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so while studying non linear dynamics I came across a simple question that I cannot find the solution to anywhere.

Consider the following system: $$\begin{align} \frac{dx}{dt} &= y + y^{2} \\ \dfrac{dy}{dt}&= \dfrac{-1}{2}x +\dfrac{1}{5}y -xy + \dfrac{6}{5}y^2 \end{align} $$

a) What are the indices of the fixed points and why?

Trying to solve this I computed the fixed points to be $(0,0)$ and $\left(\dfrac{14}{15},1\right)$

using the Jacobian I figured they were both unstable spirals (I think, even though I have difficulty understanding how that is). so how do I compute the indices, I,m not sure what they are and cant find it on the internet. Do they have a role in determining in the fixed points are enclosed by a limit cycle?

Moo
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Robbie Meaney
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  • Sorry you're correct I don't know how I obtained $\frac{14}{15}$. I don't know what the indices of a fixed point are and can't find it online so I was hoping it was common terminology that someone could explain. all I know is that I need to know them to determine which of the fixed points is enclosed by a limit cycle, given that the system is known to have at least one limit cycle. – Robbie Meaney Aug 08 '18 at 14:42
  • Would it make sense to apply index theory here in the context of this question? – Robbie Meaney Aug 08 '18 at 14:45
  • That seems to have put me on the right track, thank you very much! – Robbie Meaney Aug 08 '18 at 14:48

1 Answers1

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The equilibrium sets are given by the solutions to

$$ \dot x = y + y^2 = 0\\ \dot y = -\frac x2+\frac y5-x y +\frac 65 y^2 = 0 $$

giving two pints

$$ p_1 = (-2,-1)\\ p_2 = (0,0) $$

now calculating the jacobians at each poin

$$ J_{p_1} = \left( \begin{array}{cc} 0 & -1 \\ \frac{1}{2} & -\frac{1}{5} \\ \end{array} \right)\\ J_{p_2} = \left( \begin{array}{cc} 0 & 1 \\ -\frac{1}{2} & \frac{1}{5} \\ \end{array} \right) $$

with eigenvalues

$$ J_{p_1} \Rightarrow \left\{-\frac{1}{10}+\frac{7 i}{10},-\frac{1}{10}-\frac{7 i}{10}\right\}\;\;\mbox{sink}\\ J_{p_2} \Rightarrow \left\{\frac{1}{10}+\frac{7 i}{10},\frac{1}{10}-\frac{7 i}{10}\right\}\;\;\mbox{source} $$

Attached a stream plot showing the phase-plane

enter image description here

Cesareo
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