Is the "sensitive dependence on initial conditions" of chaotic systems referring to the seed, control parameters, or both?

Question

At the Wikipedia, one of the initial descriptions of Chaos Theory regarding dynamical systems goes as follows:

Chaos theory is a branch of mathematics and it is focused on the behavior of dynamical systems that are highly sensitive to initial conditions. 'Chaos' is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions.

Regarding “sensitive dependence on initial conditions”, I have a doubt with one specific example: I have a two dimensional dynamical system that generates by iteration two $(X_n,Y_n)$ variables and is defined with three control parameters $D,f,t$ such that $X_n=f(X_{n-1},Y_{n-1},D,f,t)$ and $Y_n=g(X_{n-1},Y_{n-1},D,f,t)$ and the initial seed is $(x_0=0,y_0=0)$.

To what part does initial conditions refer here? To the initial seed $(x_0,y_0)$, or to the initial parameters $D,f,t$? or to both?

Context: I am not sure if I have a dynamical system that is chaotic or not because (more details in this former question):

• When the seed is modified but the control parameters are not modified, the system reaches an attractor and is always the same one.

• When the control parameters are slightly modified the system reaches an attractor but then it is different each time the control parameter is modified.

So only the changes in the control parameters make the attractor change with slight changes, but the changes in the seed obtain the same attractor when the control parameters do not change.

My questions are:

1. The “initial conditions” will refer to what part? To the initial seed $(x_0,y_0)$, or to the initial parameters $D,f,t$? or to both?

2. Then, is it correct to say that the dynamical system of my example is chaotic according to the definition because the attractor changes with slight variations of the control parameters (but not the seed)?

My thoughts: I think that both is the correct answer, but not really sure. All the classic examples as far as I recall seem to rely on the slight changes of the control parameters (e.g. bifurcation diagram of the logistics map) and not on the use of different seeds. Thanks in advance.

Answer 1

The “initial conditions” will refer to what part? To the initial seed $(x_0,y_0)$, or to the initial parameters $D,f,t$? or to both?

What you call seed is typically called initial conditions in the context of dynamical systems, iterative maps, ODEs, and similar. This is what is used in all definitions of chaos I am aware of.

However, once you have a chaotic system, it will also be sensitive to changes of control parameters in almost all cases. The reason for this is that a change of control parameters leads to a difference of the state after the first time step – in your case: $(x_1,y_1)$ in your case. You can then regard the problem as a new one with these initial conditions and due to the sensitivity to initial conditions you will observe a drastically different behaviour. Note that it may also happen that a tiny change in control parameters completely changes the nature of the dynamics (which would have the same consequences).

Now to the almost all: You can devise a parameter transformation that preserves the actual dynamics, e.g., by rotating or scaling it. You can furthermore devise the transformation such that your initial condition is a fixed point of the transformation (i.e., the centre of rotation or scaling). In this case, the dynamics would not be sensitive to changes of the control parameter for that very initial condition.

Then, is it correct to say that the dynamical system of my example is chaotic according to the definition because the attractor changes with slight variations of the control parameters (but not the seed)?

First some clarification of terminology: Chaos mostly happens on attractors as well, usually called chaotic or strange attractors. In a simple chaotic scenario, you only have one attractor, and a slight change of initial conditions will bring you to a different trajectory segment of that attractor. Thus you get a quantitatively different behaviour. The qualitative behaviour is the same though (as it is on the same attractor).

It is completely normal that the attractors change upon changing the control parameters. This is what is described in bifurcation diagrams and similar. In particular, the attractor needs not depend on the initial conditions in a continuous manner. Moreover, you have regions in parameter space where these changes cascade, i.e., the parameter regions with continuous behaviour can become arbitrarily small around a certain point.

However, this is not the same as chaos. Chaos happens for a single parameter setting, i.e., a fixed system; it is not about the dependence on parameters. You cannot directly see chaos in a typical bifurcation diagram of the logistic map. You can only see that there are regions where $x$ (the state) appears not take discrete values anymore (corresponding to a periodic dynamics) but all values within certain intervals. This in turn usually means chaos (but it could also be quasiperiodicity).

Most importantly: Unless you have sensitivity on initial conditions (what you call seed), you do not have chaos. If your system exhibits a fixed-point, periodic, or quasiperiodic dynamics, this is what you have. The dependencies on control parameters may still be interesting, but it’s not chaos.