I am reading through Strogatz's book on nonlinear odes and dynamical systems. One thing that is a little confusing is his description of stabilizing higher order terms to control the dynamics of a subcritical pitchfork.
So a subcritical pitchfork bifurcation has a normal form:
$$ \dot{x} = rx + x^3 $$
The danger of a subcritical pitchfork is that the stable fixed points can disappear above some level of the parameter $r$. Further, that once the system passes beyond the stable range, that decreasing the $r$ parameter may not stabilize the system.
In describing this issue, Strogatz (p.60) says:
In real physical systems, such an explosive instability is usually opposed by the stabilizing influence of higher-order terms. Assuming that the system is still symmetric under $x \rightarrow -x$, the first stabilizing term must be $x^5$. Thus the canonical example of a system with a subcritical pitchfork bifurcation is:
$$ \dot{x} = rx + x^3 - x^5 $$
Here is the picture from the book.
So I am a little confused on how to analyze this system. So in most of the analysis of nonlinear dynamical system, we are doing taylor expansions around the fixed points, and looking at the lowest order terms--like $x, x^2$, because these terms dominate in the convergence region of the taylor expansion--say $f(x-a), |a| \leq 1$. So how does the $x^5$ term suddenly control the dynamics? I am assuming that the subcritical pitchfork pushes the system beyond the convergence region, and so the system moves into the range where higher order terms actually dominate. But the explanation from the book above does not really provide any explanation of how to evaluate systems outside of the convergence region of the taylor expansion.
I think theoretically I could make a Continuation method argument here--in the 1d case only--that the new fixed point range has to be stable because the other side of the fixed point is unstable? But I am not sure what happens in the 2d or higher cases.
Can anyone explain a bit more about how these higher order terms actually stabilize a subcritical pitchfork. I mean does this issue only apply to subcritical pitchforks, or is it the case in all systems possibly? Are there situations where the higher order dynamics are unable to stabilize a subcritical pitchfork? Any help with these basic questions would help.
The second confusing thing is what is the physical interpretation of a 5th order term? Usually in ODEs we don't have more than a 2nd and maybe a 3rd order term? The beam equation is 4th order, but I am not sure of other good examples. Even in PDEs, I think the Korteweg-de Vries equation is 3rd order or the Kuramoto-Sivashinsky equation is 4th order. But these examples are exceptions rather than the rules. So what is a 5th order term mean?