# No Constant Solution => No periodic solution

• cacosomoza
In summary, the statement "No Constant Solution => No periodic solution" means that a system without a steady state cannot exhibit periodic behavior. A constant solution is a state in which the values of all variables remain unchanged over time. This concept is relevant in scientific research and can be observed in real-world examples such as pendulums and chemical reactions.
cacosomoza
Shall f be continuous function of two real variables. Proof that if equation x''=f(x,x') has not constant solutions, then neither it has periodic solutions.

This follows fom the Poincare-Bendixon theorem : a closed orbit must enclose a stationary point.

## What does "No Constant Solution => No periodic solution" mean?

This statement means that if a system does not have a constant or stationary solution, it cannot have a periodic solution. In other words, if the system does not have a steady state, it cannot exhibit periodic behavior.

## What is a constant solution?

A constant solution is a state in which the values of all variables in a system do not change over time. This means that the system is in a steady state and is not exhibiting any periodic or oscillatory behavior.

## Can a system have a constant solution and a periodic solution at the same time?

No, it cannot. If a system has a constant solution, it cannot have a periodic solution. This is because a periodic solution implies that the system is not in a steady state, and therefore cannot have constant values for its variables.

## How is this concept relevant in scientific research?

The concept of "No Constant Solution => No periodic solution" is relevant in many areas of scientific research, such as physics, biology, and chemistry. It helps scientists understand and predict the behavior of systems, and can also aid in the design of experiments and models.

## What are some real-world examples of this concept?

One example is a pendulum. If the pendulum is at rest, it has a constant solution and is not exhibiting any periodic behavior. However, if the pendulum is in motion, it can exhibit periodic behavior. Another example is a chemical reaction. If the reactants are in a steady state, there is no periodic behavior, but if the reaction is occurring, it can exhibit oscillatory behavior.

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