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Consider the equation $\dot{x} = r+x^2$. When $0 < r \ll 1$, this system experiences a bottleneck effect. Then the time $T$ spent in this bottleneck can be approximated by:

$$T_{bn} = \int_{-\infty}^{\infty} \frac{dx}{r+x^2}$$

Now consider a two-dimensional system by:

$$\dot{x} = x(3-x-2y)\qquad \dot{y} = y(2-x-y)$$

It's easy to show that $(1,1)$ is a saddle point and $(3,0)$ and $(0,2)$ are stable fixed points. Thus $(1,1)$ should experience a bottleneck effect as well. Is there a way to expand on the one-dimensional case to approximate the time spent in the bottleneck of the two-dimensional case? Then in theory, one could build this up to any higher dimension.

Did
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Brenton
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  • You might look into linearization about fixed points. In particular, near $(1,1)$ your system will essentially follow the unstable manifold, which locally looks like the eigenvector of the Jacobian with the positive eigenvalue, escaping with a rate that scales with the size of this positive eigenvalue. – Ian Jun 03 '15 at 21:40
  • @Ian If I'm visualizing this correctly, I would imagine that leaving near the fixed point would be with a rate about $\lambda_{unstable}$, and I'd imagine entering near the fixed point would be with rate $|\lambda_{stable}|$, but then slows down immensely and picks back up as it goes near the fixed point. I think linearizing is a good idea, but I'm not seeing how that helps to tell what goes on inbetween entering and escaping – Brenton Jun 03 '15 at 21:49
  • In the first case, what you call "the time spent in the bottleneck" is actually the length of the maximal time interval of definition. By contrast, the solutions of the second system are defined on the whole time line. It is true that the solutions starting from some initial conditions close to the stable manifold pass near the fixed point and spend a large amount of time there before escaping in the direction of the unstable manifold - but one is in need of a definition of "the time spent in the bottleneck" in this situation. – Did Jun 03 '15 at 21:50
  • To be specific: the same effect occurs for the system $\dot x=-x$, $\dot y=y$. What is "the time spent in the bottleneck" then? – Did Jun 03 '15 at 21:52
  • @Did: "spend a large amount of time there before escaping in the direction of the unstable manifold" -- that time spent in the bottle neck is the same as the "large amount of time there" (that's what I want to quantify). The first approximation just says that the time in the bottleneck is so much more that we'll say that it's close to the full time scale. I'm not sure how to calculate it for the second example. – Brenton Jun 03 '15 at 22:02
  • No formal definition then? – Did Jun 04 '15 at 15:00
  • @Did I suppose I could create one. Fix $\epsilon > 0$. Let $\tau_1= \inf\left\lbrace t \in [0, \infty) | dist(\vec{x}(t), (3,0)) < \epsilon \right\rbrace$ and $\tau_2= \inf\left\lbrace t \in [\tau_1, \infty) | dist(\vec{x}(t), (3,0)) \geq \epsilon \right\rbrace$. Then $T_{bn} = \tau_2 - \tau_1$. I think something along those lines would work – Brenton Jun 04 '15 at 15:09
  • For which starting point? – Did Jun 04 '15 at 15:49
  • @Did Well this part gets tougher since you can't solve for $\vec{x}(t)$ explicitly, but I would imagine it would be an initial condition $(x_0,y_0)$ so that $\vec{x}(t)$ comes within some ball of radius $\epsilon$ of $(3,0)$. I'm sure the bottleneck time will depend on exactly how close you are to $(3,0)$ so I'm not exactly sure how to pick an initial condition so that those times can be determined – Brenton Jun 04 '15 at 15:59
  • Then I am not sure we have a question here. – Did Jun 09 '15 at 19:35
  • @Did I don't think that's fair to say. It's like in the one-dimensional case, you can approximate the time spent in the bottleneck because the time spent there is so much longer than where the trajectory moves quickly. How do you define "in the bottleneck" there? You really don't define it explicitly, but you can still approximate it. The question is if one can extend this idea to multiple dimensions, and if so, how one does that – Brenton Jun 10 '15 at 14:17
  • To me, the main trouble is that in higher dimensions, there is more than one path "passing by" the slow point and that the notion of time you want to define will depend crucially on how close to this point the path passes. Unless you suggest a way to bypass this, it seems perfectly "fair" to say that there is no question here. – Did Jun 10 '15 at 18:39

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