Determining Bifurcation of a Function

Question

Hello I am trying to find analyze the bifurcation behavior of

$\dot{N} = N(N - e^{\alpha N}) , N \geq 0, \alpha > 0$

as $\alpha$ is varied and find their stability. Playing around with different values of $\alpha$, I see that when $\alpha$ is not small(e.g. $\alpha \geq 1$ would be considered "not small" in regards to this particular system) I find that for positive values of $N$ the only fixed point will be at $N = 0$ when considering only the non negative values of $N$. This fixed point is stable because all positive values of $N$ are decreasing towards it and more rapidly so the larger $N$ is relative to $\alpha$.

On the other hand an interesting thing seems to occur for a small enough value, say suppose $\alpha = .1$, assigning random nonnegative but small such as $N = 1$ $\dot{N}$ will be negative since $\dot{N}(1) = 1 - e^.1 < 0$, positive when $N$ is "just right in terms of being small or large" e.g. $N = 10$ causes $\dot{N}$ to be positive since $\dot{N}(10) = 10(10 - e^1) > 0$ but then negative again after $N$ is large enough relative to $\alpha = .1$ since $\dot{N}(100) = 100(100 - e^{10}) < 0$. Based on this quick qualitative analysis of the simple example of the value I assigned of $\alpha = .1$, it seems there will now be 3 fixed points, 0 will still be stable but the second fixed point occurring immediately after $N$ starts to be positive will be unstable since values greater than this fixed point will increase away from it. The third fixed point occurring when $\dot{N} = 0$for when the function then starts to decrease again will also be stable since values before it are increasing but decreasing afterwards.

The main difficulty I am having in further studying the bifurcation behavior of this system is determining what is the exact bifurcation point and what type of bifurcation it would be classified as. Currently I hypothesize that the particular bifurcation point would be considered a saddle node bifurcation since the two fixed points appear and vanish before and after alpha is small enough relative to the system and that a logarithmic scale analysis would be helpful when trying to represent the bifurcation. I want to know how to solve for the exact bifurcation value in a quantitative manner, any advice would be much appreciated.

Answer 1

First, there is a constant fixed point of $N=0$ for all $\alpha$. The other fixed points are the real roots of $N - \exp(\alpha N)$, which are potentially $$r_0(\alpha) = \frac{\text W_0(-\alpha)}{-\alpha}\text{ and }r_{-1} (\alpha) = \frac{\text W_{-1}(-\alpha)}{-\alpha}$$ where $\text W(x)\exp(\text W(x)) = x$ for all $x\in\mathbb{C}$, and $\text W_n(x)$ is the $n$-th branch of $\text W(x)$.

I say potentially because, as you know, these roots are only real for a subset of possible $\alpha$. In particular, we can see that for $\alpha > 0$ close to $0$, both roots are real, and we have $$\lim\limits_{\alpha\to 0} r_0 = 1\text{ and }\lim\limits_{\alpha\to 0} r_{-1} = \infty$$ The $\alpha$ for which $\max\limits_N [N - \exp(\alpha N)] = 0$ is $\exp(-1) = \frac{1}{e}$, and $r_0(\frac{1}{e}) = r_{-1}(\frac{1}{e}) = e$.

Using this information, you can see there is a saddle-node bifurcation at $\alpha = \frac{1}{e}$. The $N=0$ branch and the upper $N = \exp(\alpha N)$ branch, $r_{-1}(\alpha)$, are both stable; the lower $N = \exp(\alpha N)$ branch, $r_0(\alpha)$, is unstable. Below is a graph of the bifurcation diagram on the $(\alpha, N)$-plane (the vertical green line is $\alpha = \frac{1}{e}$):

Here is a link to the graphing utility

Answer 2

For $\alpha < 0$ there are indeed two equilibria, one at $x_1 = 0$ (stable) and one at $x_2 = - \frac{W(-\alpha)}{\alpha}$ (unstable), where $W$ is the Lambert W function. For $\alpha = 0$ the unstable solution is at $x_2 = 1$. For $\alpha > 0$ a third equilibrium $x_3= - \frac{W_{-1}(-\alpha)}{\alpha}$ appears, which is stable. Here $W_{-1}$ is the lower branch of the Lambert W function.

Therefore, as $\alpha \to 0$, $x_3 \to \infty$. As $\alpha$ increases past $e^{-1}$, $x_2$ and $x_3$ disappear in a fold bifurcation and only the stable solution $x_1 = 0$ remains.

So there is a stable solution branch which appears from $\infty$ as $\alpha$ becomes positive, a phenomenon called "bifurcation from infinity".