Classify the bifurcation that occurs at $mu$ =0
$begingroup$
$ dx/dt=mu x+y+x^2+x^3 , dy/dt=-x+mu y+x^2y$
What I have done so far is getting the matrix A with $A_{11}=mu,A_{12}=1,A_{21}=-1,A_{22}=mu$ at $(0,0)$.I can see the bifurcation is Hopf bifurcation.But I'm not sure whether it is subcritical or degenerate?At origin,a stable spiral becomes to an unstable spiral as $mu$ changes from negative to positive.(Thus it's not supercritical.)
Can someone help me to solve this problem?
I know by phase portrait I can see it's a subcritical hopf bifurcation.But is there any proof to show that?I thought about change coordinate to polar,but I'm stuck at the middle of the calculation.
dynamical-systems non-linear-dynamics
asked Dec 11 '18 at 7:02
XYCXYC
266
$endgroup$
add a comment |
$begingroup$
$ dx/dt=mu x+y+x^2+x^3 , dy/dt=-x+mu y+x^2y$
What I have done so far is getting the matrix A with $A_{11}=mu,A_{12}=1,A_{21}=-1,A_{22}=mu$ at $(0,0)$.I can see the bifurcation is Hopf bifurcation.But I'm not sure whether it is subcritical or degenerate?At origin,a stable spiral becomes to an unstable spiral as $mu$ changes from negative to positive.(Thus it's not supercritical.)
Can someone help me to solve this problem?
I know by phase portrait I can see it's a subcritical hopf bifurcation.But is there any proof to show that?I thought about change coordinate to polar,but I'm stuck at the middle of the calculation.
dynamical-systems non-linear-dynamics
asked Dec 11 '18 at 7:02
XYCXYC
266
$endgroup$
$begingroup$
It all depends on the sign of the first Lyapunov value (coefficient). In Scholarpedia you can find explicit formulas for its calculation.
$endgroup$
– Evgeny
Dec 11 '18 at 10:37
add a comment |
$begingroup$
$ dx/dt=mu x+y+x^2+x^3 , dy/dt=-x+mu y+x^2y$
What I have done so far is getting the matrix A with $A_{11}=mu,A_{12}=1,A_{21}=-1,A_{22}=mu$ at $(0,0)$.I can see the bifurcation is Hopf bifurcation.But I'm not sure whether it is subcritical or degenerate?At origin,a stable spiral becomes to an unstable spiral as $mu$ changes from negative to positive.(Thus it's not supercritical.)
Can someone help me to solve this problem?
I know by phase portrait I can see it's a subcritical hopf bifurcation.But is there any proof to show that?I thought about change coordinate to polar,but I'm stuck at the middle of the calculation.
dynamical-systems non-linear-dynamics
asked Dec 11 '18 at 7:02
XYCXYC
266
$endgroup$
$ dx/dt=mu x+y+x^2+x^3 , dy/dt=-x+mu y+x^2y$
What I have done so far is getting the matrix A with $A_{11}=mu,A_{12}=1,A_{21}=-1,A_{22}=mu$ at $(0,0)$.I can see the bifurcation is Hopf bifurcation.But I'm not sure whether it is subcritical or degenerate?At origin,a stable spiral becomes to an unstable spiral as $mu$ changes from negative to positive.(Thus it's not supercritical.)
Can someone help me to solve this problem?
I know by phase portrait I can see it's a subcritical hopf bifurcation.But is there any proof to show that?I thought about change coordinate to polar,but I'm stuck at the middle of the calculation.
dynamical-systems non-linear-dynamics
dynamical-systems non-linear-dynamics
asked Dec 11 '18 at 7:02
XYCXYC
266
asked Dec 11 '18 at 7:02
XYCXYC
266
edited Dec 11 '18 at 7:39
XYC
asked Dec 11 '18 at 7:02
XYCXYC
266
asked Dec 11 '18 at 7:02
XYCXYC
266
asked Dec 11 '18 at 7:02
XYCXYC
266
266
$begingroup$
It all depends on the sign of the first Lyapunov value (coefficient). In Scholarpedia you can find explicit formulas for its calculation.
$endgroup$
– Evgeny
Dec 11 '18 at 10:37
add a comment |
$begingroup$
It all depends on the sign of the first Lyapunov value (coefficient). In Scholarpedia you can find explicit formulas for its calculation.
$endgroup$
– Evgeny
Dec 11 '18 at 10:37
$begingroup$
It all depends on the sign of the first Lyapunov value (coefficient). In Scholarpedia you can find explicit formulas for its calculation.
$endgroup$
– Evgeny
Dec 11 '18 at 10:37
$begingroup$
It all depends on the sign of the first Lyapunov value (coefficient). In Scholarpedia you can find explicit formulas for its calculation.
$endgroup$
– Evgeny
Dec 11 '18 at 10:37
add a comment |
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$begingroup$
It all depends on the sign of the first Lyapunov value (coefficient). In Scholarpedia you can find explicit formulas for its calculation.
$endgroup$
– Evgeny
Dec 11 '18 at 10:37