Dimensions of the logistic map state space and quantum chaos.

[Fallen Seraph]

Apologies if this is the wrong forum, but I have a pair of thematically connected questions that I can't really fit anywhere else. Please move if this is better suited to the quantum physics forums.

My first question being:

The Poincare-Bendixon theorem states that chaos can only occur for systems with state spaces that have more than two dimensions. Since chaos occurs for the logistic map it must have a state space with at least three dimensions, but I can't figure out what they are. My best guess is that the three dimensions are x_n, x_(n+1) and the control parameter. But it just doesn't seem right to give the control parameter a dimenion, and I'm really uncomfortable with giving x_(n+1) a dimension; I can only really justify it in terms of analogy to the state space of a simple harmonic oscillator. Can anyone enlighten me?


My second question being:

The problem of quantum chaos was only lightly touched on in my lectures, but it was implied that it was a serious problem, or a hot topic if you will, in chaos theory. But I can't get my head around the need for chaos on a quantum level.

As is my understanding, the gist of the problem is: "The quantum world is linear. The quantum world scales to the classical world. The classical world exhibits chaos therefore the quantum world must exhibit chaos in order to let the classical world exhibit chaos".

I suspect I may be misunderstanding the problem because as far as I can tell it shouldn't be a problem at all.

My reasoning goes thusly: "If we model a non-linear system on a computer and the program we use makes no reference to quantum corrections then as far as the computer is concerned it's doing calculations about an entirely classical world. So the occurrence of chaos without reference to the quantum world is nothing to be surprised about."

Can anyone point out my errors?

Thanks a lot.

 

[eljose79]

Hello,

Thank you for your questions. I am a scientist specializing in chaos theory and quantum physics, so I hope I can provide some insight into your questions.

Regarding your first question, you are correct in thinking that the three dimensions in the logistic map are x_n, x_(n+1), and the control parameter. The control parameter can be thought of as a measure of the overall "chaoticness" of the system. As the control parameter is varied, the system can exhibit different types of behavior, including stable fixed points, periodic behavior, and chaotic behavior. So, while it may seem strange to assign a dimension to the control parameter, it is an important factor in determining the dynamics of the system.

In regards to your second question, the issue of quantum chaos is indeed a hot topic in both chaos theory and quantum physics. The problem arises because, as you mentioned, the quantum world is inherently linear, while the classical world exhibits chaotic behavior. This seems contradictory, as the classical world is supposed to emerge from the quantum world. However, this does not necessarily mean that the quantum world must exhibit chaos in order to allow for chaos in the classical world.

There are various approaches to resolving this issue, such as the concept of quantum chaos, which explores the behavior of quantum systems that exhibit chaotic behavior. Another approach is to consider the role of decoherence, which is the process by which a quantum system becomes entangled with its environment and appears to behave classically. Ultimately, the problem of quantum chaos is still an active area of research and there is no definitive answer yet.

As for your reasoning about modeling a non-linear system on a computer, it is important to note that even classical chaos can arise from simple, deterministic systems, so it is not necessary to consider quantum effects in order to observe chaos in a computer simulation. However, understanding the relationship between chaos and quantum mechanics can provide insights into the behavior of complex systems, both classical and quantum.

I hope this helps to clarify your questions. Please let me know if you have any further inquiries. Thank you for your interest in these fascinating topics.

 

[denian]

I would like to address both of your questions and provide some insights into the dimensions of the logistic map state space and the concept of quantum chaos.

Firstly, the logistic map is a mathematical model used to describe the population dynamics of a species in a given environment. It is a one-dimensional map, meaning that it has only one variable, typically denoted by x, which represents the population size. However, the state space of the logistic map is not limited to just x, but also includes the control parameter, denoted by r. This parameter determines the rate at which the population grows or declines. So, in essence, the state space of the logistic map is two-dimensional, with x and r as its dimensions. This is in contrast to the Poincare-Bendixon theorem, which applies to continuous systems with more than two dimensions. Therefore, the logistic map does not violate the theorem, as it is a discrete system with only two dimensions in its state space.

Moving on to the concept of quantum chaos, it is indeed a hot topic in chaos theory and quantum physics. The main problem at hand is the apparent contradiction between the deterministic nature of classical mechanics and the probabilistic nature of quantum mechanics. As you correctly pointed out, the quantum world is linear, meaning that the evolution of a system can be described by a linear equation. However, the classical world exhibits chaotic behavior, which is characterized by non-linear dynamics. This raises the question of whether chaos can exist in a purely quantum world.

The answer to this question is still a subject of ongoing research and debate. Some scientists argue that chaos can exist in the quantum world, but it is different from classical chaos. This is because the uncertainty principle in quantum mechanics introduces a fundamental limit to the precision with which we can know the initial conditions of a system, which is a crucial factor in the development of chaos. Other scientists argue that chaos in the classical world emerges from the quantum world, but it is not inherent to it.

In summary, the problem of quantum chaos is a complex and ongoing topic in the scientific community. While there is no clear answer yet, it is an area of research that has the potential to deepen our understanding of both classical and quantum systems. I hope this response has shed some light on the dimensions of the logistic map state space and the concept of quantum chaos.