Chaos: difference vs differential equation

[haushofer]

Dear all,

I have a question concerning chaos. As you may well know, the logistic mapping $$x_{n+1} = rx_n (1-x_n) $$ exhibits chaos, depending on the value of r. This logistic mapping is a reparametrized version of the difference equation

$$x_{n+1} = x_n + k x_n (1 - \frac{x_n}{M}) $$

describing exponential growth with a maximum. The continuous limit of this difference equation, where x=x(t), is the differential equation

$$\frac{dx}{dt} = kx \Bigl(1 - \frac{x}{M}\Bigr) $$

In the continuous limit this differential equation somehow does not exhibit chaotic behaviour anymore. Intuitively I can understand this, because the solution of this differential equation, when x_0 < M, describes an S-like curve between x(t)=x_0 and x(t)=M, and hence one does not have periodic solutions anymore. My question is: why exactly does the chaotic behaviour disappear in the continuous limit? Are there some theorems about this (e.g. theorems forbidding chaotic behaviour for scalar differential equations, as opposed to sets of differential equations like the one of Lorenz)? I tried to look for some resources and found a lot of notes describing the logistic mapping, but hardly any of them is describing the continuous limit (i.e., chaotic behaviour if one goes from the difference equation to the differential equation). Any help is appreciated!

 

[zwierz]

"Chaos" is a very general and informal term which assigns that trajectories of a dynamical system have a complicated behaviour. In different systems this term has different sense.The term "chaos" is specified for every concrete system or concrete classes of systems by mathematical definition. If there is no definition then nothing to speak about. For example a shift of the circle
$x\mapsto x+\omega\pmod {2\pi},\quad \omega\notin 2\pi\mathrm{Q}$ is a chaotic mapping (it is ergodic) but erogdicity of hyprerbolic systems has completely different nature.The trajectories of the shift of the circle do not run from each other. In such a sense the shift is not a chaotic mapping
I do not think that you find an answer your question because "continuous limit" is also a very informal term from physics books.
Speaking informally, mappings are associated with Poincare mappings in continuous systems rather than with "continuous limit"

 

[haushofer]

With the continuous limit, I mean the limit in which the time steps $$\Delta t \rightarrow 0 $$. If one originally had time steps of one, $$\Delta t = 1$$, then

$$x_{n+1} - x_n = \frac{x_{n+1} - x_n}{\Delta t}$$

and hence in the limit \Delta t \rightarrow 0 the difference becomes a derivative and the difference equation becomes a differential equation. This is what I refer to as the continuous limit, but maybe that's sloppy language :)

 

[zwierz]

again . mathematically correct way to shift to the discrete dynamics is considering of Poincare map in continuous dynamical system, not the argument you are speaking about

 

[haushofer]

Why not?

 

[zwierz]

Because chaos in finite dimensional systems is manifested only for $t\in [0,\infty)$ but for infinite time your discrete approximation does not have any relation to original continuous system.

 

[haushofer]

Well, that's basically my question: what is the relation? :P

I'm now reading up some stuff, in Strogatz, and start to see that first order scalar differential equations cannot exhibit periodic behaviour due to topological reasons. This, in combination which your remark, answers my question I guess.

 

[zwierz]

first order scalar differential equations cannot exhibit periodic behaviour due to topological reasons

It depends on how to understand this phrase

Consider a dynamical system on the circle
$\dot x=1$ here $x\in S^1=\mathbb{R}/(2\pi\mathbb{Z})$ is the angle variable. All the solutions are periodic
By the way it is an ergodic dynamical system with respect to the standard Lebesgue measure in $S^1$

 

[haushofer]

Yes, I didn't mention that that x is defined on the line, and not on the circle. For that second topology, it's a different matter.