 2 days ago
aaaaaa
407
407
3
We do have a bifurcation here since equilibrium's stability has changed. However, it's not a Hopf bufurcation: since the system is linear for all values of $a$, there is no limit cycle (isolated closed trajectory). Also, typically at critical value for Hopf bifurcation we have a degenerate focus, not a center as here. I think you can take a look at simplest bifurcation diagrams in Strogatz's book.
– Evgeny
2 days ago
@Evgeny was the book you refer to is Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering by Steven H. Strogatz? and just to clarify, so Hopf bifurcation could only occur when the system is nonlinear?
– aaaaaa
yesterday
2
Yeah, that was exactly the book I was referring to. And yes, Hopf bifurcation (changing stability of equilibrium by emerging/colliding with limit cycle) can happen only when system is nonlinear: linear systems don't have limit cycles.
– Evgeny
yesterday
add a comment |
3
We do have a bifurcation here since equilibrium's stability has changed. However, it's not a Hopf bufurcation: since the system is linear for all values of $a$, there is no limit cycle (isolated closed trajectory). Also, typically at critical value for Hopf bifurcation we have a degenerate focus, not a center as here. I think you can take a look at simplest bifurcation diagrams in Strogatz's book.
– Evgeny
2 days ago
@Evgeny was the book you refer to is Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering by Steven H. Strogatz? and just to clarify, so Hopf bifurcation could only occur when the system is nonlinear?
– aaaaaa
yesterday
2
Yeah, that was exactly the book I was referring to. And yes, Hopf bifurcation (changing stability of equilibrium by emerging/colliding with limit cycle) can happen only when system is nonlinear: linear systems don't have limit cycles.
– Evgeny
yesterday
3
3
We do have a bifurcation here since equilibrium's stability has changed. However, it's not a Hopf bufurcation: since the system is linear for all values of $a$, there is no limit cycle (isolated closed trajectory). Also, typically at critical value for Hopf bifurcation we have a degenerate focus, not a center as here. I think you can take a look at simplest bifurcation diagrams in Strogatz's book.
– Evgeny
2 days ago
We do have a bifurcation here since equilibrium's stability has changed. However, it's not a Hopf bufurcation: since the system is linear for all values of $a$, there is no limit cycle (isolated closed trajectory). Also, typically at critical value for Hopf bifurcation we have a degenerate focus, not a center as here. I think you can take a look at simplest bifurcation diagrams in Strogatz's book.
– Evgeny
2 days ago
@Evgeny was the book you refer to is Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering by Steven H. Strogatz? and just to clarify, so Hopf bifurcation could only occur when the system is nonlinear?
– aaaaaa
yesterday
@Evgeny was the book you refer to is Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering by Steven H. Strogatz? and just to clarify, so Hopf bifurcation could only occur when the system is nonlinear?
– aaaaaa
yesterday
2
2
Yeah, that was exactly the book I was referring to. And yes, Hopf bifurcation (changing stability of equilibrium by emerging/colliding with limit cycle) can happen only when system is nonlinear: linear systems don't have limit cycles.
– Evgeny
yesterday
Yeah, that was exactly the book I was referring to. And yes, Hopf bifurcation (changing stability of equilibrium by emerging/colliding with limit cycle) can happen only when system is nonlinear: linear systems don't have limit cycles.
– Evgeny
yesterday
add a comment |
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3
We do have a bifurcation here since equilibrium's stability has changed. However, it's not a Hopf bufurcation: since the system is linear for all values of $a$, there is no limit cycle (isolated closed trajectory). Also, typically at critical value for Hopf bifurcation we have a degenerate focus, not a center as here. I think you can take a look at simplest bifurcation diagrams in Strogatz's book.
– Evgeny
2 days ago
@Evgeny was the book you refer to is Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering by Steven H. Strogatz? and just to clarify, so Hopf bifurcation could only occur when the system is nonlinear?
– aaaaaa
yesterday
2
Yeah, that was exactly the book I was referring to. And yes, Hopf bifurcation (changing stability of equilibrium by emerging/colliding with limit cycle) can happen only when system is nonlinear: linear systems don't have limit cycles.
– Evgeny
yesterday