Non-Autonomous System: Definition & Explanation

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In summary, a non-autonomous system is a system where the acceleration is explicitly dependent on time. This can result in the phase trajectory crossing itself, as there is no unique solution to the equations of motion. An example of a non-autonomous system is one where q'' has an explicit time dependence, such as q'' = qt. In general, there is no unique solution to this type of equation given q(0) and q'(0). This is because the right-hand side of the equation is dependent on time, making it non-autonomous. However, non-autonomous systems can be converted to autonomous systems with an additional dimension.
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I want to get some terminology straight.

A non-autonomous system is as far as I can understand a system, where the acceleration depends explicitly on time. This again means that the phase trajectory can cut itself since there is no unique solution to the equations of motion for non-autonomous systems. I think that is how my teacher said it.

I just want to get straight what a non-autonomous system is. Is it everything where q'' has an explicit time dependence? So what if we had the equation of motion:

q'' = qt

Does that equation describe a non-autonomous system? And is there in general no unique solution to the above equation given q(0),q'(0)? And why is that?
 
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An autonomous system is one whose right-hand side is independent of time. Both autonomous and non-autonomous systems typically have unique solutions. However, in non-autonomous system, its phase space does not describe the entire state of the system. A non-autonomous system can be trivially converted to an N+1 dimension autonomous system.
 

1. What is a non-autonomous system?

A non-autonomous system is a type of mathematical model that describes the behavior of a system over a period of time. Unlike an autonomous system, which has a fixed set of rules and does not change over time, a non-autonomous system can change and evolve over time.

2. How is a non-autonomous system different from an autonomous system?

The main difference between a non-autonomous system and an autonomous system is that the former can change and evolve over time, while the latter has a fixed set of rules and does not change. In other words, an autonomous system is self-contained and does not rely on external factors, while a non-autonomous system is influenced by external factors.

3. What are some examples of non-autonomous systems?

Non-autonomous systems can be found in various fields, such as biology, physics, and economics. Examples include biological systems that are affected by external factors like temperature and weather, as well as economic systems that are influenced by market trends and government policies.

4. How are non-autonomous systems used in research?

Non-autonomous systems are commonly used in research to study complex systems that are influenced by external factors. By incorporating these external factors into the model, researchers can better understand the behavior of the system and make predictions about its future behavior.

5. What are the advantages of using non-autonomous systems?

Non-autonomous systems have several advantages, including their ability to model complex systems that are affected by external factors, their flexibility in adapting to changes over time, and their ability to make predictions about the future behavior of a system. They are also useful in studying real-world systems, as they can more accurately reflect the complexities and uncertainties present in these systems.

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