Discussion between LPZ and Luthier415Hz

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11:09

Q: Let $T_{-1}(x)=x^3+x$. Prove that $T_{-1}$ is not structurally stable.

Let $T_{-1}(x)=x^3+x$. Prove that $T_{-1}$ is not structurally stable. Let $f:J\to J$. $f$ is said to be $C^r$-structurally stable on $J$ if there exists $\epsilon>0$ such that whenever $d_r(f,g)<\epsilon$ for $g:J\to J$, it follows that $f$ is topologically conjugate to $g$. Let us see if $T$ ...

dynamical-systems

LPZ

consider instead $x^3+(1+\epsilon)x$

Luthier415Hz

Thanks, how did you choose that?

LPZ

Intuitively, the structural instability is due to the behaviour at the fixed point $x=0$. The fixed point is critical, which is not stable by a $C^1$ perturbation by modifying the local slope. The simplest way to do that is to add a small linear term.

Luthier415Hz

Do you always add this linear term to the smallest polynomial?

LPZ

I don't get your question. I just perturbed $T_{-1}\to T_{-1}+\epsilon x$ which changes the slope by $\epsilon$ at the origin without changing the fixed point. The idea is actually pretty standard, it's a pitchfork bifurcation

Luthier415Hz

11:09

Thanks, this was not given in Devaney's respective chapter on this problem. So adding a small linear perturbation $\epsilon x$ is always the right method for any type of function.

LPZ

it won't always work, but it helps for simple bifurcations like this one

Luthier415Hz

Thanks for the tip. But if I get another type of function, for instance quartic, like x^4+x^2 , would it be right to add the same type of linear perturbation?

(I just try to see how to do in worse cases)

LPZ

actually your example $x^4+x^2$ is stable, perhaps you mean $x^4+x$? If so, then yes, a linear perturbation still does the trick

Luthier415Hz

That was fast. How could you see it was stable?

Another example, the quadratic map, if you want to add a perturbation, you add only \epsilon, right?

LPZ

I'm only looking at the dynamic in $\mathbb R$. It's essentially like $x^2$, with the stable fixed point at $0$ and an unstable fixed point at the other real root of $x^4+x^2=x$

Luthier415Hz

11:16

OK. Can adding the multiplicative by epsilon to the smallest polynomial term may work for any nonstable map?

LPZ

No, still the same, $x^2$ is not stable, you can have a saddle-node bifurcation by adding a linear term $x^2+\epsilon x$

Luthier415Hz

ahh

Ok...

LPZ

Saddle-node bifurcation

In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems. In discrete dynamical systems, the same bifurcation is often instead called a fold bifurcation. Another name is blue sky bifurcation in reference to the sudden creation of two fixed points.If the phase space is one-dimensional, one of the equilibrium points is...

Luthier415Hz

Thanks, I will check it out.

Have a good day!

LPZ

Oh ok I get what you meant. Actually, the pattern is more to add a linear term. The idea is that generically, having $f'(p)=1$ is not stable, which is why you add linear terms to make it generic.

And I meant $x^2+x$ for the quadratic example, not $x^2$.

Thanks you too

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